Combined Effects of Heat and Mass Transfer on MHD Free

Research ArticleCombined Effects of Heat and Mass Transfer on MHD FreeConvective Flow of Maxwell Fluid with Variable Temperatureand Concentration

Muhammad Bilal Riaz 12 Maryam Asgir34 A A Zafar3 and Shaowen Yao 5

1Department of Mathematics School of Science University of Management and Technology C-II Johar TownLahore 54770 Pakistan2Institute for Groundwater Studies (IGS) University of the Free State Bloemfotain South Africa3Department of Mathematics Government College University Lahore Lahore54000 Pakistan4Department of Mathematics Riphah International University QIE Township Lahore 54000 Pakistan5School of Mathematics and Information Science Henan Polytechnic University Jiaozuo China

Correspondence should be addressed to Shaowen Yao yaoshaowenhpueducn

Received 18 November 2020 Revised 15 March 2021 Accepted 26 March 2021 Published 14 April 2021

Academic Editor Fateh Mebarek-Oudina

Copyright copy 2021Muhammad Bilal Riaz et al is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Heat andmass transfer combined effects onMHDnatural convection for a viscoelastic fluid flow are investigatede dynamics ofthe fluid are controlled by the motion of the plate with arbitrary velocity along with varying temperature and mass diffusion enon-dimensional forms of the governing equations of the model are developed along with generalized boundary conditions andthe resulting forms are solved by the classical integral (Laplace) transform techniquemethod and closed-form solutions aredeveloped Obtained generalized results are very important due to their vast applications in the field of engineering and appliedsciences few of them are highlighted here as limiting cases Moreover parametric analysis of system parametersPr S Kc GT Gc M Sc λ is done via graphical simulations

1 Introduction

In science and in many engineering applications such as incondensation evaporation and chemical process manytransport processes are influenced by the combined action ofthe buoyancy forces from both heat and mass diffusion Heatand mass transfer combined effects are studied extensivelydue to their significant role in chemical processing equip-ment oceanic circulation emergency cooling system of ad-vanced nuclear reactors cooling process of plastic sheetsformation and dissipation of the fog processing and dryingthe food temperature distribution andmoisture of agriculturefields and production of polymer In recent years a lot ofpractical applications attracted many scientists and engineersto pay a considerable amount of focus to learn the heat andmass effects either analytically or numerically [1ndash4]In

industrial and engineering processes most fluids are non-Newtonian Since the non-Newtonian fluids deal morecomplexities due to the rheological behavior than Newtonianfluids distinct models were proposed e influence of heatand mass transfer in the non-Newtonian fluid is an importantsubject from the theoretical as well as practical point of viewdue to its abundant applications in industry and engineeringCommon examples include polymer extrusion the emer-gency cooling system of nuclear reactors food processingthermal welding to name a few Convective flow is a self-sustained flow with the effect of the temperature gradient Inliterature different theories are made to see the occurrence ofheat and mass transfer in convective flows of different fluidsMebarek-Oudina et al [5] investigated the natural convectiveheat transfer phenomenon of water-based hybrid nanofluid ina porous medium along with the magnetic field Das et al [6]

HindawiMathematical Problems in EngineeringVolume 2021 Article ID 6641835 36 pageshttpsdoiorg10115520216641835

considered the natural convective flow of the electricallyconducting fluid past on vertical plate embedded in a per-meable medium and explored the impacts of heat and masstransfer e heat transfer phenomenon of Casson nanofluidflow is taken into account by Abo-Dahab et al [7] eyanalyzed the problem with the convective boundary condi-tions and discussed the influence of chemical reaction andheat source Sajad et al[8] studied the heat transfer andmagnetic effect on hybrid nanofluid Nazish et al [9] exploredthe influence of heat and concentrationmass transform withthe existence of fields developed by magnetic in the Maxwellfluid model Ahmad [10] explored the heat transfer for theMaxwell fluid on the stretching plate with the slip boundaryon the velocity ey explored the numerical solutions andshowed the heat flux effect using the Nusselt number and thePrandtl number A computational analysis is performed tostudy the effects of the transverse magnetic field at the un-steady PoiseuillendashRayleighndashBenard flow by Marzougui et al[11] e thermal propertiesrsquo effects on the soil temperatureare modeled and investigated numerically by Belatrache [12]To have more insight about heat and mass transfer mecha-nisms in fluid flow and their applications readers are referredto review references [13ndash16]On the other hand many re-searchers paid a significant amount of attention to the study ofMHD free convective flows due to its numerous applicationsin solar and stellar structure radio propagation MHDpumps MHD bearings aerodynamics polymer technologypetroleum industry crude oil purification glass fiber drawingetc In light of these applications many researchers such asRajput [17] Gupta [18] and El Amin [19] studied the MHDflow of different fluids ey found the exact solution forvelocity concentration and temperature by the Laplacetransform method Heat and mass transfer simultaneouseffects on MHD flow of Maxwell fluid have been investigatedby Nadeem et al [20] Recently the study of the unsteadyboundary layer heat transfer of Maxwell viscoelastic fluid wascarried out by Zhao et al [21] Ahmad [22] studied MHDviscous with constant density electro-conducting fluid in theexistence of the radiation thermal diffusion free convectionand mass transfer flow ese results motivated Chaudhryet al [23] and they used classical integral transform to obtainthe exact solutions of naturalMHD convective flow past on anaccelerated surface submerged in a permeable medium Das[24] developed the closed-form solution of the unsteady

MHD natural convection flow on a moving vertical plateaccompanied by mass transfer and thermal radiation Car-rying on Das et al [25] investigated the time-dependentMHD natural convection flow past a moving vertical platedipped in a porous medium and studied the different aspectsof heat and mass transfer ey discussed the problem withthe uniform oscillating and impulsive motions of the platebesides considering the constant heat and mass diffusion andimplemented the Laplace integral transform to develop theanalytic solutionsMotivated by these investigations the ob-jective of this manuscript is to study the combined effect ofheat and mass transfer on MHD Maxwell fluid Laplace in-tegral transformation is used to obtain the unique solution oftemperature velocity and concentration under the impact ofgeneralized boundary conditions on temperature velocityand concentration e importance of the problem is high-lighted by showing its impactapplications in the field ofengineering and applied sciences e paper is organized intosix sections After the introductory section in Section 2 thedimensionless governing equations are developed In Section3 Laplace integral transform is implemented to find the exactsolution of the temperature velocity and concentration fieldIn Section 4 some applications in different fields are discussedas limiting cases to justify our results In Section 5 the effect ofphysical parameters is analyzed graphically e concludingobservation is listed at the end

2 Problem Formulation

We studied here the motion of the viscoelastic in-compressible electronically conducting Maxwell fluiddue to plate motion with arbitrary velocity u0fprime(tprime) eplate is along x minus axis and y minus axis is considered normalon the plate In the first instance at t 0 the plate andfluid are at temperature Tinfinprime and concentration Cinfinprime Withthe time t 0+ the plate starts to move in its own axisen the level of temperature and concentration takes up

to Tinfin^

prime1113882 1113883 + Tw

^

prime1113882 1113883hprime(tprime) and Cinfin^

prime1113882 1113883 + Cw

^

prime1113882 1113883gprime(tprime) where

fprime(tprime) hprime(tprime) and gprime(tprime) are piecewise continuous func-tions that vanish at t 0 Details of different parametersare given in Table 1 Momentum energy and concen-tration equations are formed as follows

2 Mathematical Problems in Engineering

1 + λprimez

ztprime1113888 1113889

zuprime χprime tprime( 1113857

ztprime ]

z2uprime χprime tprime( 1113857

zχprime2

+ gBT 1 + λprimez

ztprime1113888 1113889 φprime minus φinfinprime( 1113857 + gBC

1 + λprimez

ztprime1113888 1113889 Cprime minus Cinfinprime( 1113857 minus

σB20

ρ1 + λprime

z

ztprime1113888 1113889uprime χprime tprime( 1113857

ρCp

zφprime χprime tprime( 1113857

ztprime K

z2φprime χprime tprime( 1113857

zχprime2minus Sprime φprime minus φinfinprime( 1113857

zCprime χprime tprime( 1113857

ztprime D

z2Cprime χprime tprime( 1113857

zχprime2minus Kcprime Cprime minus Cinfinprime( 1113857

(1)

e imposed initial and boundary conditions are

tprime le 0 uprime χprime tprime( 1113857 0

φprime χprime tprime( 1113857 φinfinprime

Cprime χprime tprime( 1113857 Cinfinprime χprime ge 0

tprime ge 0 uprime 0 tprime( 1113857 u0fprime tprime( 1113857

φprime 0 tprime( 1113857 φinfinprime + φwprimehprime tprime( 1113857

Cprime 0 tprime( 1113857 Cinfinprime + Cwprime gprime tprime( 1113857

uprime χprime tprime( 1113857⟶ 0

φprime χprime tprime( 1113857⟶ φinfinprime

Cprime χprime tprime( 1113857⟶ Cinfinprime χprime ⟶infin

(2)

For dimensionless problem we use the following relations

Table 1 Nomenclature

Symbol Quantityu Velocity of fluidB0 Magnetic field parameterq Laplace transforms parameterD Mass diffusivityBT ermal expansion parameterBC Concentration expansion coefficientK ermal conductivityρ Density of fluidλ Relaxation timeσ Electric conductivity coefficientμ Dynamic viscosityυ Kinematic viscositycp Specific heatS Heat source parameterKc Chemical reaction coefficientg Gravitational accelerationSC Schmidt numberM Parameter due to magnetic fieldPr Prandtl numberGT Grashof number due to thermal effectGC Grashof number due to concentration

Mathematical Problems in Engineering 3

t u20]

tprime

y u0

]χprime

Sc ]D

Kc ]u20Kcprime

λ u20]λprime

uprime u0u

C Cprime minus Cinfinprime

Cwprime

T φprime minus φinfinprime

φwprime

S ]

ρCpu20Sprime

Pr μCp

K

Kpprime

u20

]2KpGc

g BC]( 1113857

u30

Cwprime

GT gBT]

u30φwprime

M σB

20]

ρu20

f ]u2o

fprime

g ]u2o

gprime

h ]u2o

hprime

(3)

After non-dimensionalizing the governing equationsbecome

1 + λz

zt1113888 1113889

zu(y t)

zt

z2u(y t)

zy2 + GT 1 + λ

z

zt1113888 1113889T(y t)

+ GC 1 + λz

zt1113888 1113889C(y t)

minus M 1 + λz

zt1113888 1113889u(y t)

(4)

zT(y t)

zt

1Pr

z2T(y t)

zy2 minus ST(y t) (5)

zC(y t)

zt

1Sc

z2C(y t)

zy2 minus KcC(y t) (6)

along the following initial and boundary conditions

u(y 0) 0 T(y 0) 0 C(y 0) 0 (7)

u(0 t) f(t) T(0 t) h(t) C(0 t) g(t) (8)

u(y t)⟶ 0 T(y t)⟶ 0 C(y t)⟶ 0 asy⟶infin

(9)

3 Solution of the Problem

31 Concentration Transforming equation (7) after apply-ing the Laplace integral transform and utilizing the corre-sponding initial condition we get

z2C(y q)

zy2 minus Sc Kc + q( 1113857C(y q) 0 (10)

e above differential equation solution is

C(y q) C1eminus y

Sc Kc+q( )

1113968

+ C2ey

Sc Kc+q( )

1113968

(11)

e solution of equation (11) with the transformed formof boundary conditions becomes

C(y q) G(q)eminus y

Sc Kc+q( )

1113968

(12)


Applying the Laplace inverse on equation (12) and usingthe Lminus 1 G(q)1113864 1113865 gprime(t) with g(0) 0 convolution theoremand equation (A22) the generalized solution for concen-tration is

C(y t) 1113946t

0gprime(t minus s)Φ y

Sc

1113968 s Kc( 1113857ds (13)

and Φ is specified in equation (A23)

32 Temperature Distribution Implementing the Laplacetransform on equation (5) and using the concerned theinitial condition we get

z2T(y q)

zy2 minus Pr(S + q)T(y q) 0 (14)

e solution is

T(y q) C1eminus y

Pr(S+q)

radic

+ C2ey

Pr(S+q)

radic

(15)

After implementing the boundary conditions equation (15)becomes

T(y q) H(q)eminus y

Pr(S+q)

radic

(16)

e Laplace inverse on equation (16) and using theLminus 1 H(q)1113864 1113865 hprime(t) with h(0) 0 convolution theorem andequation (A24) the generalized solution for temperatureobtained is

T(y t) 1113946t

0hprime(t minus s)Ψ y

Pr

1113968 s S( 1113857ds (17)

where Ψ is defined in equation (A25)

33 Velocity Employing the Laplace transform on equation(4) and using the corresponding initial condition on velocityform the following differential equation

z2u(y q)

zy2 minus ((1 + λq)(q + M))u(y q)

minus GT(1 + λq)T(y q)) minus Gc(1 + λq)C(y q)

(18)

In order to solve equation (18) we use the value ofC(y q) T(y q) from equation (12) and equation (16)respectively With boundary conditions use on velocity thefollowing solution is obtained

u(y q) F(q)eminus y

(q+M)(1+λq)

radic

+GTH(q)(1 + λq) e

minus y(q+M)(1+λq)

radic

minus eminus y

Pr(S+q)

radic

1113874 1113875

Pr(S + q) minus (1 + λq)(q + M)( 1113857

+GcG(q)(1 + λq) e

minus y(q+M)(1+λq)

radic

minus eminus y

Sc Kc+q( )

1113968

1113874 1113875

Sc Kc + q( 1113857 minus (1 + λq)(q + M)( 1113857

(19)

Further simplification reduces equation (19)

u(y q) F(q)eminus y

λ q+α1( )

2+α23( 1113857

1113969

minus

GTH(q)(1 + λq) eminus y

λ q+α1( )

2+α231113858 1113859

1113969

minus eminus y

Pr(S+q)

radic⎛⎝ ⎞⎠

λ q + α4( 11138572

minus α261113872 1113873

minus

GcG(q)(1 + λq) eminus y

λ q+α1( )

2+α231113858 1113859

1113969

minus eminus y

Sc Kc+q( )

1113968⎛⎝ ⎞⎠

λ q + α7( 11138572

minus α291113872 1113873

(20)

where 2α1 (λM + 1λ) α2 (Mλ) α23 α2 minus α21 2α4

(λM + 1 minus Prλ) α5 (M minus PrSλ) α26 α24 minus α5 2α7

(λM + 1 minus Scλ) α8 (M minus ScKcλ) α29 α27 minus α8

Generalized expression for velocity field is acquired byemploying the inverse Laplace transform on equation (20)

u(y t) I1 +GT

λI2 minus

GT

λI3 +

Gc

λI4 minus

Gc

λI5 (21)

where

I1 Lminus 1

F(q)eminus y

λ q+α1( )

2+α23( 1113857

1113969

⎛⎝ ⎞⎠

Lminus 1

B1(y q)( 1113857lowast Lminus 1

B2(q)( 1113857

B1(y t)lowastB2(t)

(22)

where B1(y q) (eminusλ

radicy

((q+α1)2+α23)

1113968

((q + α1)2 + α23)

1113969

)

B2(q) F(q)

((q + α1)2 + α23)

1113969

By using equation (A20) and equation (A21) expres-sions for the B1(y t) and B2(t) are evaluated as follows


B1(y t)

0 0lt tltyλ

radic

eminus α1t

I0 iα3

t2

minus λy2

1113969

1113874 1113875 tgtyλ

radic

⎧⎪⎪⎨

⎪⎪⎩

(23)

B2(t) eminus α1t

iα3( 1113857I1 iα3t( 1113857 + δ(t)( 11138571113872 1113873lowast

fprime(t) + α1f(t)( 1113857

+ α23eminus α1t

I0 iα3t( 11138571113872 1113873lowastf(t)

(24)

I2 Lminus 1 H(q)(1 + λq)e

minus yPr(S+q)

radic

q + α4( 11138572

minus α261113872 1113873⎛⎝ ⎞⎠

h(t) + λhprime(t)( 1113857lowast

eminus α4tcosh α6t( 11138571113872

+S minus α4( 1113857

α6e

minus α4tsinh α6t( 11138571113889

lowasterfcy

Pr

1113968

2t

radic1113888 1113889eminus St

(25)


minus y

λ q+α1( )

2+α23( 1113857

1113969

q + α4( 11138572

minus α261113872 1113873

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

B1(y t)lowastB3(t)

(26)

where B1(y t) is given in equation (23) and

B3(t) hprime(t) + λH(t)lowasthprime(t)( 1113857

lowastAlowast

+ Blowaste

minus α4+α6( )t+ Clowaste

minus α4minus α6( )t1113874 1113875

lowaste

minus α1tI0 iα3t( 1113857

(27)

where

Alowast

α21 + α231113872 1113873

α24 minus α26

Blowast

2 α4 minus α6( 1113857A

lowastminus 2α1 minus α4 minus α6( 1113857

2α6

Clowast

1 minus Alowast

minus Blowast

(28)

I4 Lminus 1 G(q)(1 + λq)e

minus ySc Kc+q( )

1113968

q + α7( 11138572

minus α291113872 1113873⎛⎝ ⎞⎠

g(t) + λgprime(t)( 1113857lowast


+Kc minus α7( 1113857

α9e

minus α7tsinh α9t( 11138571113889

lowasterfcy

Sc

1113968

2t

radic1113888 1113889eminus Kct

(29)

Similarly


minus y

λ q+α1( )

2+α23( 1113857

1113969

q + α7( 11138572

minus α291113872 1113873

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

B1(y t)lowastB4(t)

(30)


B4(t) gprime(t) + λH(t)lowastgprime(t)( 1113857

lowastClowast

+ Dlowaste

minus α7+α9( )t+ Elowaste

minus α7minus α9( )t1113874 1113875

lowaste


Dlowast

α21 + α231113872 1113873

α27 minus α29

Elowast

2 α7 minus α9( 1113857A


2α9

Flowast

1 minus Dlowast

minus Elowast

(31)

e above results are obtained for generalized time-dependentboundary conditions on velocity concentration and temper-ature ese results have many applications in engineering andapplied science Now we will consider and discuss its fewapplications

4 Applications

41 Application 1 f(t) H(t) g(t) H(t) h(t) H(t)is function value shows the motion of the fluid is becauseof the motion of an infinite plate in its plane with constantvelocityis function has importance in a lot of engineeringproblems such as signal waves driving forces that act for ashort time only and impulsive forces acting for an instancesuch as a hammer blowSubstituting the value of G(q)

(1q) into equation (12) and applying the Laplace inversethe expression for concentration is

C(y t) δ(t) + Kct( 1113857lowasterfc

ySc

1113968

2t


(32)

where δ() is known as Dirac delta functionEmbedding the value of H(q) (1q) into equation (16)

and taking Laplace inverse make the expression oftemperature

T(y t) (δ(t) + St)lowasterfc

yPr

1113968

2t


(33)

e equation of velocity


u(y t) II1 +

GT

λI

I2 minus

GT

λI

I3 +

Gc

λI

I4 minus

Gc

λI

I5 (34)

where

II1 B1(y t)

lowastB

I2(t) (35)

BI2(t) is obtained as

BI2(t) e

minus α1tiα3( 1113857I1 iα3t( 1113857 + δ(t)( 11138571113872 1113873

lowast δ(t) + α1H(t)( 1113857

+ α23eminus α1t

I0 iα3t( 1113857lowastH(t)

(36)

and for B1(y t) (see equation (A1))After substituting the value of H(q) (1q) into

equation (25)

II2 (H(t) + λδ(t)) e

minus α4tcosh α6t( 11138571113872

+S minus α4( 1113857

α6e

minus α4tsinh α6t( 11138571113889

lowasterfcy

Pr

1113968

2t


(37)

Equation (26) takes the form after employing the value ofH(q) (1q)

II3 B1(y t)

lowastB

I3(t) (38)

where

BI3(t) δ(t) + λH(t)

lowastδ(t)( 1113857

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(39)

and for B1(y t) (see equation (A1))After substituting the value of G(q) (1q) into equa-

tion (29)


minus α7tcosh α9t( 1113857 +Kc minus α7( 1113857

α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


(40)

Similarly equation (30) after substituting the value ofG(q) (1q)

II5 B1(y t)

lowastB

I4(t) (41)

and



lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(42)

and for B1(y t) (see equation (A1)

Similar result for concentration was obtained by NehadAli Shah [26] (equation (35)) us our result supports theresult already present in literature

42 Application 2 f(t) t g(t) t h(t) t e impor-tant concepts of engineering are based around linearfunctions ey are often used in engineering to explain dataand evaluate the lines that best fit the given data sets It has alot of applications in engineering and it can be representedin a variety of ways One of the particular interests is directvariation which forms many engineering applications suchas Hookersquos law and Ohmrsquos law To learn about slope en-gineers use linear functions to interpret and understandgraphs that describe displacement velocity and accelera-tioney use these functions to analyze data to learn how todesign their engineering products more efficiently reliablyand safelyFor the choice of F(q) G(q) H(q) equal to(1q2) in the appropriate equations and employing theLaplace inverse the expression of C(y t) T(y t) u(y t)and then III

1 III2 III

3 III4 and III

5 changes into respectively

C(y t) 1 + Kct( 1113857lowasterfc

ySc

1113968

2t


T(y t) (1 + St)lowasterfc

yPr

1113968

2t


u(y t) III1 +

GT

λI

II2 minus

GT

λI

II3 +

Gc

λI

II4 minus

Gc

λI

II5

(43)

where

III1 B1(y t)

lowastB

II2 (t) (44)

where B1(y t) (see equation (A1)) and BII2 (t) (see equation

(A2))

III2 (t + λ)

lowaste

minus α4tcosh α6t( 1113857 +S minus α4( 1113857

α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


III3 B1(y t)

lowastB

II3 (t)

(45)


(A3))

III4 (t + λ)

lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


III5 B1(y t)

lowastB

II4 (t)

(46)



(A4))

43 Application 3 f(t) sin t g(t) sin t h(t) sin te choice of this function shows the fluid motion due to theoscillation of the plate It has a lot of applications in physicssuch as wave motion other oscillatory motions and engi-neering It is used to model the behavior that repeatsTrigonometric functions are used to calculate angles in manyengineering problems In civil and mechanical engineeringtrigonometry is used to calculate torque and forces onobjects which help build bridges and girders In the con-struction of bridges we need to consider the forces whichkeep the bridges at their balance and trigonometry helps usto calculate the static force which keeps the bridges static Inengineering trigonometry is used to decompose the forcesinto horizontal and vertical components that can be ana-lyzede expression for concentration after putting thevalue of G(q) (1q2 + 1) into equation (12) is

C(y t) cos t + Kc sin t( 1113857lowasterfc

ySc

1113968

2t


(47)

the expression for temperature become after putting thevalue of H(q) (1q2 + 1) into equation (16) is

T(y t) (cos t + S sin t)lowasterfc

yPr

1113968

2t


(48)

and velocity change after substitute the value ofF(q) (1q2 + 1) into equation (21) is

u(y t) IIII1 +

GT

λI

III2 minus

GT

λI

III3 +

Gc

λI

III4 minus

Gc

λI

III5 (49)

where

IIII1 B1(y t)

lowastB

III2 (t) (50)

where B1(y t) (see equation (A1)) and BIII2 (t) (see equation

(A5))

IIII2 (sin t + λ cos t)

lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


IIII3 B1(y t)

lowastB

III3 (t)

(51)


(A6))


lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


IIII5 B1(y t)

lowastB

III4 (t)

(52)


(A7))

44 Application 4 f(t) et g(t) et h(t) et e ex-ponent functions are used for real-world application as forcalculating area volume determining growth or decay andimpacts of force In engineering it helps them to designbuild and improve the machinery structure and equip-ment For example in sound engineering it is used tocalculate sound waves In basic engineering it is used tocompute the tensile strength which finds out the amount ofstress that a structure can withstand In aeronautical engi-neering it is used to predict how airplanes rockets and jetswill perform during flight To determine the kinetic andpotential energy pressure heat and airflow of waves be-havior it is very helpful Nuclear power sources are one ofthe important things developed by nuclear engineers eyused the exponents to work with extremely small numbers tomake the big things happen Substituting the value of G(q)

(1q minus 1) into equation (12) the concentration equationafter implementing the Laplace inverse becomes

C(y t) et

+ Kcet

1113872 1113873lowasterfc

ySc

1113968

2t


(53)

and equation of temperature distribution after putting thevalue of H(q) (1q minus 1) into equation (16) and applyingLaplace inverse

T(y t) et

+ Set


yPr

1113968

2t


(54)

e expression for velocity is

u(y t) IIV1 +

GT

λI

IV2 minus

GT

λI

IV3 +

Gc

λI

IV4 minus

Gc

λI

IV5 (55)

where

IIV1 B1(y t)

lowastB

IV2 (t) (56)

and B1(y t) (see equation (A1)) and BIV2 (t) (see equation

(A8))


IIV2 e

t+ λe

t1113872 1113873

lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


IIV3 B1(y t)

lowastB

IV3 (t)

(57)


(A9))

IIV4 e

t+ λe

t1113872 1113873

lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


(58)

Similarly

IIV5 B1(y t)

lowastB

IV4 (t) (59)


(A10))

45 Application 5 f(t) tet g(t) tet h(t) tet Insertingthe G(q) (1(q minus 1)2) into equation (12) and applying theLaplace inverse we get

C(y t) et

+ Kc + 1( 1113857tet


ySc

1113968

2t


(60)

and insert the H(q) (1(q minus 1)2) into equation (16) andtake Laplace inverse

T(y t) et

+(S + 1)tet


yPr

1113968

2t


u(y t) IV1 +

GT

λI

V2 minus

GT

λI

V3 +

Gc

λI

V4 minus

Gc

λI

V5

(61)

e IV1 takes the form after embedding the F(q) (1

(q minus 1)2)

IV1 B1(y t)

lowastB

V2 (t) (62)

where B1(y t) (see equation (A1)) and BV2 (t) (see equation

(A11))

IV2 te

t+ λe

t(t + 1)1113872 1113873

lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


IV3 B1(y t)

lowastB

V3 (t)

(63)


(A12))

IV4 te

t+ λe

t(t + 1)1113872 1113873

lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


IV5 B1(y t)

lowastB

V4 (t)

(64)


(A13))

46 Application 6 f(t) et sin t g(t) et sin t h(t) et

sin t e choice of the value of G(q) (1(q minus 1)2 + 1) H

(q) (1(q minus 1)2 + 1) F(q) (1(q minus 1)2 + 1) makes theexpression

C(y t) et cos t + 1 + Kc( 1113857sin t( 11138571113872 1113873

lowasterfcy

Sc

1113968

2t


T(y t) et(cos t +(1 + S)sin t)1113872 1113873

lowasterfcy

Pr

1113968

2t


u(y t) IVI1 +

GT

λI

VI2 minus

GT

λI

VI3 +

Gc

λI

VI4 minus

Gc

λI

VI5

IVI1 B1(y t)

lowastB

VI2 (t)

(65)

where B1(y t) (see equation (A1)) and BVI2 (t) (see equation

(A14))

IVI2 e

t sin t(1 + λ) + et cos t1113872 1113873

lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


IVI3 B1(y t)

lowastB

VI3 (t)

(66)


(A15))

IVI4 e

t sin t(1 + λ) + et cos t1113872 1113873

lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


(67)

Similarly

IVI5 B1(y t)

lowastB

VI4 (t) (68)

where B1(y t) (see (A1)) and BVI4 (t) (see equation (A16))


47Application 7f(t) t sin t g(t) t sin t h(t) t sint By putting the value of G(q) (2q(q2 + 1)2) intoequation (12) and employing Laplace inverse

C(y t) t cos t + Kct + 1( 1113857sin t( 1113857lowasterfc

ySc

1113968

2t


(69)

By putting the value of H(q) (2q(q2 + 1)2) intoequation (16) and Laplace inverse gives the expression

T(y t) (t cos t +(St + 1)sin t)lowasterfc

yPr

1113968

2t


u(y t) IVII1 +

GT

λI

VII2 minus

GT

λI

VII3 +

Gc

λI

VII4 minus

Gc

λI

VII5

(70)

where

IVII1 B1(y t)

lowastB

VII2 (t) (71)

and B1(y t) (see equation (A1)) and BVII2 (t) (see equation

(A17))

IVII2 ((t + λ)sin t + λ cos t)

lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


IVII3 B1(y t)

lowastB

VII3 (t)

(72)

where B1(y t) (see equation (A1)) and BVII3 (t) (see equa-

tion (A18))


lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


IVII5 B1(y t)

lowastB

VII4 (t)

(73)

and for B1(y t) (see equation (A1)) and BVII4 (t) (see

equation (A19))ese are solutions for the choice of samefunction for f(t) g(t) and h(t) from the list of functionsH(t) t sin t et tet t sin t sin tet We can consider theproblem with the different choice of function forf(t) g(t) h(t) eg f(t) t g(t) et h(t) H(t) andfind its solution For the validation of results if we take λ

0 GT 1 S 0 h(t) 1 minus aebt g(t) 1 with choice off(t) H(t)tα (αgt 0) or sin t in our system of equations(4)ndash(8) the results obtained are the same as the result

obtained by Nehad Ali shah [27] (choosing the ϵ 0 N

GC in equation (9))

5 Results and Discussion

e heat and mass transfer study of Maxwell fluid is dis-cussed here e solutions for dimensionless velocityconcentration and temperature are assessed by the Laplacetransform method e application of these solutions indifferent fields of engineering sciences is also discussed Itbrings to attention that these results are helpful to solve thecomplicated problems of engineering and applied sciencee behavior of these solutions for velocity concentrationand temperature profile is depicted graphically e impactsof different pertinent parameters λ M Sc Kc Pr GT GC S

on fluid flow are also deliberated using plots and theirphysical aspects described To avoid repetition only themost significant graphical representations regarding theeffects of the concerned parameter will be included here

e variation in the behavior of velocity and concen-tration with varying values of Schmidt number Sc is illus-trated inFigures 1ndash3 respectively An increase in Schmidtnumber results in the decline in the thickness of theboundary layer of concentration Since the Schmidt numberis defined as the ratio between kinematic viscosity and massdiffusivity it reduced the concentration as well as velocityprofile In reality the increase occurring in momentumdiffusivity causes a decline in the fluid velocity

e deviation in temperature profile for varying values ofPrandtl number Pr is demonstrated in Figures 4 and 5 It isdepicted that the thermal boundary layer thickness decreasesrapidly with the increase in the values of Pr For a small valueof Pr heat diffuses very quickly in comparison to the ve-locitye reason is the thermal boundary layer thickness inliquid metals is higher than the momentum boundary layerFinally Pr can be practiced to expand the percentage ofcooling

Similar effects can be seen for the heat absorption co-efficient S on temperature profile with different values of thefunction g(t) at different time scales depicted in Figure 6and 7 e thermal buoyancy forces decrease with the in-crease of S which decreases the fluid temperature

Impacts of magnetic parameter M displayed in Figure 8depict the velocity decline with the increase of magneticparameter values Physically when magnetic force is appliedto the velocity field it generates the drag force known as theLorentz force which opposes themotion of the fluid Figure 9shows the impact of Pr on velocitye increase of Pr resultsin a decrease in velocity e velocity boundary layer getsthicker due to the low rate of thermal diffusion Basically inheat transfer problems Pr control the relative thicknessmomentum boundary layer

In Figure 10 the study of the effects of GT on velocitydescribes the increase in behavior with increment in thevalues of GT Physically the result of more induced fluidflows is due to a rise in buoyancy effects which is the result ofthe increase in GT e depiction of GC on velocity isportrayed in Figure 11 We can see the rise in velocity withthe rise in the value of GC e natural convection and


f (t) = 1 g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

15

1

05

0

(a)

f (t) = t g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

2

15

1

05

0

(b)

f (t) = sint g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

15

1

05

0

(c)

f (t) = et g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

15

1

05

0

(d)

f (t) = tet g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

(e)

f (t) = et sint g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

5

4

3

2

1

0

Velo

city

(f )

Figure 1 Continued


f (t) = t sint g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

15

1

05

0

(g)

Figure 1 Profile of velocity for different values of Sc and M 20 λ 06 S 105 Kc 05 GT 50 GC 20 Pr 071

Conc

entr

atio

n

15

1

05

0

ndash050 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = H (t)

(a)

Conc

entr

atio

n

15

1

05

00 1 2 3 4 5

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = H (t)

(b)

005

004

003

002

001

0

ndash0010 1 2 3

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = t

(c)

02

015

01

005

00 1 2 3 4

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = t

(d)

Figure 2 Continued


005

004

003

002

001

0

ndash0010 1 2 3

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = sin t

(e)

02

015

01

005

00 1 2 3 4

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = sin t

(f )

Figure 2 Profile of concentration for different values of Sc and variation of time

04

03

02

01

0

ndash010 1 2 3

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = et

Conc

entr

atio

n

(a)

08

06

04

02

0

ndash020 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = etCo

ncen

trat

ion

(b)

004

0

006

002

ndash0020 1 2 3

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = tet

Conc

entr

atio

n

(c)

03

02

01

0

ndash010 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = tet

Conc

entr

atio

n

(d)

Figure 3 Continued


006

004

002

0

ndash0020 1 2 3

y

Sc = 14Sc = 096Sc = 022

Sc = 060

g (t) = et sintCo

ncen

trat

ion

(e)

03

02

01

00 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = et sint

Conc

entr

atio

n

(f )

001

8 times 10ndash3

6 times 10ndash3

4 times 10ndash3

2 times 10ndash3

0

ndash2 times 10ndash3

0 1 2 3y

Sc = 14Sc = 096Sc = 022

Sc = 060

g (t) = t sint

Conc

entr

atio

n

(g)

0

008

006

004

002

ndash0020 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = t sintCo

ncen

trat

ion

(h)


Tem

pera

ture

15

1

05

0

ndash051 2 30

y

Pr = 70Pr = 50Pr = 071

Pr = 15

g (t) = H (t)

(a)

Tem

pera

ture

15

1

05

0

ndash051 2 30

y

Pr = 70Pr = 50Pr = 071

Pr = 15

g (t) = H (t)

(b)

Figure 4 Continued


0

008

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t

(c)

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t

(d)

0

008

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = sint

(e)

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = sint

(f )

05

04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et

(g)

Tem

pera

ture

1

08

06

04

02

00

y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et

(h)

Figure 4 Profile of temperature for different values of Pr and variation of time


0

008

01

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = tet

(a)

04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = tet

(b)

0

008

01

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et sint

(c)

04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et sint

(d)

Tem

pera

ture

0

002

0015

001

5 times 10ndash3


y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t sint

(e)

0

008

01

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t sint

(f )



Tem

pera

ture

15

1

05

00

y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = H (t)

(a)

Tem

pera

ture

12

1

08

06

04

02

00

y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = H (t)

(b)

0

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t

(c)

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t

(d)

0

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = sin t

(e)

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = sint

(f )

Figure 6 Continued


0y

S = 45S = 30S = 05

S = 15

05 1 15 2

05

04

03

02

01

0

Tem

pera

ture

g (t) = et

(g)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

Tem

pera

ture

1

08

06

04

02

0

g (t) = et

(h)

Figure 6 Profile of temperature for different values of S and variation of time

0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = tet

(a)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

04

03

02

01

0

ndash01

Tem

pera

ture

g (t) = tet

(b)

0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = et sint

(c)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

04

03

02

01

0

Tem

pera

ture

g (t) = et sint

(d)

Figure 7 Continued


Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

015

001

002

5 times 10ndash3

0

g (t) = t sint

(e)

0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t sint

(f )


0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = 1 g(t) = 1 h(t) = 1

(a)

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = t g(t) = 1 h(t) = 1

(b)

0 1 2 3 4 5y

Velo

city

2

15

1

05

0

M = 05M = 10

M = 15M = 25

f (t) = et g(t) = 1 h(t) = 1

(c)

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = sin t g(t) = 1 h(t) = 1

(d)

Figure 8 Continued


acceleration in fluid flow is due to the fact that buoyancyforces bcomes powerful than viscous forces

Figures 12ndash14 show the chemical reaction Kc pa-rameter effects on velocity and concentration descriptione increase in Kc decreases the concentration and ve-locity profiles Basically chemical molecular diffusivityand species concentration drop with the higher values ofKc e distribution in concentration falls at all the fluidflow field points with the rise in Kc e curve for velocityfor different values of the Maxwell fluid parameter λ isplotted in Figure 15 It is observed that an increase in λproduces a significant increase in the momentumboundary layer of the fluid which then increases the ve-locity e rise in λ will therefore correspond to a fall influid viscosity resulting in it accelerating the flow andhence velocity rising Further an increase in λ causes a risein velocity near the plate surface Although the trend isreversed away from the plate the Newtonian fluid(λ⟶ 0) has a higher velocity

In Figure 16 velocity profiles are plotted against theheat absorption parameter S values ese curves showthat the velocity is a decreasing function of parameterS Furthermore due to the absorption of heat the fluidtemperature diminishes and the thermal buoyancy forcediminishes ese results have seen a fall in the velocity ofa fluid with the increase in the values of S

Figure 17 shows the behavior of velocity profile fordifferent values of the parameters with a different choice ofthe functionf(t) g(t) h(t) For validation of our results weconsider some special cases of temperature profile alreadyexisting in literature and their graphical illustration isdepicted in Figures 18ndash20 Figure 18 shows the temperaturedecrease with the increase in Pr for the variation of time withg(t) 1 minus eminus t e effects of the heat absorption parametercan be observed in Figure (19) which depicts the decline intemperature e impacts of g(t) 1 minus aeminus bt for differentchoices of a and b are explained in Figure 20 We see thedecline in temperature with the increasing values of a and b

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = tet g(t) = 1 h(t) = 1

(e)

0 1 2 3 4 5y

6

4

2

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = et sin t g(t) = 1 h(t) = 1

(f )

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = t sin t g(t) = 1 h(t) = 1

(g)

Figure 8 Profile of velocity for different values of M and Pr 071 λ 06 S 05 Sc 060 Kc 05 GT 50 GC 20


3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = sin t g (t) = 1 h (t) = 1

(c)

2

1

0

ndash1

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = et g (t) = 1 h (t) = 1

(d)

20 1 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

f (t) = tet g (t) = 1 h (t) = 1

(e)

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

5

4

3

2

1

0

Velo

city

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 9 Continued


2

1

0

ndash1

Velo

city

20 1 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = t sin t g (t) = 1 h (t) = 1

(g)

Figure 9 Profile of velocity for different values of Pr and M 05 λ 06 S 05 Sc 060 Kc 05 GT 50 GC 20

f (t) = 1 g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(a)

f (t) = t g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(b)

4

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

f (t) = sin t g (t) = 1 h (t) = 1

(c)

f (t) = et g (t) = 1 h (t) = 1

4

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(d)

Figure 10 Continued


f (t) = tet g (t) = 1 h (t) = 1

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

(e)

f (t) = et sin t g (t) = 1 h (t) = 1

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

5

4

3

2

1

0

Velo

city

(f )

f (t) = t sin t g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(g)

Figure 10 Profile of velocity for different values of GT and M 02 λ 06 S 105 Kc 05 Sc 060 GC 20 Pr 071

0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

4

3

2

1

0

Velo

city

f (t) = 1 g (t) = 1 h (t) = 1

(a)

GC = 80GC = 60GC = 20

GC = 40

0 1 2 3 54y

4

3

2

1

0

Velo

city

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 11 Continued


0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

4

3

2

1

0

Velo

city

f (t) = sin t g (t) = 1 h (t) = 1

(c)

GC = 80GC = 60GC = 20

GC = 40

0 1 2 3 54y

4

3

2

1

0

Velo

city

f (t) = et g (t) = 1 h (t) = 1

(d)

0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

f (t) = tet g (t) = 1 h (t) = 1

(e)

GC = 80GC = 60GC = 20

GC = 40

5

4

3

2

1

0

Velo

city f (t) = et sin t g (t) = 1 h (t) = 1

0 1 2 3 54y

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

f (t) = t sint g (t) = 1 h (t) = 1

(g)

Figure 11 Profile of velocity for different values of GC and M 02 λ 06 S 05 Kc 05 Sc 060 GT 50 Pr 071


3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = t g (t) = 1 h (t) = 1

(b)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = sin t g (t) = 1 h (t) = 1

(c)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = et g (t) = 1 h (t) = 1

(d)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

f (t) = tet g (t) = 1 h (t) = 1

(e)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

5

4

3

2

1

0

Velo

city

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 12 Continued


0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0f (t) = t sin t g (t) = 1 h (t) = 1

(g)

Figure 12 Profile of velocity for different values of Kc and M 05 λ 06 S 05 Sc 060 GT 50 GC 20 Pr 071

Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(a)

Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(b)

005

004

003

002

001

0

Conc

entr

atio

n

g (t) = t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(c)

Conc

entr

atio

n

02

015

01

005

0

g (t) = t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(d)

Figure 13 Continued


005

004

003

002

001

0

Conc

entr

atio

ng (t) = sin t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(e)

Conc

entr

atio

n

02

015

01

005

0

g (t) = sin t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(f )

03

04

02

01

0

Conc

entr

atio

n

g (t) = et

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(g)

Conc

entr

atio

n

08

06

04

02

0

g (t) = et

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(h)

Figure 13 Profile of concentration for different values of Kc and variation of time

0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = tet

(a)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = tet

(b)

Figure 14 Continued


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = et sint

(c)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = et sint

(d)

001

8 times 10ndash3

6 times 10ndash3

4 times 10ndash3

2 times 10ndash3

0

Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = t sin t

(e)

0

008

006

004

002

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = t sin t

(f )


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = 1 g (t) = 1 h (t) = 1

(a)

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 15 Continued


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = sin t g (t) = 1 h (t) = 1

(c)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 105

0

ndash1 times 105

ndash2 times 105

ndash3 times 105

f (t) = et g (t) = 1 h (t) = 1

(d)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 106

0

ndash1 times 106

ndash2 times 106

ndash3 times 106

f (t) = tet g (t) = 1 h (t) = 1

(e)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

100

0

ndash100

ndash200

ndash300

ndash400

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t sin t g (t) = 1 h (t) = 1

(g)

Figure 15 Profile of velocity for different values of λ and M 20 GC 20 S 15 Kc 05 Sc 060 GT 100 Pr 05


3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = sin t g (t) = 1 h (t) = 1

(c)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et g (t) = 1 h (t) = 1

(d)

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = tet g (t) = 1 h (t) = 1

(e)

5

4

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 16 Continued


3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t sin t g (t) = 1 h (t) = 1

(g)

Figure 16 Profile of velocity for different values of S and M 20 GC 20 λ 06 Kc 05 Sc 060 GT 100 Pr 05

3

2

1

00 1 2 3 54

y

M = 20M = 15M = 05

M = 10

f (t) = 1 g (t) = sin t h (t) = t sin t

Vel

ocity

(a)

1

0

ndash1

ndash2

ndash30 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

f (t) = 1 g (t) = sin t h (t) = tet

Vel

ocity

(b)

2

0

ndash2

ndash40 1 2 3 54

y

Gc = 80Gc = 60Gc = 20

Gc = 40

f (t) = t sin t g (t) = t h (t) = tet

Vel

ocity

(c)

6

4

2

00 1 2 3 54

y

GT = 10GT = 60GT = 10

GT = 35

f (t) = 1 g (t) = sin t h (t) = t

Vel

ocity

(d)

Figure 17 Continued


2

1

0

ndash1

ndash2

ndash30 1 2 3 54

y

Sc = 078Sc = 060Sc = 022

Sc = 030

f (t) = et g (t) = sin t h (t) = tetV

eloc

ity

(e)

3

2

1

0

ndash10 1 2 3 54

y

S = 40S = 30S = 10

S = 20

f (t) = t g (t) = et h (t) = 1

Vel

ocity

(f )

Figure 17 Profile of velocity for different values of M S Sc Kc GT GC and different choice of function for f(t) g(t) h(t)

04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = 1 ndash endasht

(a)

06

04

02

0

ndash02

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(b)

Figure 18 Temperature profile for different values of Pr with S 05 and g(t) 1 minus eminus t


6 Conclusions

A thorough investigation of MHDMaxwell fluid motion hasbeen studied here under the effects of different parameterse exact solutions are obtained for concentration tem-perature and velocity which satisfied the described initialand boundary conditions Laplace transform is employed toobtain the exact solution and the behavior of different pa-rameters on the flow of fluid along with different boundaryconditions is investigated Effects of chemical reaction co-efficient Schmidt number and different boundary condi-tions on concentration effects of Prandtl number heat

source Newtonian heating etc on temperature Magneticparameter Schmidt number thermal Grashof number re-laxation parameter mass Grashof number Prandtl numberheat source and chemical reaction impacts on the fluidmotion are discussed e results obtained are as follows

(1) e boundary layer thickness of concentration de-creases with the increase in the mass diffusivity Sc

and chemical reaction parameter KC(2) e thermal boundary layer decreases with the in-

crease in momentum boundary layer due to Pr andheat absorption S

04

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(a)

06

04

02

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(b)

Figure 19 Temperature profile for different values of S with Pr 071 and g(t) 1 minus eminus t

08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2

g (t) = 1 ndash aendashbt

(a)

1

08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(b)

Figure 20 Temperature profile for different values of a b with Pr 071 S 05 and g(t) 1 minus aeminus bt


(3) Lorentz force effects due to M momentumboundary layer effects due to Pr mass diffusivityeffects due to Sc chemical reaction Kc and heatabsorption decrease the velocity with the increase ofthese parameters

(4) Increase in buoyancy forces due to GT and GC

stimulates the speed of fluid flow

(5) A rise in relaxation time λ reduces the fluid viscosityand results in acceleration of fluid flow

Appendix

B1(y t)

0 0lt tltyλ

eminus α1t

I0 iα3

t2

minus λy2

1113969

1113874 1113875 tgtyλ

⎧⎪⎪⎨

⎪⎪⎩(A1)

BII2 (t) e


lowast 1 + α1t( 1113857 + α23eminus α1t

I0 iα3t( 11138571113872 1113873lowastt (A2)

BII3 (t) (1 + λH(t))

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste

minus α1tI0 iα3t( 1113857 (A3)

BII4 (t) (1 + λH(t)

lowastDlowast

+ Elowaste

minus α7+α9( )t+ Flowaste

minus α7minus α9( )t1113874 1113875

lowaste


BIII2 (t) e


lowast cos t + α1 sin t( 1113857 + α23eminus α1t

I0 iα3t( 11138571113872 1113873lowast sin t (A5)

BIII3 (t) cos t + λH(t)

lowast cos t( 1113857lowast

Alowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste




Dlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIV2 (t) e


lowastH(t) + α1( 1113857

lowaste

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t (A8)

BIV3 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BIV4 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste

minus α1tI0 iα3t( 1113857 (A10)

BV2 (t) e


lowaste

t+ 1 + α1( 1113857te

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowastte

t (A11)

BV3 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A12)

BV4 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A13)

BVI2 (t) e


lowaste

t cos t + 1 + α1( 1113857et sin t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t sin t(A14)

BVI3 (t) e

t cos t + et sin t + λH(t)

lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A15)


BVI4 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A16)

BVII2 (t) e


lowastH(t) + 1 + α1( 1113857t sin t( 1113857 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowastt sin t

(A17)

BVII3 (t) (t sin t + t cos t) + λH(t)

lowast(sin t + λ cos t)( 1113857

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A18)



lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A19)

Lminus 1 e

minus bq2minus a2

radic

q2

minus a2

1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

0 0lt tlt b bgt 0

I0 a

t2

minus b2

1113969

1113888 1113889 tgt bRe(q)gtRe(a)

⎧⎪⎪⎨

⎪⎪⎩(A20)

Lminus 1

[F(q + m)] f(t)eminus mt

(A21)

Lminus 1 exp(minus y

q + a

radic)

q minus b1113890 1113891 e

btΦ(y t a + b) (A22)

Φ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A23)


q + a

radic)

q minus b1113890 1113891 e

btΨ(y t a + b) (A24)

Ψ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A25)

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

e authors are thankful and grateful to their respectivedepartments and universities for supporting and facilitatingthe research work

References

[1] C-Y Cheng ldquoNatural convection heat andmass transfer froma sphere in micropolar fluids with constant wall temperatureand concentrationrdquo International Communications in Heatand Mass Transfer vol 35 no 6 pp 750ndash755 2008

[2] K R Cramer and S I Pai Magneto Fluid Dynamics forEngineers and Applied Physicists pp 204ndash237 McGraw-HillBook Co New York New York NY USA 1973

[3] N F M Noor S Abbasbandy and I Hashim ldquoHeat and masstransfer of thermophoreticMHD flow over an inclined radiateisothermal permeable surface in the presence of heat sourcesinkrdquo International Journal of Heat andMass Transfer vol 55no 7-8 pp 2122ndash2128 2012

[4] S M Mousazadeh M M Shahmardan T TavangarK Hosseinzadeh and D D Ganji ldquoNumerical investigationon convective heat transfer over two heated wall-mountedcubes in tandem and staggered arrangementrdquoBeoretical andApplied Mechanics Letters vol 8 no 3 pp 171ndash183 2018

[5] F M Oudina F Redouane F Redouane and C RajashekharldquoConvection heat transfer of MgO-Agwater magneto-hybridnanoliquid flow into a special porous enclosurerdquo AlgerianJournal of Renewable Energy and Sustainable Developmentvol 2 no 2 pp 84ndash95 2020

[6] S S Das A Satapathy J K Das and J P Panda ldquoMasstransfer effects on MHD flow and heat transfer past a verticalporous plate through a porous medium under oscillatorysuction and heat sourcerdquo International Journal of Heat andMass Transfer vol 52 no 25-26 pp 5962ndash5969 2009


[7] S M Abo-Dahab M A Abdelhafez F Mebarek-Oudina andS M Bilal ldquoMHD Casson nanofluid flow over nonlinearlyheated porous medium in presence of extending surface effectwith suctioninjectionrdquo Indian Journal of Physics vol 232021

[8] S Sajad A Nori K Hosseinzadeh and D D Ganji ldquoHy-drothermal analysis of MHD squeezing mixture fluid sus-pended by hybrid nano particles between two parallel platesrdquoCase Studies in Bermal Engineering vol 21 Article ID100650 2020

[9] N Iftikhar S M Husnine and M B Riaz ldquoHeat and masstransfer in MHDMaxwell fluid over an infinite vertical platerdquoJournal of Prime Research in Mathematics vol 15 pp 63ndash802019

[10] M Ahmed ldquoMegahed Variable fluid properties and variableheat flux effects on the flow and heat transfer in a non-Newtonian Maxwell fluid over an unsteady stretching sheetwith slip velocityrdquo Chinese Physics B vol 922 Article ID094701 2013

[11] S Marzougui M Bouabid F Mebarek-Oudina N Abu-Hamdeh M Magherbi and K Ramesh ldquoA computationalanalysis of heat transport irreversibility phenomenon in amagnetized porous channelrdquo International Journal of Nu-merical Methods for Heat amp Fluid Flow vol 43 2020

[12] D Belatrache N Saifi A Harrouz and S BentoubaldquoModelling and numerical investigation of the thermalproperties effect on the soil temperature in Adrar regionrdquoAlgerian Journal of Renewable Energy and Sustainable De-velopment vol 2 no 2 pp 165ndash174 2020

[13] K Hosseinzadeh A R Mogharrebi A Asadi M Paikar andD D Ganji ldquoEffect of fin and hybrid nano-particles on solidprocess in hexagonal triplex latent heat thermal energystorage systemrdquo Journal of Molecular Liquids vol 300 ArticleID 112347 2020

[14] M Gholinia S Gholinia K Hosseinzadeh and D D GanjildquoInvestigation on ethylene glycol nano fluid flow over avertical permeable circular cylinder under effect of magneticfieldrdquo Results in Physics vol 9 pp 1525ndash1533 2018

[15] J Rahimi D D Ganji M Khaki and K HosseinzadehldquoSolution of the boundary layer flow of an Eyring-Powell non-Newtonian fluid over a linear stretching sheet by collocationmethodrdquo Alexandria Engineering Journal vol 56 no 4pp 621ndash627 2017

[16] K Hosseinzadeh M A E Moghaddam A AsadiA R Mogharrebi and D D Ganji ldquoEffect of internal finsalong with Hybrid Nano-Particles on solid process in starshape triplex Latent Heat ermal Energy Storage System bynumerical simulationrdquo Renewable Energy vol 154 pp 497ndash507 2020

[17] U S Rajput and S Kumar ldquoRadiation effects on MHD flowpast an impulsively started vertical plate with variable heatand mass transferrdquo IJAMM International Journal of AppliedMathematics and Mechanics vol 8 pp 66ndash85 2012

[18] A S Gupta ldquoSteady and transient free convection of anelectrically conducting fluid from a vertical plate in thepresence of magnetic fieldrdquo Archives of Applied Science Re-search vol 8 pp 319ndash333 1961

[19] M F El-Amin ldquoMHD free convection and mass transfer flowin micropolar fluid with constant suctionrdquo Journal of Mag-netism and Magnetic Materials vol 243 no 3 pp 567ndash5742006

[20] S Nadeem R U Haq and Z H Khan ldquoNumerical study ofMHD boundary layer flow of a Maxwell fluid past a stretchingsheet in the presence of nanoparticlesrdquo Journal of the Taiwan

Institute of Chemical Engineers vol 45 no 1 pp 121ndash1262014

[21] J Zhao L Zheng X Zhang and F Liu ldquoUnsteady naturalconvection boundary layer heat transfer of fractional Maxwellviscoelastic fluid over a vertical platerdquo International Journal ofHeat and Mass Transfer vol 97 pp 760ndash766 2016

[22] N Ahmad ldquoSoret and radiation effects on transientMHD freeconvection from an impulsively started infinite verticl platerdquoBe Journal of Heat Transfer vol 134 Article ID 062701 2012

[23] R C Chaudhary and A Jain ldquoAn exact solution of magne-tohydrodynamic convection flow past an accelerated surfaceembedded in a porousmediumrdquo International Journal of Heatand Mass Transfer vol 53 no 7-8 pp 1609ndash1611 2010

[24] K Das ldquoExacat solution of MHD free convection flow andmass transfer near a moving vertical plate in presesnce ofthermal radiationrdquo African Journal of Mathematical Physicsvol 8 pp 29ndash41 2010

[25] K Das and S Jana ldquoHeat and mass transfer effects on un-steady MHD free convection flow near a moving plate inporous mediumrdquo Bulletin of Society of Mathematiciansvol 17 pp 15ndash32 2010

[26] C Fetecau and N A Shah ldquoDumitru vieru general solutionsfor hydromagnetic free convection flow over an infinite platewith Newtonian heating mass diffusion and chemical reac-tionrdquo Communications in Beoretical Physics vol 68 p 62017

[27] N A Shah ldquoHeat and mass transfer in hydromagnetic flowsof viscous fluids over a flat platerdquo Phd thesis GovernmentCollege University Lahore Lahore Pakistan 2019


considered the natural convective flow of the electricallyconducting fluid past on vertical plate embedded in a per-meable medium and explored the impacts of heat and masstransfer e heat transfer phenomenon of Casson nanofluidflow is taken into account by Abo-Dahab et al [7] eyanalyzed the problem with the convective boundary condi-tions and discussed the influence of chemical reaction andheat source Sajad et al[8] studied the heat transfer andmagnetic effect on hybrid nanofluid Nazish et al [9] exploredthe influence of heat and concentrationmass transform withthe existence of fields developed by magnetic in the Maxwellfluid model Ahmad [10] explored the heat transfer for theMaxwell fluid on the stretching plate with the slip boundaryon the velocity ey explored the numerical solutions andshowed the heat flux effect using the Nusselt number and thePrandtl number A computational analysis is performed tostudy the effects of the transverse magnetic field at the un-steady PoiseuillendashRayleighndashBenard flow by Marzougui et al[11] e thermal propertiesrsquo effects on the soil temperatureare modeled and investigated numerically by Belatrache [12]To have more insight about heat and mass transfer mecha-nisms in fluid flow and their applications readers are referredto review references [13ndash16]On the other hand many re-searchers paid a significant amount of attention to the study ofMHD free convective flows due to its numerous applicationsin solar and stellar structure radio propagation MHDpumps MHD bearings aerodynamics polymer technologypetroleum industry crude oil purification glass fiber drawingetc In light of these applications many researchers such asRajput [17] Gupta [18] and El Amin [19] studied the MHDflow of different fluids ey found the exact solution forvelocity concentration and temperature by the Laplacetransform method Heat and mass transfer simultaneouseffects on MHD flow of Maxwell fluid have been investigatedby Nadeem et al [20] Recently the study of the unsteadyboundary layer heat transfer of Maxwell viscoelastic fluid wascarried out by Zhao et al [21] Ahmad [22] studied MHDviscous with constant density electro-conducting fluid in theexistence of the radiation thermal diffusion free convectionand mass transfer flow ese results motivated Chaudhryet al [23] and they used classical integral transform to obtainthe exact solutions of naturalMHD convective flow past on anaccelerated surface submerged in a permeable medium Das[24] developed the closed-form solution of the unsteady

MHD natural convection flow on a moving vertical plateaccompanied by mass transfer and thermal radiation Car-rying on Das et al [25] investigated the time-dependentMHD natural convection flow past a moving vertical platedipped in a porous medium and studied the different aspectsof heat and mass transfer ey discussed the problem withthe uniform oscillating and impulsive motions of the platebesides considering the constant heat and mass diffusion andimplemented the Laplace integral transform to develop theanalytic solutionsMotivated by these investigations the ob-jective of this manuscript is to study the combined effect ofheat and mass transfer on MHD Maxwell fluid Laplace in-tegral transformation is used to obtain the unique solution oftemperature velocity and concentration under the impact ofgeneralized boundary conditions on temperature velocityand concentration e importance of the problem is high-lighted by showing its impactapplications in the field ofengineering and applied sciences e paper is organized intosix sections After the introductory section in Section 2 thedimensionless governing equations are developed In Section3 Laplace integral transform is implemented to find the exactsolution of the temperature velocity and concentration fieldIn Section 4 some applications in different fields are discussedas limiting cases to justify our results In Section 5 the effect ofphysical parameters is analyzed graphically e concludingobservation is listed at the end

2 Problem Formulation

We studied here the motion of the viscoelastic in-compressible electronically conducting Maxwell fluiddue to plate motion with arbitrary velocity u0fprime(tprime) eplate is along x minus axis and y minus axis is considered normalon the plate In the first instance at t 0 the plate andfluid are at temperature Tinfinprime and concentration Cinfinprime Withthe time t 0+ the plate starts to move in its own axisen the level of temperature and concentration takes up

to Tinfin^

prime1113882 1113883 + Tw

^

prime1113882 1113883hprime(tprime) and Cinfin^

prime1113882 1113883 + Cw

^

prime1113882 1113883gprime(tprime) where

fprime(tprime) hprime(tprime) and gprime(tprime) are piecewise continuous func-tions that vanish at t 0 Details of different parametersare given in Table 1 Momentum energy and concen-tration equations are formed as follows


1 + λprimez

ztprime1113888 1113889


ztprime ]


zχprime2

+ gBT 1 + λprimez


1 + λprimez


σB20

ρ1 + λprime

z


ρCp


ztprime K




ztprime D



(1)











(2)





t u20]

tprime

y u0

]χprime

Sc ]D

Kc ]u20Kcprime

λ u20]λprime

uprime u0u


Cwprime


φwprime

S ]

ρCpu20Sprime

Pr μCp

K

Kpprime

u20

]2KpGc

g BC]( 1113857

u30

Cwprime

GT gBT]

u30φwprime

M σB

20]

ρu20

f ]u2o

fprime

g ]u2o

gprime

h ]u2o

hprime

(3)


1 + λz

zt1113888 1113889

zu(y t)

zt

z2u(y t)

zy2 + GT 1 + λ

z

zt1113888 1113889T(y t)

+ GC 1 + λz

zt1113888 1113889C(y t)

minus M 1 + λz

zt1113888 1113889u(y t)

(4)

zT(y t)

zt

1Pr

z2T(y t)


zC(y t)

zt

1Sc

z2C(y t)



u(y 0) 0 T(y 0) 0 C(y 0) 0 (7)

u(0 t) f(t) T(0 t) h(t) C(0 t) g(t) (8)


(9)



z2C(y q)



C(y q) C1eminus y

Sc Kc+q( )

1113968

+ C2ey

Sc Kc+q( )

1113968

(11)


C(y q) G(q)eminus y

Sc Kc+q( )

1113968

(12)



C(y t) 1113946t


Sc

1113968 s Kc( 1113857ds (13)



z2T(y q)


e solution is

T(y q) C1eminus y

Pr(S+q)

radic

+ C2ey

Pr(S+q)

radic

(15)


T(y q) H(q)eminus y

Pr(S+q)

radic

(16)


T(y t) 1113946t


Pr

1113968 s S( 1113857ds (17)



z2u(y q)



(18)


u(y q) F(q)eminus y

(q+M)(1+λq)

radic

+GTH(q)(1 + λq) e

minus y(q+M)(1+λq)

radic

minus eminus y

Pr(S+q)

radic

1113874 1113875

Pr(S + q) minus (1 + λq)(q + M)( 1113857

+GcG(q)(1 + λq) e

minus y(q+M)(1+λq)

radic

minus eminus y

Sc Kc+q( )

1113968

1113874 1113875

Sc Kc + q( 1113857 minus (1 + λq)(q + M)( 1113857

(19)


u(y q) F(q)eminus y

λ q+α1( )

2+α23( 1113857

1113969

minus


λ q+α1( )

2+α231113858 1113859

1113969

minus eminus y

Pr(S+q)

radic⎛⎝ ⎞⎠

λ q + α4( 11138572

minus α261113872 1113873

minus


λ q+α1( )

2+α231113858 1113859

1113969

minus eminus y

Sc Kc+q( )

1113968⎛⎝ ⎞⎠

λ q + α7( 11138572

minus α291113872 1113873

(20)





u(y t) I1 +GT

λI2 minus

GT

λI3 +

Gc

λI4 minus

Gc

λI5 (21)

where

I1 Lminus 1

F(q)eminus y

λ q+α1( )

2+α23( 1113857

1113969

⎛⎝ ⎞⎠

Lminus 1


B2(q)( 1113857

B1(y t)lowastB2(t)

(22)


radicy

((q+α1)2+α23)

1113968

((q + α1)2 + α23)

1113969

)

B2(q) F(q)

((q + α1)2 + α23)

1113969



B1(y t)

0 0lt tltyλ

radic

eminus α1t

I0 iα3

t2

minus λy2

1113969

1113874 1113875 tgtyλ

radic

⎧⎪⎪⎨

⎪⎪⎩

(23)

B2(t) eminus α1t

iα3( 1113857I1 iα3t( 1113857 + δ(t)( 11138571113872 1113873lowast

fprime(t) + α1f(t)( 1113857

+ α23eminus α1t

I0 iα3t( 11138571113872 1113873lowastf(t)

(24)


minus yPr(S+q)

radic

q + α4( 11138572

minus α261113872 1113873⎛⎝ ⎞⎠



+S minus α4( 1113857

α6e

minus α4tsinh α6t( 11138571113889

lowasterfcy

Pr

1113968

2t


(25)


minus y

λ q+α1( )

2+α23( 1113857

1113969

q + α4( 11138572

minus α261113872 1113873

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

B1(y t)lowastB3(t)

(26)



lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(27)

where

Alowast

α21 + α231113872 1113873

α24 minus α26

Blowast

2 α4 minus α6( 1113857A


2α6

Clowast

1 minus Alowast

minus Blowast

(28)


minus ySc Kc+q( )

1113968

q + α7( 11138572

minus α291113872 1113873⎛⎝ ⎞⎠



+Kc minus α7( 1113857

α9e

minus α7tsinh α9t( 11138571113889

lowasterfcy

Sc

1113968

2t


(29)

Similarly


minus y

λ q+α1( )

2+α23( 1113857

1113969

q + α7( 11138572

minus α291113872 1113873

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

B1(y t)lowastB4(t)

(30)



lowastClowast

+ Dlowaste


minus α7minus α9( )t1113874 1113875

lowaste


Dlowast

α21 + α231113872 1113873

α27 minus α29

Elowast

2 α7 minus α9( 1113857A


2α9

Flowast

1 minus Dlowast

minus Elowast

(31)


4 Applications




ySc

1113968

2t


(32)




yPr

1113968

2t


(33)



u(y t) II1 +

GT

λI

I2 minus

GT

λI

I3 +

Gc

λI

I4 minus

Gc

λI

I5 (34)

where

II1 B1(y t)

lowastB

I2(t) (35)


BI2(t) e


lowast δ(t) + α1H(t)( 1113857

+ α23eminus α1t


(36)


equation (25)


minus α4tcosh α6t( 11138571113872

+S minus α4( 1113857

α6e

minus α4tsinh α6t( 11138571113889

lowasterfcy

Pr

1113968

2t


(37)


II3 B1(y t)

lowastB

I3(t) (38)

where



lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(39)


tion (29)



α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


(40)


II5 B1(y t)

lowastB

I4(t) (41)

and



lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(42)




1 III2 III

3 III4 and III



ySc

1113968

2t



yPr

1113968

2t


u(y t) III1 +

GT

λI

II2 minus

GT

λI

II3 +

Gc

λI

II4 minus

Gc

λI

II5

(43)

where

III1 B1(y t)

lowastB

II2 (t) (44)


(A2))

III2 (t + λ)

lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


III3 B1(y t)

lowastB

II3 (t)

(45)


(A3))

III4 (t + λ)

lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


III5 B1(y t)

lowastB

II4 (t)

(46)



(A4))



ySc

1113968

2t


(47)



yPr

1113968

2t


(48)


u(y t) IIII1 +

GT

λI

III2 minus

GT

λI

III3 +

Gc

λI

III4 minus

Gc

λI

III5 (49)

where

IIII1 B1(y t)

lowastB

III2 (t) (50)


(A5))


lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


IIII3 B1(y t)

lowastB

III3 (t)

(51)


(A6))


lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


IIII5 B1(y t)

lowastB

III4 (t)

(52)


(A7))



C(y t) et

+ Kcet


ySc

1113968

2t


(53)


T(y t) et

+ Set


yPr

1113968

2t


(54)


u(y t) IIV1 +

GT

λI

IV2 minus

GT

λI

IV3 +

Gc

λI

IV4 minus

Gc

λI

IV5 (55)

where

IIV1 B1(y t)

lowastB

IV2 (t) (56)


(A8))


IIV2 e

t+ λe

t1113872 1113873

lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


IIV3 B1(y t)

lowastB

IV3 (t)

(57)


(A9))

IIV4 e

t+ λe

t1113872 1113873

lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


(58)

Similarly

IIV5 B1(y t)

lowastB

IV4 (t) (59)


(A10))


C(y t) et

+ Kc + 1( 1113857tet


ySc

1113968

2t


(60)


T(y t) et

+(S + 1)tet


yPr

1113968

2t


u(y t) IV1 +

GT

λI

V2 minus

GT

λI

V3 +

Gc

λI

V4 minus

Gc

λI

V5

(61)


(q minus 1)2)

IV1 B1(y t)

lowastB

V2 (t) (62)


(A11))

IV2 te

t+ λe

t(t + 1)1113872 1113873

lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


IV3 B1(y t)

lowastB

V3 (t)

(63)


(A12))

IV4 te

t+ λe

t(t + 1)1113872 1113873

lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


IV5 B1(y t)

lowastB

V4 (t)

(64)


(A13))





lowasterfcy

Sc

1113968

2t


T(y t) et(cos t +(1 + S)sin t)1113872 1113873

lowasterfcy

Pr

1113968

2t


u(y t) IVI1 +

GT

λI

VI2 minus

GT

λI

VI3 +

Gc

λI

VI4 minus

Gc

λI

VI5

IVI1 B1(y t)

lowastB

VI2 (t)

(65)


(A14))

IVI2 e

t sin t(1 + λ) + et cos t1113872 1113873

lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


IVI3 B1(y t)

lowastB

VI3 (t)

(66)


(A15))

IVI4 e

t sin t(1 + λ) + et cos t1113872 1113873

lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


(67)

Similarly

IVI5 B1(y t)

lowastB

VI4 (t) (68)





ySc

1113968

2t


(69)



yPr

1113968

2t


u(y t) IVII1 +

GT

λI

VII2 minus

GT

λI

VII3 +

Gc

λI

VII4 minus

Gc

λI

VII5

(70)

where

IVII1 B1(y t)

lowastB

VII2 (t) (71)


(A17))


lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


IVII3 B1(y t)

lowastB

VII3 (t)

(72)


tion (A18))


lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


IVII5 B1(y t)

lowastB

VII4 (t)

(73)





GC in equation (9))










f (t) = 1 g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

15

1

05

0

(a)

f (t) = t g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

2

15

1

05

0

(b)

f (t) = sint g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

15

1

05

0

(c)

f (t) = et g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

15

1

05

0

(d)

f (t) = tet g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

(e)

f (t) = et sint g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

5

4

3

2

1

0

Velo

city

(f )

Figure 1 Continued


f (t) = t sint g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

15

1

05

0

(g)


Conc

entr

atio

n

15

1

05

0

ndash050 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = H (t)

(a)

Conc

entr

atio

n

15

1

05

00 1 2 3 4 5

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = H (t)

(b)

005

004

003

002

001

0

ndash0010 1 2 3

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = t

(c)

02

015

01

005

00 1 2 3 4

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = t

(d)

Figure 2 Continued


005

004

003

002

001

0

ndash0010 1 2 3

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = sin t

(e)

02

015

01

005

00 1 2 3 4

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = sin t

(f )


04

03

02

01

0

ndash010 1 2 3

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = et

Conc

entr

atio

n

(a)

08

06

04

02

0

ndash020 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = etCo

ncen

trat

ion

(b)

004

0

006

002

ndash0020 1 2 3

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = tet

Conc

entr

atio

n

(c)

03

02

01

0

ndash010 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = tet

Conc

entr

atio

n

(d)

Figure 3 Continued


006

004

002

0

ndash0020 1 2 3

y

Sc = 14Sc = 096Sc = 022

Sc = 060

g (t) = et sintCo

ncen

trat

ion

(e)

03

02

01

00 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = et sint

Conc

entr

atio

n

(f )

001

8 times 10ndash3

6 times 10ndash3

4 times 10ndash3

2 times 10ndash3

0


0 1 2 3y

Sc = 14Sc = 096Sc = 022

Sc = 060

g (t) = t sint

Conc

entr

atio

n

(g)

0

008

006

004

002

ndash0020 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = t sintCo

ncen

trat

ion

(h)


Tem

pera

ture

15

1

05

0

ndash051 2 30

y

Pr = 70Pr = 50Pr = 071

Pr = 15

g (t) = H (t)

(a)

Tem

pera

ture

15

1

05

0

ndash051 2 30

y

Pr = 70Pr = 50Pr = 071

Pr = 15

g (t) = H (t)

(b)

Figure 4 Continued


0

008

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t

(c)

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t

(d)

0

008

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = sint

(e)

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = sint

(f )

05

04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et

(g)

Tem

pera

ture

1

08

06

04

02

00

y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et

(h)



0

008

01

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = tet

(a)

04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = tet

(b)

0

008

01

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et sint

(c)

04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et sint

(d)

Tem

pera

ture

0

002

0015

001

5 times 10ndash3


y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t sint

(e)

0

008

01

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t sint

(f )



Tem

pera

ture

15

1

05

00

y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = H (t)

(a)

Tem

pera

ture

12

1

08

06

04

02

00

y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = H (t)

(b)

0

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t

(c)

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t

(d)

0

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = sin t

(e)

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = sint

(f )

Figure 6 Continued


0y

S = 45S = 30S = 05

S = 15

05 1 15 2

05

04

03

02

01

0

Tem

pera

ture

g (t) = et

(g)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

Tem

pera

ture

1

08

06

04

02

0

g (t) = et

(h)


0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = tet

(a)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

04

03

02

01

0

ndash01

Tem

pera

ture

g (t) = tet

(b)

0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = et sint

(c)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

04

03

02

01

0

Tem

pera

ture

g (t) = et sint

(d)

Figure 7 Continued


Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

015

001

002

5 times 10ndash3

0

g (t) = t sint

(e)

0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t sint

(f )


0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = 1 g(t) = 1 h(t) = 1

(a)

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = t g(t) = 1 h(t) = 1

(b)

0 1 2 3 4 5y

Velo

city

2

15

1

05

0

M = 05M = 10

M = 15M = 25

f (t) = et g(t) = 1 h(t) = 1

(c)

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = sin t g(t) = 1 h(t) = 1

(d)

Figure 8 Continued






0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = tet g(t) = 1 h(t) = 1

(e)

0 1 2 3 4 5y

6

4

2

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = et sin t g(t) = 1 h(t) = 1

(f )

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = t sin t g(t) = 1 h(t) = 1

(g)



3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = sin t g (t) = 1 h (t) = 1

(c)

2

1

0

ndash1

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = et g (t) = 1 h (t) = 1

(d)

20 1 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

f (t) = tet g (t) = 1 h (t) = 1

(e)

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

5

4

3

2

1

0

Velo

city

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 9 Continued


2

1

0

ndash1

Velo

city

20 1 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


f (t) = 1 g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(a)

f (t) = t g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(b)

4

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

f (t) = sin t g (t) = 1 h (t) = 1

(c)

f (t) = et g (t) = 1 h (t) = 1

4

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(d)

Figure 10 Continued


f (t) = tet g (t) = 1 h (t) = 1

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

(e)

f (t) = et sin t g (t) = 1 h (t) = 1

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

5

4

3

2

1

0

Velo

city

(f )

f (t) = t sin t g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(g)


0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

4

3

2

1

0

Velo

city

f (t) = 1 g (t) = 1 h (t) = 1

(a)

GC = 80GC = 60GC = 20

GC = 40

0 1 2 3 54y

4

3

2

1

0

Velo

city

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 11 Continued


0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

4

3

2

1

0

Velo

city

f (t) = sin t g (t) = 1 h (t) = 1

(c)

GC = 80GC = 60GC = 20

GC = 40

0 1 2 3 54y

4

3

2

1

0

Velo

city

f (t) = et g (t) = 1 h (t) = 1

(d)

0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

f (t) = tet g (t) = 1 h (t) = 1

(e)

GC = 80GC = 60GC = 20

GC = 40

5

4

3

2

1

0

Velo


0 1 2 3 54y

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

f (t) = t sint g (t) = 1 h (t) = 1

(g)



3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = t g (t) = 1 h (t) = 1

(b)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = sin t g (t) = 1 h (t) = 1

(c)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = et g (t) = 1 h (t) = 1

(d)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

f (t) = tet g (t) = 1 h (t) = 1

(e)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

5

4

3

2

1

0

Velo

city

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 12 Continued


0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0f (t) = t sin t g (t) = 1 h (t) = 1

(g)


Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(a)

Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(b)

005

004

003

002

001

0

Conc

entr

atio

n

g (t) = t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(c)

Conc

entr

atio

n

02

015

01

005

0

g (t) = t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(d)

Figure 13 Continued


005

004

003

002

001

0

Conc

entr

atio

ng (t) = sin t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(e)

Conc

entr

atio

n

02

015

01

005

0

g (t) = sin t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(f )

03

04

02

01

0

Conc

entr

atio

n

g (t) = et

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(g)

Conc

entr

atio

n

08

06

04

02

0

g (t) = et

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(h)


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = tet

(a)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = tet

(b)

Figure 14 Continued


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = et sint

(c)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = et sint

(d)

001

8 times 10ndash3

6 times 10ndash3

4 times 10ndash3

2 times 10ndash3

0

Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = t sin t

(e)

0

008

006

004

002

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = t sin t

(f )


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = 1 g (t) = 1 h (t) = 1

(a)

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 15 Continued


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = sin t g (t) = 1 h (t) = 1

(c)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 105

0

ndash1 times 105

ndash2 times 105

ndash3 times 105

f (t) = et g (t) = 1 h (t) = 1

(d)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 106

0

ndash1 times 106

ndash2 times 106

ndash3 times 106

f (t) = tet g (t) = 1 h (t) = 1

(e)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

100

0

ndash100

ndash200

ndash300

ndash400

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t sin t g (t) = 1 h (t) = 1

(g)



3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = sin t g (t) = 1 h (t) = 1

(c)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et g (t) = 1 h (t) = 1

(d)

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = tet g (t) = 1 h (t) = 1

(e)

5

4

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 16 Continued


3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


3

2

1

00 1 2 3 54

y

M = 20M = 15M = 05

M = 10


Vel

ocity

(a)

1

0

ndash1

ndash2

ndash30 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

f (t) = 1 g (t) = sin t h (t) = tet

Vel

ocity

(b)

2

0

ndash2

ndash40 1 2 3 54

y

Gc = 80Gc = 60Gc = 20

Gc = 40


Vel

ocity

(c)

6

4

2

00 1 2 3 54

y

GT = 10GT = 60GT = 10

GT = 35

f (t) = 1 g (t) = sin t h (t) = t

Vel

ocity

(d)

Figure 17 Continued


2

1

0

ndash1

ndash2

ndash30 1 2 3 54

y

Sc = 078Sc = 060Sc = 022

Sc = 030


eloc

ity

(e)

3

2

1

0

ndash10 1 2 3 54

y

S = 40S = 30S = 10

S = 20

f (t) = t g (t) = et h (t) = 1

Vel

ocity

(f )


04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(a)

06

04

02

0

ndash02

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(b)



6 Conclusions






04

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(a)

06

04

02

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(b)


08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(a)

1

08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(b)







Appendix

B1(y t)

0 0lt tltyλ

eminus α1t

I0 iα3

t2

minus λy2

1113969

1113874 1113875 tgtyλ

⎧⎪⎪⎨

⎪⎪⎩(A1)

BII2 (t) e



I0 iα3t( 11138571113872 1113873lowastt (A2)

BII3 (t) (1 + λH(t))

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BII4 (t) (1 + λH(t)

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIII2 (t) e






Alowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste




Dlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIV2 (t) e


lowastH(t) + α1( 1113857

lowaste

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t (A8)

BIV3 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BIV4 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste

minus α1tI0 iα3t( 1113857 (A10)

BV2 (t) e


lowaste

t+ 1 + α1( 1113857te

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowastte

t (A11)

BV3 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A12)

BV4 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A13)

BVI2 (t) e


lowaste

t cos t + 1 + α1( 1113857et sin t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t sin t(A14)

BVI3 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A15)


BVI4 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A16)

BVII2 (t) e



minus α1tI0 iα3t( 11138571113872 1113873

lowastt sin t

(A17)



lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A18)



lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A19)

Lminus 1 e

minus bq2minus a2

radic

q2

minus a2

1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

0 0lt tlt b bgt 0

I0 a

t2

minus b2

1113969

1113888 1113889 tgt bRe(q)gtRe(a)

⎧⎪⎪⎨

⎪⎪⎩(A20)

Lminus 1


(A21)


q + a

radic)

q minus b1113890 1113891 e


Φ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A23)


q + a

radic)

q minus b1113890 1113891 e


Ψ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A25)

Data Availability




Acknowledgments


References































1 + λprimez

ztprime1113888 1113889


ztprime ]


zχprime2

+ gBT 1 + λprimez


1 + λprimez


σB20

ρ1 + λprime

z


ρCp


ztprime K




ztprime D



(1)











(2)





t u20]

tprime

y u0

]χprime

Sc ]D

Kc ]u20Kcprime

λ u20]λprime

uprime u0u


Cwprime


φwprime

S ]

ρCpu20Sprime

Pr μCp

K

Kpprime

u20

]2KpGc

g BC]( 1113857

u30

Cwprime

GT gBT]

u30φwprime

M σB

20]

ρu20

f ]u2o

fprime

g ]u2o

gprime

h ]u2o

hprime

(3)


1 + λz

zt1113888 1113889

zu(y t)

zt

z2u(y t)

zy2 + GT 1 + λ

z

zt1113888 1113889T(y t)

+ GC 1 + λz

zt1113888 1113889C(y t)

minus M 1 + λz

zt1113888 1113889u(y t)

(4)

zT(y t)

zt

1Pr

z2T(y t)


zC(y t)

zt

1Sc

z2C(y t)



u(y 0) 0 T(y 0) 0 C(y 0) 0 (7)

u(0 t) f(t) T(0 t) h(t) C(0 t) g(t) (8)


(9)



z2C(y q)



C(y q) C1eminus y

Sc Kc+q( )

1113968

+ C2ey

Sc Kc+q( )

1113968

(11)


C(y q) G(q)eminus y

Sc Kc+q( )

1113968

(12)



C(y t) 1113946t


Sc

1113968 s Kc( 1113857ds (13)



z2T(y q)


e solution is

T(y q) C1eminus y

Pr(S+q)

radic

+ C2ey

Pr(S+q)

radic

(15)


T(y q) H(q)eminus y

Pr(S+q)

radic

(16)


T(y t) 1113946t


Pr

1113968 s S( 1113857ds (17)



z2u(y q)



(18)


u(y q) F(q)eminus y

(q+M)(1+λq)

radic

+GTH(q)(1 + λq) e

minus y(q+M)(1+λq)

radic

minus eminus y

Pr(S+q)

radic

1113874 1113875

Pr(S + q) minus (1 + λq)(q + M)( 1113857

+GcG(q)(1 + λq) e

minus y(q+M)(1+λq)

radic

minus eminus y

Sc Kc+q( )

1113968

1113874 1113875

Sc Kc + q( 1113857 minus (1 + λq)(q + M)( 1113857

(19)


u(y q) F(q)eminus y

λ q+α1( )

2+α23( 1113857

1113969

minus


λ q+α1( )

2+α231113858 1113859

1113969

minus eminus y

Pr(S+q)

radic⎛⎝ ⎞⎠

λ q + α4( 11138572

minus α261113872 1113873

minus


λ q+α1( )

2+α231113858 1113859

1113969

minus eminus y

Sc Kc+q( )

1113968⎛⎝ ⎞⎠

λ q + α7( 11138572

minus α291113872 1113873

(20)





u(y t) I1 +GT

λI2 minus

GT

λI3 +

Gc

λI4 minus

Gc

λI5 (21)

where

I1 Lminus 1

F(q)eminus y

λ q+α1( )

2+α23( 1113857

1113969

⎛⎝ ⎞⎠

Lminus 1


B2(q)( 1113857

B1(y t)lowastB2(t)

(22)


radicy

((q+α1)2+α23)

1113968

((q + α1)2 + α23)

1113969

)

B2(q) F(q)

((q + α1)2 + α23)

1113969



B1(y t)

0 0lt tltyλ

radic

eminus α1t

I0 iα3

t2

minus λy2

1113969

1113874 1113875 tgtyλ

radic

⎧⎪⎪⎨

⎪⎪⎩

(23)

B2(t) eminus α1t

iα3( 1113857I1 iα3t( 1113857 + δ(t)( 11138571113872 1113873lowast

fprime(t) + α1f(t)( 1113857

+ α23eminus α1t

I0 iα3t( 11138571113872 1113873lowastf(t)

(24)


minus yPr(S+q)

radic

q + α4( 11138572

minus α261113872 1113873⎛⎝ ⎞⎠



+S minus α4( 1113857

α6e

minus α4tsinh α6t( 11138571113889

lowasterfcy

Pr

1113968

2t


(25)


minus y

λ q+α1( )

2+α23( 1113857

1113969

q + α4( 11138572

minus α261113872 1113873

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

B1(y t)lowastB3(t)

(26)



lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(27)

where

Alowast

α21 + α231113872 1113873

α24 minus α26

Blowast

2 α4 minus α6( 1113857A


2α6

Clowast

1 minus Alowast

minus Blowast

(28)


minus ySc Kc+q( )

1113968

q + α7( 11138572

minus α291113872 1113873⎛⎝ ⎞⎠



+Kc minus α7( 1113857

α9e

minus α7tsinh α9t( 11138571113889

lowasterfcy

Sc

1113968

2t


(29)

Similarly


minus y

λ q+α1( )

2+α23( 1113857

1113969

q + α7( 11138572

minus α291113872 1113873

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

B1(y t)lowastB4(t)

(30)



lowastClowast

+ Dlowaste


minus α7minus α9( )t1113874 1113875

lowaste


Dlowast

α21 + α231113872 1113873

α27 minus α29

Elowast

2 α7 minus α9( 1113857A


2α9

Flowast

1 minus Dlowast

minus Elowast

(31)


4 Applications




ySc

1113968

2t


(32)




yPr

1113968

2t


(33)



u(y t) II1 +

GT

λI

I2 minus

GT

λI

I3 +

Gc

λI

I4 minus

Gc

λI

I5 (34)

where

II1 B1(y t)

lowastB

I2(t) (35)


BI2(t) e


lowast δ(t) + α1H(t)( 1113857

+ α23eminus α1t


(36)


equation (25)


minus α4tcosh α6t( 11138571113872

+S minus α4( 1113857

α6e

minus α4tsinh α6t( 11138571113889

lowasterfcy

Pr

1113968

2t


(37)


II3 B1(y t)

lowastB

I3(t) (38)

where



lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(39)


tion (29)



α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


(40)


II5 B1(y t)

lowastB

I4(t) (41)

and



lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(42)




1 III2 III

3 III4 and III



ySc

1113968

2t



yPr

1113968

2t


u(y t) III1 +

GT

λI

II2 minus

GT

λI

II3 +

Gc

λI

II4 minus

Gc

λI

II5

(43)

where

III1 B1(y t)

lowastB

II2 (t) (44)


(A2))

III2 (t + λ)

lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


III3 B1(y t)

lowastB

II3 (t)

(45)


(A3))

III4 (t + λ)

lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


III5 B1(y t)

lowastB

II4 (t)

(46)



(A4))



ySc

1113968

2t


(47)



yPr

1113968

2t


(48)


u(y t) IIII1 +

GT

λI

III2 minus

GT

λI

III3 +

Gc

λI

III4 minus

Gc

λI

III5 (49)

where

IIII1 B1(y t)

lowastB

III2 (t) (50)


(A5))


lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


IIII3 B1(y t)

lowastB

III3 (t)

(51)


(A6))


lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


IIII5 B1(y t)

lowastB

III4 (t)

(52)


(A7))



C(y t) et

+ Kcet


ySc

1113968

2t


(53)


T(y t) et

+ Set


yPr

1113968

2t


(54)


u(y t) IIV1 +

GT

λI

IV2 minus

GT

λI

IV3 +

Gc

λI

IV4 minus

Gc

λI

IV5 (55)

where

IIV1 B1(y t)

lowastB

IV2 (t) (56)


(A8))


IIV2 e

t+ λe

t1113872 1113873

lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


IIV3 B1(y t)

lowastB

IV3 (t)

(57)


(A9))

IIV4 e

t+ λe

t1113872 1113873

lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


(58)

Similarly

IIV5 B1(y t)

lowastB

IV4 (t) (59)


(A10))


C(y t) et

+ Kc + 1( 1113857tet


ySc

1113968

2t


(60)


T(y t) et

+(S + 1)tet


yPr

1113968

2t


u(y t) IV1 +

GT

λI

V2 minus

GT

λI

V3 +

Gc

λI

V4 minus

Gc

λI

V5

(61)


(q minus 1)2)

IV1 B1(y t)

lowastB

V2 (t) (62)


(A11))

IV2 te

t+ λe

t(t + 1)1113872 1113873

lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


IV3 B1(y t)

lowastB

V3 (t)

(63)


(A12))

IV4 te

t+ λe

t(t + 1)1113872 1113873

lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


IV5 B1(y t)

lowastB

V4 (t)

(64)


(A13))





lowasterfcy

Sc

1113968

2t


T(y t) et(cos t +(1 + S)sin t)1113872 1113873

lowasterfcy

Pr

1113968

2t


u(y t) IVI1 +

GT

λI

VI2 minus

GT

λI

VI3 +

Gc

λI

VI4 minus

Gc

λI

VI5

IVI1 B1(y t)

lowastB

VI2 (t)

(65)


(A14))

IVI2 e

t sin t(1 + λ) + et cos t1113872 1113873

lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


IVI3 B1(y t)

lowastB

VI3 (t)

(66)


(A15))

IVI4 e

t sin t(1 + λ) + et cos t1113872 1113873

lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


(67)

Similarly

IVI5 B1(y t)

lowastB

VI4 (t) (68)





ySc

1113968

2t


(69)



yPr

1113968

2t


u(y t) IVII1 +

GT

λI

VII2 minus

GT

λI

VII3 +

Gc

λI

VII4 minus

Gc

λI

VII5

(70)

where

IVII1 B1(y t)

lowastB

VII2 (t) (71)


(A17))


lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


IVII3 B1(y t)

lowastB

VII3 (t)

(72)


tion (A18))


lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


IVII5 B1(y t)

lowastB

VII4 (t)

(73)





GC in equation (9))










f (t) = 1 g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

15

1

05

0

(a)

f (t) = t g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

2

15

1

05

0

(b)

f (t) = sint g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

15

1

05

0

(c)

f (t) = et g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

15

1

05

0

(d)

f (t) = tet g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

(e)

f (t) = et sint g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

5

4

3

2

1

0

Velo

city

(f )

Figure 1 Continued


f (t) = t sint g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

15

1

05

0

(g)


Conc

entr

atio

n

15

1

05

0

ndash050 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = H (t)

(a)

Conc

entr

atio

n

15

1

05

00 1 2 3 4 5

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = H (t)

(b)

005

004

003

002

001

0

ndash0010 1 2 3

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = t

(c)

02

015

01

005

00 1 2 3 4

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = t

(d)

Figure 2 Continued


005

004

003

002

001

0

ndash0010 1 2 3

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = sin t

(e)

02

015

01

005

00 1 2 3 4

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = sin t

(f )


04

03

02

01

0

ndash010 1 2 3

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = et

Conc

entr

atio

n

(a)

08

06

04

02

0

ndash020 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = etCo

ncen

trat

ion

(b)

004

0

006

002

ndash0020 1 2 3

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = tet

Conc

entr

atio

n

(c)

03

02

01

0

ndash010 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = tet

Conc

entr

atio

n

(d)

Figure 3 Continued


006

004

002

0

ndash0020 1 2 3

y

Sc = 14Sc = 096Sc = 022

Sc = 060

g (t) = et sintCo

ncen

trat

ion

(e)

03

02

01

00 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = et sint

Conc

entr

atio

n

(f )

001

8 times 10ndash3

6 times 10ndash3

4 times 10ndash3

2 times 10ndash3

0


0 1 2 3y

Sc = 14Sc = 096Sc = 022

Sc = 060

g (t) = t sint

Conc

entr

atio

n

(g)

0

008

006

004

002

ndash0020 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = t sintCo

ncen

trat

ion

(h)


Tem

pera

ture

15

1

05

0

ndash051 2 30

y

Pr = 70Pr = 50Pr = 071

Pr = 15

g (t) = H (t)

(a)

Tem

pera

ture

15

1

05

0

ndash051 2 30

y

Pr = 70Pr = 50Pr = 071

Pr = 15

g (t) = H (t)

(b)

Figure 4 Continued


0

008

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t

(c)

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t

(d)

0

008

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = sint

(e)

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = sint

(f )

05

04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et

(g)

Tem

pera

ture

1

08

06

04

02

00

y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et

(h)



0

008

01

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = tet

(a)

04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = tet

(b)

0

008

01

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et sint

(c)

04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et sint

(d)

Tem

pera

ture

0

002

0015

001

5 times 10ndash3


y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t sint

(e)

0

008

01

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t sint

(f )



Tem

pera

ture

15

1

05

00

y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = H (t)

(a)

Tem

pera

ture

12

1

08

06

04

02

00

y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = H (t)

(b)

0

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t

(c)

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t

(d)

0

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = sin t

(e)

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = sint

(f )

Figure 6 Continued


0y

S = 45S = 30S = 05

S = 15

05 1 15 2

05

04

03

02

01

0

Tem

pera

ture

g (t) = et

(g)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

Tem

pera

ture

1

08

06

04

02

0

g (t) = et

(h)


0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = tet

(a)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

04

03

02

01

0

ndash01

Tem

pera

ture

g (t) = tet

(b)

0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = et sint

(c)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

04

03

02

01

0

Tem

pera

ture

g (t) = et sint

(d)

Figure 7 Continued


Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

015

001

002

5 times 10ndash3

0

g (t) = t sint

(e)

0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t sint

(f )


0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = 1 g(t) = 1 h(t) = 1

(a)

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = t g(t) = 1 h(t) = 1

(b)

0 1 2 3 4 5y

Velo

city

2

15

1

05

0

M = 05M = 10

M = 15M = 25

f (t) = et g(t) = 1 h(t) = 1

(c)

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = sin t g(t) = 1 h(t) = 1

(d)

Figure 8 Continued






0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = tet g(t) = 1 h(t) = 1

(e)

0 1 2 3 4 5y

6

4

2

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = et sin t g(t) = 1 h(t) = 1

(f )

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = t sin t g(t) = 1 h(t) = 1

(g)



3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = sin t g (t) = 1 h (t) = 1

(c)

2

1

0

ndash1

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = et g (t) = 1 h (t) = 1

(d)

20 1 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

f (t) = tet g (t) = 1 h (t) = 1

(e)

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

5

4

3

2

1

0

Velo

city

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 9 Continued


2

1

0

ndash1

Velo

city

20 1 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


f (t) = 1 g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(a)

f (t) = t g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(b)

4

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

f (t) = sin t g (t) = 1 h (t) = 1

(c)

f (t) = et g (t) = 1 h (t) = 1

4

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(d)

Figure 10 Continued


f (t) = tet g (t) = 1 h (t) = 1

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

(e)

f (t) = et sin t g (t) = 1 h (t) = 1

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

5

4

3

2

1

0

Velo

city

(f )

f (t) = t sin t g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(g)


0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

4

3

2

1

0

Velo

city

f (t) = 1 g (t) = 1 h (t) = 1

(a)

GC = 80GC = 60GC = 20

GC = 40

0 1 2 3 54y

4

3

2

1

0

Velo

city

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 11 Continued


0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

4

3

2

1

0

Velo

city

f (t) = sin t g (t) = 1 h (t) = 1

(c)

GC = 80GC = 60GC = 20

GC = 40

0 1 2 3 54y

4

3

2

1

0

Velo

city

f (t) = et g (t) = 1 h (t) = 1

(d)

0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

f (t) = tet g (t) = 1 h (t) = 1

(e)

GC = 80GC = 60GC = 20

GC = 40

5

4

3

2

1

0

Velo


0 1 2 3 54y

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

f (t) = t sint g (t) = 1 h (t) = 1

(g)



3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = t g (t) = 1 h (t) = 1

(b)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = sin t g (t) = 1 h (t) = 1

(c)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = et g (t) = 1 h (t) = 1

(d)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

f (t) = tet g (t) = 1 h (t) = 1

(e)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

5

4

3

2

1

0

Velo

city

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 12 Continued


0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0f (t) = t sin t g (t) = 1 h (t) = 1

(g)


Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(a)

Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(b)

005

004

003

002

001

0

Conc

entr

atio

n

g (t) = t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(c)

Conc

entr

atio

n

02

015

01

005

0

g (t) = t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(d)

Figure 13 Continued


005

004

003

002

001

0

Conc

entr

atio

ng (t) = sin t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(e)

Conc

entr

atio

n

02

015

01

005

0

g (t) = sin t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(f )

03

04

02

01

0

Conc

entr

atio

n

g (t) = et

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(g)

Conc

entr

atio

n

08

06

04

02

0

g (t) = et

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(h)


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = tet

(a)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = tet

(b)

Figure 14 Continued


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = et sint

(c)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = et sint

(d)

001

8 times 10ndash3

6 times 10ndash3

4 times 10ndash3

2 times 10ndash3

0

Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = t sin t

(e)

0

008

006

004

002

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = t sin t

(f )


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = 1 g (t) = 1 h (t) = 1

(a)

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 15 Continued


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = sin t g (t) = 1 h (t) = 1

(c)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 105

0

ndash1 times 105

ndash2 times 105

ndash3 times 105

f (t) = et g (t) = 1 h (t) = 1

(d)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 106

0

ndash1 times 106

ndash2 times 106

ndash3 times 106

f (t) = tet g (t) = 1 h (t) = 1

(e)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

100

0

ndash100

ndash200

ndash300

ndash400

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t sin t g (t) = 1 h (t) = 1

(g)



3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = sin t g (t) = 1 h (t) = 1

(c)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et g (t) = 1 h (t) = 1

(d)

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = tet g (t) = 1 h (t) = 1

(e)

5

4

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 16 Continued


3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


3

2

1

00 1 2 3 54

y

M = 20M = 15M = 05

M = 10


Vel

ocity

(a)

1

0

ndash1

ndash2

ndash30 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

f (t) = 1 g (t) = sin t h (t) = tet

Vel

ocity

(b)

2

0

ndash2

ndash40 1 2 3 54

y

Gc = 80Gc = 60Gc = 20

Gc = 40


Vel

ocity

(c)

6

4

2

00 1 2 3 54

y

GT = 10GT = 60GT = 10

GT = 35

f (t) = 1 g (t) = sin t h (t) = t

Vel

ocity

(d)

Figure 17 Continued


2

1

0

ndash1

ndash2

ndash30 1 2 3 54

y

Sc = 078Sc = 060Sc = 022

Sc = 030


eloc

ity

(e)

3

2

1

0

ndash10 1 2 3 54

y

S = 40S = 30S = 10

S = 20

f (t) = t g (t) = et h (t) = 1

Vel

ocity

(f )


04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(a)

06

04

02

0

ndash02

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(b)



6 Conclusions






04

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(a)

06

04

02

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(b)


08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(a)

1

08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(b)







Appendix

B1(y t)

0 0lt tltyλ

eminus α1t

I0 iα3

t2

minus λy2

1113969

1113874 1113875 tgtyλ

⎧⎪⎪⎨

⎪⎪⎩(A1)

BII2 (t) e



I0 iα3t( 11138571113872 1113873lowastt (A2)

BII3 (t) (1 + λH(t))

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BII4 (t) (1 + λH(t)

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIII2 (t) e






Alowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste




Dlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIV2 (t) e


lowastH(t) + α1( 1113857

lowaste

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t (A8)

BIV3 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BIV4 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste

minus α1tI0 iα3t( 1113857 (A10)

BV2 (t) e


lowaste

t+ 1 + α1( 1113857te

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowastte

t (A11)

BV3 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A12)

BV4 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A13)

BVI2 (t) e


lowaste

t cos t + 1 + α1( 1113857et sin t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t sin t(A14)

BVI3 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A15)


BVI4 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A16)

BVII2 (t) e



minus α1tI0 iα3t( 11138571113872 1113873

lowastt sin t

(A17)



lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A18)



lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A19)

Lminus 1 e

minus bq2minus a2

radic

q2

minus a2

1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

0 0lt tlt b bgt 0

I0 a

t2

minus b2

1113969

1113888 1113889 tgt bRe(q)gtRe(a)

⎧⎪⎪⎨

⎪⎪⎩(A20)

Lminus 1


(A21)


q + a

radic)

q minus b1113890 1113891 e


Φ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A23)


q + a

radic)

q minus b1113890 1113891 e


Ψ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A25)

Data Availability




Acknowledgments


References































t u20]

tprime

y u0

]χprime

Sc ]D

Kc ]u20Kcprime

λ u20]λprime

uprime u0u


Cwprime


φwprime

S ]

ρCpu20Sprime

Pr μCp

K

Kpprime

u20

]2KpGc

g BC]( 1113857

u30

Cwprime

GT gBT]

u30φwprime

M σB

20]

ρu20

f ]u2o

fprime

g ]u2o

gprime

h ]u2o

hprime

(3)


1 + λz

zt1113888 1113889

zu(y t)

zt

z2u(y t)

zy2 + GT 1 + λ

z

zt1113888 1113889T(y t)

+ GC 1 + λz

zt1113888 1113889C(y t)

minus M 1 + λz

zt1113888 1113889u(y t)

(4)

zT(y t)

zt

1Pr

z2T(y t)


zC(y t)

zt

1Sc

z2C(y t)



u(y 0) 0 T(y 0) 0 C(y 0) 0 (7)

u(0 t) f(t) T(0 t) h(t) C(0 t) g(t) (8)


(9)



z2C(y q)



C(y q) C1eminus y

Sc Kc+q( )

1113968

+ C2ey

Sc Kc+q( )

1113968

(11)


C(y q) G(q)eminus y

Sc Kc+q( )

1113968

(12)



C(y t) 1113946t


Sc

1113968 s Kc( 1113857ds (13)



z2T(y q)


e solution is

T(y q) C1eminus y

Pr(S+q)

radic

+ C2ey

Pr(S+q)

radic

(15)


T(y q) H(q)eminus y

Pr(S+q)

radic

(16)


T(y t) 1113946t


Pr

1113968 s S( 1113857ds (17)



z2u(y q)



(18)


u(y q) F(q)eminus y

(q+M)(1+λq)

radic

+GTH(q)(1 + λq) e

minus y(q+M)(1+λq)

radic

minus eminus y

Pr(S+q)

radic

1113874 1113875

Pr(S + q) minus (1 + λq)(q + M)( 1113857

+GcG(q)(1 + λq) e

minus y(q+M)(1+λq)

radic

minus eminus y

Sc Kc+q( )

1113968

1113874 1113875

Sc Kc + q( 1113857 minus (1 + λq)(q + M)( 1113857

(19)


u(y q) F(q)eminus y

λ q+α1( )

2+α23( 1113857

1113969

minus


λ q+α1( )

2+α231113858 1113859

1113969

minus eminus y

Pr(S+q)

radic⎛⎝ ⎞⎠

λ q + α4( 11138572

minus α261113872 1113873

minus


λ q+α1( )

2+α231113858 1113859

1113969

minus eminus y

Sc Kc+q( )

1113968⎛⎝ ⎞⎠

λ q + α7( 11138572

minus α291113872 1113873

(20)





u(y t) I1 +GT

λI2 minus

GT

λI3 +

Gc

λI4 minus

Gc

λI5 (21)

where

I1 Lminus 1

F(q)eminus y

λ q+α1( )

2+α23( 1113857

1113969

⎛⎝ ⎞⎠

Lminus 1


B2(q)( 1113857

B1(y t)lowastB2(t)

(22)


radicy

((q+α1)2+α23)

1113968

((q + α1)2 + α23)

1113969

)

B2(q) F(q)

((q + α1)2 + α23)

1113969



B1(y t)

0 0lt tltyλ

radic

eminus α1t

I0 iα3

t2

minus λy2

1113969

1113874 1113875 tgtyλ

radic

⎧⎪⎪⎨

⎪⎪⎩

(23)

B2(t) eminus α1t

iα3( 1113857I1 iα3t( 1113857 + δ(t)( 11138571113872 1113873lowast

fprime(t) + α1f(t)( 1113857

+ α23eminus α1t

I0 iα3t( 11138571113872 1113873lowastf(t)

(24)


minus yPr(S+q)

radic

q + α4( 11138572

minus α261113872 1113873⎛⎝ ⎞⎠



+S minus α4( 1113857

α6e

minus α4tsinh α6t( 11138571113889

lowasterfcy

Pr

1113968

2t


(25)


minus y

λ q+α1( )

2+α23( 1113857

1113969

q + α4( 11138572

minus α261113872 1113873

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

B1(y t)lowastB3(t)

(26)



lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(27)

where

Alowast

α21 + α231113872 1113873

α24 minus α26

Blowast

2 α4 minus α6( 1113857A


2α6

Clowast

1 minus Alowast

minus Blowast

(28)


minus ySc Kc+q( )

1113968

q + α7( 11138572

minus α291113872 1113873⎛⎝ ⎞⎠



+Kc minus α7( 1113857

α9e

minus α7tsinh α9t( 11138571113889

lowasterfcy

Sc

1113968

2t


(29)

Similarly


minus y

λ q+α1( )

2+α23( 1113857

1113969

q + α7( 11138572

minus α291113872 1113873

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

B1(y t)lowastB4(t)

(30)



lowastClowast

+ Dlowaste


minus α7minus α9( )t1113874 1113875

lowaste


Dlowast

α21 + α231113872 1113873

α27 minus α29

Elowast

2 α7 minus α9( 1113857A


2α9

Flowast

1 minus Dlowast

minus Elowast

(31)


4 Applications




ySc

1113968

2t


(32)




yPr

1113968

2t


(33)



u(y t) II1 +

GT

λI

I2 minus

GT

λI

I3 +

Gc

λI

I4 minus

Gc

λI

I5 (34)

where

II1 B1(y t)

lowastB

I2(t) (35)


BI2(t) e


lowast δ(t) + α1H(t)( 1113857

+ α23eminus α1t


(36)


equation (25)


minus α4tcosh α6t( 11138571113872

+S minus α4( 1113857

α6e

minus α4tsinh α6t( 11138571113889

lowasterfcy

Pr

1113968

2t


(37)


II3 B1(y t)

lowastB

I3(t) (38)

where



lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(39)


tion (29)



α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


(40)


II5 B1(y t)

lowastB

I4(t) (41)

and



lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(42)




1 III2 III

3 III4 and III



ySc

1113968

2t



yPr

1113968

2t


u(y t) III1 +

GT

λI

II2 minus

GT

λI

II3 +

Gc

λI

II4 minus

Gc

λI

II5

(43)

where

III1 B1(y t)

lowastB

II2 (t) (44)


(A2))

III2 (t + λ)

lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


III3 B1(y t)

lowastB

II3 (t)

(45)


(A3))

III4 (t + λ)

lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


III5 B1(y t)

lowastB

II4 (t)

(46)



(A4))



ySc

1113968

2t


(47)



yPr

1113968

2t


(48)


u(y t) IIII1 +

GT

λI

III2 minus

GT

λI

III3 +

Gc

λI

III4 minus

Gc

λI

III5 (49)

where

IIII1 B1(y t)

lowastB

III2 (t) (50)


(A5))


lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


IIII3 B1(y t)

lowastB

III3 (t)

(51)


(A6))


lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


IIII5 B1(y t)

lowastB

III4 (t)

(52)


(A7))



C(y t) et

+ Kcet


ySc

1113968

2t


(53)


T(y t) et

+ Set


yPr

1113968

2t


(54)


u(y t) IIV1 +

GT

λI

IV2 minus

GT

λI

IV3 +

Gc

λI

IV4 minus

Gc

λI

IV5 (55)

where

IIV1 B1(y t)

lowastB

IV2 (t) (56)


(A8))


IIV2 e

t+ λe

t1113872 1113873

lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


IIV3 B1(y t)

lowastB

IV3 (t)

(57)


(A9))

IIV4 e

t+ λe

t1113872 1113873

lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


(58)

Similarly

IIV5 B1(y t)

lowastB

IV4 (t) (59)


(A10))


C(y t) et

+ Kc + 1( 1113857tet


ySc

1113968

2t


(60)


T(y t) et

+(S + 1)tet


yPr

1113968

2t


u(y t) IV1 +

GT

λI

V2 minus

GT

λI

V3 +

Gc

λI

V4 minus

Gc

λI

V5

(61)


(q minus 1)2)

IV1 B1(y t)

lowastB

V2 (t) (62)


(A11))

IV2 te

t+ λe

t(t + 1)1113872 1113873

lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


IV3 B1(y t)

lowastB

V3 (t)

(63)


(A12))

IV4 te

t+ λe

t(t + 1)1113872 1113873

lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


IV5 B1(y t)

lowastB

V4 (t)

(64)


(A13))





lowasterfcy

Sc

1113968

2t


T(y t) et(cos t +(1 + S)sin t)1113872 1113873

lowasterfcy

Pr

1113968

2t


u(y t) IVI1 +

GT

λI

VI2 minus

GT

λI

VI3 +

Gc

λI

VI4 minus

Gc

λI

VI5

IVI1 B1(y t)

lowastB

VI2 (t)

(65)


(A14))

IVI2 e

t sin t(1 + λ) + et cos t1113872 1113873

lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


IVI3 B1(y t)

lowastB

VI3 (t)

(66)


(A15))

IVI4 e

t sin t(1 + λ) + et cos t1113872 1113873

lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


(67)

Similarly

IVI5 B1(y t)

lowastB

VI4 (t) (68)





ySc

1113968

2t


(69)



yPr

1113968

2t


u(y t) IVII1 +

GT

λI

VII2 minus

GT

λI

VII3 +

Gc

λI

VII4 minus

Gc

λI

VII5

(70)

where

IVII1 B1(y t)

lowastB

VII2 (t) (71)


(A17))


lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


IVII3 B1(y t)

lowastB

VII3 (t)

(72)


tion (A18))


lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


IVII5 B1(y t)

lowastB

VII4 (t)

(73)





GC in equation (9))










f (t) = 1 g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

15

1

05

0

(a)

f (t) = t g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

2

15

1

05

0

(b)

f (t) = sint g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

15

1

05

0

(c)

f (t) = et g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

15

1

05

0

(d)

f (t) = tet g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

(e)

f (t) = et sint g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

5

4

3

2

1

0

Velo

city

(f )

Figure 1 Continued


f (t) = t sint g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

15

1

05

0

(g)


Conc

entr

atio

n

15

1

05

0

ndash050 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = H (t)

(a)

Conc

entr

atio

n

15

1

05

00 1 2 3 4 5

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = H (t)

(b)

005

004

003

002

001

0

ndash0010 1 2 3

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = t

(c)

02

015

01

005

00 1 2 3 4

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = t

(d)

Figure 2 Continued


005

004

003

002

001

0

ndash0010 1 2 3

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = sin t

(e)

02

015

01

005

00 1 2 3 4

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = sin t

(f )


04

03

02

01

0

ndash010 1 2 3

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = et

Conc

entr

atio

n

(a)

08

06

04

02

0

ndash020 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = etCo

ncen

trat

ion

(b)

004

0

006

002

ndash0020 1 2 3

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = tet

Conc

entr

atio

n

(c)

03

02

01

0

ndash010 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = tet

Conc

entr

atio

n

(d)

Figure 3 Continued


006

004

002

0

ndash0020 1 2 3

y

Sc = 14Sc = 096Sc = 022

Sc = 060

g (t) = et sintCo

ncen

trat

ion

(e)

03

02

01

00 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = et sint

Conc

entr

atio

n

(f )

001

8 times 10ndash3

6 times 10ndash3

4 times 10ndash3

2 times 10ndash3

0


0 1 2 3y

Sc = 14Sc = 096Sc = 022

Sc = 060

g (t) = t sint

Conc

entr

atio

n

(g)

0

008

006

004

002

ndash0020 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = t sintCo

ncen

trat

ion

(h)


Tem

pera

ture

15

1

05

0

ndash051 2 30

y

Pr = 70Pr = 50Pr = 071

Pr = 15

g (t) = H (t)

(a)

Tem

pera

ture

15

1

05

0

ndash051 2 30

y

Pr = 70Pr = 50Pr = 071

Pr = 15

g (t) = H (t)

(b)

Figure 4 Continued


0

008

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t

(c)

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t

(d)

0

008

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = sint

(e)

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = sint

(f )

05

04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et

(g)

Tem

pera

ture

1

08

06

04

02

00

y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et

(h)



0

008

01

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = tet

(a)

04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = tet

(b)

0

008

01

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et sint

(c)

04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et sint

(d)

Tem

pera

ture

0

002

0015

001

5 times 10ndash3


y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t sint

(e)

0

008

01

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t sint

(f )



Tem

pera

ture

15

1

05

00

y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = H (t)

(a)

Tem

pera

ture

12

1

08

06

04

02

00

y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = H (t)

(b)

0

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t

(c)

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t

(d)

0

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = sin t

(e)

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = sint

(f )

Figure 6 Continued


0y

S = 45S = 30S = 05

S = 15

05 1 15 2

05

04

03

02

01

0

Tem

pera

ture

g (t) = et

(g)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

Tem

pera

ture

1

08

06

04

02

0

g (t) = et

(h)


0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = tet

(a)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

04

03

02

01

0

ndash01

Tem

pera

ture

g (t) = tet

(b)

0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = et sint

(c)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

04

03

02

01

0

Tem

pera

ture

g (t) = et sint

(d)

Figure 7 Continued


Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

015

001

002

5 times 10ndash3

0

g (t) = t sint

(e)

0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t sint

(f )


0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = 1 g(t) = 1 h(t) = 1

(a)

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = t g(t) = 1 h(t) = 1

(b)

0 1 2 3 4 5y

Velo

city

2

15

1

05

0

M = 05M = 10

M = 15M = 25

f (t) = et g(t) = 1 h(t) = 1

(c)

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = sin t g(t) = 1 h(t) = 1

(d)

Figure 8 Continued






0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = tet g(t) = 1 h(t) = 1

(e)

0 1 2 3 4 5y

6

4

2

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = et sin t g(t) = 1 h(t) = 1

(f )

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = t sin t g(t) = 1 h(t) = 1

(g)



3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = sin t g (t) = 1 h (t) = 1

(c)

2

1

0

ndash1

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = et g (t) = 1 h (t) = 1

(d)

20 1 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

f (t) = tet g (t) = 1 h (t) = 1

(e)

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

5

4

3

2

1

0

Velo

city

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 9 Continued


2

1

0

ndash1

Velo

city

20 1 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


f (t) = 1 g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(a)

f (t) = t g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(b)

4

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

f (t) = sin t g (t) = 1 h (t) = 1

(c)

f (t) = et g (t) = 1 h (t) = 1

4

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(d)

Figure 10 Continued


f (t) = tet g (t) = 1 h (t) = 1

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

(e)

f (t) = et sin t g (t) = 1 h (t) = 1

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

5

4

3

2

1

0

Velo

city

(f )

f (t) = t sin t g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(g)


0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

4

3

2

1

0

Velo

city

f (t) = 1 g (t) = 1 h (t) = 1

(a)

GC = 80GC = 60GC = 20

GC = 40

0 1 2 3 54y

4

3

2

1

0

Velo

city

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 11 Continued


0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

4

3

2

1

0

Velo

city

f (t) = sin t g (t) = 1 h (t) = 1

(c)

GC = 80GC = 60GC = 20

GC = 40

0 1 2 3 54y

4

3

2

1

0

Velo

city

f (t) = et g (t) = 1 h (t) = 1

(d)

0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

f (t) = tet g (t) = 1 h (t) = 1

(e)

GC = 80GC = 60GC = 20

GC = 40

5

4

3

2

1

0

Velo


0 1 2 3 54y

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

f (t) = t sint g (t) = 1 h (t) = 1

(g)



3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = t g (t) = 1 h (t) = 1

(b)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = sin t g (t) = 1 h (t) = 1

(c)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = et g (t) = 1 h (t) = 1

(d)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

f (t) = tet g (t) = 1 h (t) = 1

(e)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

5

4

3

2

1

0

Velo

city

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 12 Continued


0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0f (t) = t sin t g (t) = 1 h (t) = 1

(g)


Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(a)

Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(b)

005

004

003

002

001

0

Conc

entr

atio

n

g (t) = t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(c)

Conc

entr

atio

n

02

015

01

005

0

g (t) = t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(d)

Figure 13 Continued


005

004

003

002

001

0

Conc

entr

atio

ng (t) = sin t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(e)

Conc

entr

atio

n

02

015

01

005

0

g (t) = sin t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(f )

03

04

02

01

0

Conc

entr

atio

n

g (t) = et

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(g)

Conc

entr

atio

n

08

06

04

02

0

g (t) = et

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(h)


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = tet

(a)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = tet

(b)

Figure 14 Continued


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = et sint

(c)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = et sint

(d)

001

8 times 10ndash3

6 times 10ndash3

4 times 10ndash3

2 times 10ndash3

0

Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = t sin t

(e)

0

008

006

004

002

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = t sin t

(f )


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = 1 g (t) = 1 h (t) = 1

(a)

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 15 Continued


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = sin t g (t) = 1 h (t) = 1

(c)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 105

0

ndash1 times 105

ndash2 times 105

ndash3 times 105

f (t) = et g (t) = 1 h (t) = 1

(d)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 106

0

ndash1 times 106

ndash2 times 106

ndash3 times 106

f (t) = tet g (t) = 1 h (t) = 1

(e)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

100

0

ndash100

ndash200

ndash300

ndash400

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t sin t g (t) = 1 h (t) = 1

(g)



3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = sin t g (t) = 1 h (t) = 1

(c)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et g (t) = 1 h (t) = 1

(d)

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = tet g (t) = 1 h (t) = 1

(e)

5

4

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 16 Continued


3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


3

2

1

00 1 2 3 54

y

M = 20M = 15M = 05

M = 10


Vel

ocity

(a)

1

0

ndash1

ndash2

ndash30 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

f (t) = 1 g (t) = sin t h (t) = tet

Vel

ocity

(b)

2

0

ndash2

ndash40 1 2 3 54

y

Gc = 80Gc = 60Gc = 20

Gc = 40


Vel

ocity

(c)

6

4

2

00 1 2 3 54

y

GT = 10GT = 60GT = 10

GT = 35

f (t) = 1 g (t) = sin t h (t) = t

Vel

ocity

(d)

Figure 17 Continued


2

1

0

ndash1

ndash2

ndash30 1 2 3 54

y

Sc = 078Sc = 060Sc = 022

Sc = 030


eloc

ity

(e)

3

2

1

0

ndash10 1 2 3 54

y

S = 40S = 30S = 10

S = 20

f (t) = t g (t) = et h (t) = 1

Vel

ocity

(f )


04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(a)

06

04

02

0

ndash02

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(b)



6 Conclusions






04

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(a)

06

04

02

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(b)


08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(a)

1

08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(b)







Appendix

B1(y t)

0 0lt tltyλ

eminus α1t

I0 iα3

t2

minus λy2

1113969

1113874 1113875 tgtyλ

⎧⎪⎪⎨

⎪⎪⎩(A1)

BII2 (t) e



I0 iα3t( 11138571113872 1113873lowastt (A2)

BII3 (t) (1 + λH(t))

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BII4 (t) (1 + λH(t)

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIII2 (t) e






Alowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste




Dlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIV2 (t) e


lowastH(t) + α1( 1113857

lowaste

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t (A8)

BIV3 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BIV4 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste

minus α1tI0 iα3t( 1113857 (A10)

BV2 (t) e


lowaste

t+ 1 + α1( 1113857te

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowastte

t (A11)

BV3 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A12)

BV4 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A13)

BVI2 (t) e


lowaste

t cos t + 1 + α1( 1113857et sin t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t sin t(A14)

BVI3 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A15)


BVI4 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A16)

BVII2 (t) e



minus α1tI0 iα3t( 11138571113872 1113873

lowastt sin t

(A17)



lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A18)



lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A19)

Lminus 1 e

minus bq2minus a2

radic

q2

minus a2

1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

0 0lt tlt b bgt 0

I0 a

t2

minus b2

1113969

1113888 1113889 tgt bRe(q)gtRe(a)

⎧⎪⎪⎨

⎪⎪⎩(A20)

Lminus 1


(A21)


q + a

radic)

q minus b1113890 1113891 e


Φ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A23)


q + a

radic)

q minus b1113890 1113891 e


Ψ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A25)

Data Availability




Acknowledgments


References
































C(y t) 1113946t


Sc

1113968 s Kc( 1113857ds (13)



z2T(y q)


e solution is

T(y q) C1eminus y

Pr(S+q)

radic

+ C2ey

Pr(S+q)

radic

(15)


T(y q) H(q)eminus y

Pr(S+q)

radic

(16)


T(y t) 1113946t


Pr

1113968 s S( 1113857ds (17)



z2u(y q)



(18)


u(y q) F(q)eminus y

(q+M)(1+λq)

radic

+GTH(q)(1 + λq) e

minus y(q+M)(1+λq)

radic

minus eminus y

Pr(S+q)

radic

1113874 1113875

Pr(S + q) minus (1 + λq)(q + M)( 1113857

+GcG(q)(1 + λq) e

minus y(q+M)(1+λq)

radic

minus eminus y

Sc Kc+q( )

1113968

1113874 1113875

Sc Kc + q( 1113857 minus (1 + λq)(q + M)( 1113857

(19)


u(y q) F(q)eminus y

λ q+α1( )

2+α23( 1113857

1113969

minus


λ q+α1( )

2+α231113858 1113859

1113969

minus eminus y

Pr(S+q)

radic⎛⎝ ⎞⎠

λ q + α4( 11138572

minus α261113872 1113873

minus


λ q+α1( )

2+α231113858 1113859

1113969

minus eminus y

Sc Kc+q( )

1113968⎛⎝ ⎞⎠

λ q + α7( 11138572

minus α291113872 1113873

(20)





u(y t) I1 +GT

λI2 minus

GT

λI3 +

Gc

λI4 minus

Gc

λI5 (21)

where

I1 Lminus 1

F(q)eminus y

λ q+α1( )

2+α23( 1113857

1113969

⎛⎝ ⎞⎠

Lminus 1


B2(q)( 1113857

B1(y t)lowastB2(t)

(22)


radicy

((q+α1)2+α23)

1113968

((q + α1)2 + α23)

1113969

)

B2(q) F(q)

((q + α1)2 + α23)

1113969



B1(y t)

0 0lt tltyλ

radic

eminus α1t

I0 iα3

t2

minus λy2

1113969

1113874 1113875 tgtyλ

radic

⎧⎪⎪⎨

⎪⎪⎩

(23)

B2(t) eminus α1t

iα3( 1113857I1 iα3t( 1113857 + δ(t)( 11138571113872 1113873lowast

fprime(t) + α1f(t)( 1113857

+ α23eminus α1t

I0 iα3t( 11138571113872 1113873lowastf(t)

(24)


minus yPr(S+q)

radic

q + α4( 11138572

minus α261113872 1113873⎛⎝ ⎞⎠



+S minus α4( 1113857

α6e

minus α4tsinh α6t( 11138571113889

lowasterfcy

Pr

1113968

2t


(25)


minus y

λ q+α1( )

2+α23( 1113857

1113969

q + α4( 11138572

minus α261113872 1113873

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

B1(y t)lowastB3(t)

(26)



lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(27)

where

Alowast

α21 + α231113872 1113873

α24 minus α26

Blowast

2 α4 minus α6( 1113857A


2α6

Clowast

1 minus Alowast

minus Blowast

(28)


minus ySc Kc+q( )

1113968

q + α7( 11138572

minus α291113872 1113873⎛⎝ ⎞⎠



+Kc minus α7( 1113857

α9e

minus α7tsinh α9t( 11138571113889

lowasterfcy

Sc

1113968

2t


(29)

Similarly


minus y

λ q+α1( )

2+α23( 1113857

1113969

q + α7( 11138572

minus α291113872 1113873

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

B1(y t)lowastB4(t)

(30)



lowastClowast

+ Dlowaste


minus α7minus α9( )t1113874 1113875

lowaste


Dlowast

α21 + α231113872 1113873

α27 minus α29

Elowast

2 α7 minus α9( 1113857A


2α9

Flowast

1 minus Dlowast

minus Elowast

(31)


4 Applications




ySc

1113968

2t


(32)




yPr

1113968

2t


(33)



u(y t) II1 +

GT

λI

I2 minus

GT

λI

I3 +

Gc

λI

I4 minus

Gc

λI

I5 (34)

where

II1 B1(y t)

lowastB

I2(t) (35)


BI2(t) e


lowast δ(t) + α1H(t)( 1113857

+ α23eminus α1t


(36)


equation (25)


minus α4tcosh α6t( 11138571113872

+S minus α4( 1113857

α6e

minus α4tsinh α6t( 11138571113889

lowasterfcy

Pr

1113968

2t


(37)


II3 B1(y t)

lowastB

I3(t) (38)

where



lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(39)


tion (29)



α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


(40)


II5 B1(y t)

lowastB

I4(t) (41)

and



lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(42)




1 III2 III

3 III4 and III



ySc

1113968

2t



yPr

1113968

2t


u(y t) III1 +

GT

λI

II2 minus

GT

λI

II3 +

Gc

λI

II4 minus

Gc

λI

II5

(43)

where

III1 B1(y t)

lowastB

II2 (t) (44)


(A2))

III2 (t + λ)

lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


III3 B1(y t)

lowastB

II3 (t)

(45)


(A3))

III4 (t + λ)

lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


III5 B1(y t)

lowastB

II4 (t)

(46)



(A4))



ySc

1113968

2t


(47)



yPr

1113968

2t


(48)


u(y t) IIII1 +

GT

λI

III2 minus

GT

λI

III3 +

Gc

λI

III4 minus

Gc

λI

III5 (49)

where

IIII1 B1(y t)

lowastB

III2 (t) (50)


(A5))


lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


IIII3 B1(y t)

lowastB

III3 (t)

(51)


(A6))


lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


IIII5 B1(y t)

lowastB

III4 (t)

(52)


(A7))



C(y t) et

+ Kcet


ySc

1113968

2t


(53)


T(y t) et

+ Set


yPr

1113968

2t


(54)


u(y t) IIV1 +

GT

λI

IV2 minus

GT

λI

IV3 +

Gc

λI

IV4 minus

Gc

λI

IV5 (55)

where

IIV1 B1(y t)

lowastB

IV2 (t) (56)


(A8))


IIV2 e

t+ λe

t1113872 1113873

lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


IIV3 B1(y t)

lowastB

IV3 (t)

(57)


(A9))

IIV4 e

t+ λe

t1113872 1113873

lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


(58)

Similarly

IIV5 B1(y t)

lowastB

IV4 (t) (59)


(A10))


C(y t) et

+ Kc + 1( 1113857tet


ySc

1113968

2t


(60)


T(y t) et

+(S + 1)tet


yPr

1113968

2t


u(y t) IV1 +

GT

λI

V2 minus

GT

λI

V3 +

Gc

λI

V4 minus

Gc

λI

V5

(61)


(q minus 1)2)

IV1 B1(y t)

lowastB

V2 (t) (62)


(A11))

IV2 te

t+ λe

t(t + 1)1113872 1113873

lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


IV3 B1(y t)

lowastB

V3 (t)

(63)


(A12))

IV4 te

t+ λe

t(t + 1)1113872 1113873

lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


IV5 B1(y t)

lowastB

V4 (t)

(64)


(A13))





lowasterfcy

Sc

1113968

2t


T(y t) et(cos t +(1 + S)sin t)1113872 1113873

lowasterfcy

Pr

1113968

2t


u(y t) IVI1 +

GT

λI

VI2 minus

GT

λI

VI3 +

Gc

λI

VI4 minus

Gc

λI

VI5

IVI1 B1(y t)

lowastB

VI2 (t)

(65)


(A14))

IVI2 e

t sin t(1 + λ) + et cos t1113872 1113873

lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


IVI3 B1(y t)

lowastB

VI3 (t)

(66)


(A15))

IVI4 e

t sin t(1 + λ) + et cos t1113872 1113873

lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


(67)

Similarly

IVI5 B1(y t)

lowastB

VI4 (t) (68)





ySc

1113968

2t


(69)



yPr

1113968

2t


u(y t) IVII1 +

GT

λI

VII2 minus

GT

λI

VII3 +

Gc

λI

VII4 minus

Gc

λI

VII5

(70)

where

IVII1 B1(y t)

lowastB

VII2 (t) (71)


(A17))


lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


IVII3 B1(y t)

lowastB

VII3 (t)

(72)


tion (A18))


lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


IVII5 B1(y t)

lowastB

VII4 (t)

(73)





GC in equation (9))










f (t) = 1 g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

15

1

05

0

(a)

f (t) = t g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

2

15

1

05

0

(b)

f (t) = sint g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

15

1

05

0

(c)

f (t) = et g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

15

1

05

0

(d)

f (t) = tet g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

(e)

f (t) = et sint g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

5

4

3

2

1

0

Velo

city

(f )

Figure 1 Continued


f (t) = t sint g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

15

1

05

0

(g)


Conc

entr

atio

n

15

1

05

0

ndash050 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = H (t)

(a)

Conc

entr

atio

n

15

1

05

00 1 2 3 4 5

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = H (t)

(b)

005

004

003

002

001

0

ndash0010 1 2 3

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = t

(c)

02

015

01

005

00 1 2 3 4

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = t

(d)

Figure 2 Continued


005

004

003

002

001

0

ndash0010 1 2 3

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = sin t

(e)

02

015

01

005

00 1 2 3 4

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = sin t

(f )


04

03

02

01

0

ndash010 1 2 3

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = et

Conc

entr

atio

n

(a)

08

06

04

02

0

ndash020 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = etCo

ncen

trat

ion

(b)

004

0

006

002

ndash0020 1 2 3

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = tet

Conc

entr

atio

n

(c)

03

02

01

0

ndash010 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = tet

Conc

entr

atio

n

(d)

Figure 3 Continued


006

004

002

0

ndash0020 1 2 3

y

Sc = 14Sc = 096Sc = 022

Sc = 060

g (t) = et sintCo

ncen

trat

ion

(e)

03

02

01

00 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = et sint

Conc

entr

atio

n

(f )

001

8 times 10ndash3

6 times 10ndash3

4 times 10ndash3

2 times 10ndash3

0


0 1 2 3y

Sc = 14Sc = 096Sc = 022

Sc = 060

g (t) = t sint

Conc

entr

atio

n

(g)

0

008

006

004

002

ndash0020 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = t sintCo

ncen

trat

ion

(h)


Tem

pera

ture

15

1

05

0

ndash051 2 30

y

Pr = 70Pr = 50Pr = 071

Pr = 15

g (t) = H (t)

(a)

Tem

pera

ture

15

1

05

0

ndash051 2 30

y

Pr = 70Pr = 50Pr = 071

Pr = 15

g (t) = H (t)

(b)

Figure 4 Continued


0

008

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t

(c)

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t

(d)

0

008

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = sint

(e)

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = sint

(f )

05

04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et

(g)

Tem

pera

ture

1

08

06

04

02

00

y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et

(h)



0

008

01

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = tet

(a)

04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = tet

(b)

0

008

01

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et sint

(c)

04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et sint

(d)

Tem

pera

ture

0

002

0015

001

5 times 10ndash3


y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t sint

(e)

0

008

01

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t sint

(f )



Tem

pera

ture

15

1

05

00

y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = H (t)

(a)

Tem

pera

ture

12

1

08

06

04

02

00

y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = H (t)

(b)

0

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t

(c)

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t

(d)

0

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = sin t

(e)

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = sint

(f )

Figure 6 Continued


0y

S = 45S = 30S = 05

S = 15

05 1 15 2

05

04

03

02

01

0

Tem

pera

ture

g (t) = et

(g)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

Tem

pera

ture

1

08

06

04

02

0

g (t) = et

(h)


0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = tet

(a)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

04

03

02

01

0

ndash01

Tem

pera

ture

g (t) = tet

(b)

0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = et sint

(c)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

04

03

02

01

0

Tem

pera

ture

g (t) = et sint

(d)

Figure 7 Continued


Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

015

001

002

5 times 10ndash3

0

g (t) = t sint

(e)

0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t sint

(f )


0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = 1 g(t) = 1 h(t) = 1

(a)

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = t g(t) = 1 h(t) = 1

(b)

0 1 2 3 4 5y

Velo

city

2

15

1

05

0

M = 05M = 10

M = 15M = 25

f (t) = et g(t) = 1 h(t) = 1

(c)

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = sin t g(t) = 1 h(t) = 1

(d)

Figure 8 Continued






0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = tet g(t) = 1 h(t) = 1

(e)

0 1 2 3 4 5y

6

4

2

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = et sin t g(t) = 1 h(t) = 1

(f )

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = t sin t g(t) = 1 h(t) = 1

(g)



3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = sin t g (t) = 1 h (t) = 1

(c)

2

1

0

ndash1

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = et g (t) = 1 h (t) = 1

(d)

20 1 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

f (t) = tet g (t) = 1 h (t) = 1

(e)

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

5

4

3

2

1

0

Velo

city

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 9 Continued


2

1

0

ndash1

Velo

city

20 1 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


f (t) = 1 g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(a)

f (t) = t g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(b)

4

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

f (t) = sin t g (t) = 1 h (t) = 1

(c)

f (t) = et g (t) = 1 h (t) = 1

4

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(d)

Figure 10 Continued


f (t) = tet g (t) = 1 h (t) = 1

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

(e)

f (t) = et sin t g (t) = 1 h (t) = 1

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

5

4

3

2

1

0

Velo

city

(f )

f (t) = t sin t g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(g)


0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

4

3

2

1

0

Velo

city

f (t) = 1 g (t) = 1 h (t) = 1

(a)

GC = 80GC = 60GC = 20

GC = 40

0 1 2 3 54y

4

3

2

1

0

Velo

city

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 11 Continued


0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

4

3

2

1

0

Velo

city

f (t) = sin t g (t) = 1 h (t) = 1

(c)

GC = 80GC = 60GC = 20

GC = 40

0 1 2 3 54y

4

3

2

1

0

Velo

city

f (t) = et g (t) = 1 h (t) = 1

(d)

0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

f (t) = tet g (t) = 1 h (t) = 1

(e)

GC = 80GC = 60GC = 20

GC = 40

5

4

3

2

1

0

Velo


0 1 2 3 54y

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

f (t) = t sint g (t) = 1 h (t) = 1

(g)



3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = t g (t) = 1 h (t) = 1

(b)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = sin t g (t) = 1 h (t) = 1

(c)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = et g (t) = 1 h (t) = 1

(d)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

f (t) = tet g (t) = 1 h (t) = 1

(e)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

5

4

3

2

1

0

Velo

city

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 12 Continued


0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0f (t) = t sin t g (t) = 1 h (t) = 1

(g)


Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(a)

Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(b)

005

004

003

002

001

0

Conc

entr

atio

n

g (t) = t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(c)

Conc

entr

atio

n

02

015

01

005

0

g (t) = t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(d)

Figure 13 Continued


005

004

003

002

001

0

Conc

entr

atio

ng (t) = sin t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(e)

Conc

entr

atio

n

02

015

01

005

0

g (t) = sin t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(f )

03

04

02

01

0

Conc

entr

atio

n

g (t) = et

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(g)

Conc

entr

atio

n

08

06

04

02

0

g (t) = et

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(h)


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = tet

(a)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = tet

(b)

Figure 14 Continued


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = et sint

(c)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = et sint

(d)

001

8 times 10ndash3

6 times 10ndash3

4 times 10ndash3

2 times 10ndash3

0

Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = t sin t

(e)

0

008

006

004

002

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = t sin t

(f )


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = 1 g (t) = 1 h (t) = 1

(a)

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 15 Continued


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = sin t g (t) = 1 h (t) = 1

(c)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 105

0

ndash1 times 105

ndash2 times 105

ndash3 times 105

f (t) = et g (t) = 1 h (t) = 1

(d)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 106

0

ndash1 times 106

ndash2 times 106

ndash3 times 106

f (t) = tet g (t) = 1 h (t) = 1

(e)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

100

0

ndash100

ndash200

ndash300

ndash400

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t sin t g (t) = 1 h (t) = 1

(g)



3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = sin t g (t) = 1 h (t) = 1

(c)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et g (t) = 1 h (t) = 1

(d)

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = tet g (t) = 1 h (t) = 1

(e)

5

4

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 16 Continued


3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


3

2

1

00 1 2 3 54

y

M = 20M = 15M = 05

M = 10


Vel

ocity

(a)

1

0

ndash1

ndash2

ndash30 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

f (t) = 1 g (t) = sin t h (t) = tet

Vel

ocity

(b)

2

0

ndash2

ndash40 1 2 3 54

y

Gc = 80Gc = 60Gc = 20

Gc = 40


Vel

ocity

(c)

6

4

2

00 1 2 3 54

y

GT = 10GT = 60GT = 10

GT = 35

f (t) = 1 g (t) = sin t h (t) = t

Vel

ocity

(d)

Figure 17 Continued


2

1

0

ndash1

ndash2

ndash30 1 2 3 54

y

Sc = 078Sc = 060Sc = 022

Sc = 030


eloc

ity

(e)

3

2

1

0

ndash10 1 2 3 54

y

S = 40S = 30S = 10

S = 20

f (t) = t g (t) = et h (t) = 1

Vel

ocity

(f )


04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(a)

06

04

02

0

ndash02

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(b)



6 Conclusions






04

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(a)

06

04

02

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(b)


08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(a)

1

08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(b)







Appendix

B1(y t)

0 0lt tltyλ

eminus α1t

I0 iα3

t2

minus λy2

1113969

1113874 1113875 tgtyλ

⎧⎪⎪⎨

⎪⎪⎩(A1)

BII2 (t) e



I0 iα3t( 11138571113872 1113873lowastt (A2)

BII3 (t) (1 + λH(t))

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BII4 (t) (1 + λH(t)

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIII2 (t) e






Alowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste




Dlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIV2 (t) e


lowastH(t) + α1( 1113857

lowaste

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t (A8)

BIV3 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BIV4 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste

minus α1tI0 iα3t( 1113857 (A10)

BV2 (t) e


lowaste

t+ 1 + α1( 1113857te

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowastte

t (A11)

BV3 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A12)

BV4 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A13)

BVI2 (t) e


lowaste

t cos t + 1 + α1( 1113857et sin t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t sin t(A14)

BVI3 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A15)


BVI4 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A16)

BVII2 (t) e



minus α1tI0 iα3t( 11138571113872 1113873

lowastt sin t

(A17)



lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A18)



lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A19)

Lminus 1 e

minus bq2minus a2

radic

q2

minus a2

1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

0 0lt tlt b bgt 0

I0 a

t2

minus b2

1113969

1113888 1113889 tgt bRe(q)gtRe(a)

⎧⎪⎪⎨

⎪⎪⎩(A20)

Lminus 1


(A21)


q + a

radic)

q minus b1113890 1113891 e


Φ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A23)


q + a

radic)

q minus b1113890 1113891 e


Ψ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A25)

Data Availability




Acknowledgments


References































B1(y t)

0 0lt tltyλ

radic

eminus α1t

I0 iα3

t2

minus λy2

1113969

1113874 1113875 tgtyλ

radic

⎧⎪⎪⎨

⎪⎪⎩

(23)

B2(t) eminus α1t

iα3( 1113857I1 iα3t( 1113857 + δ(t)( 11138571113872 1113873lowast

fprime(t) + α1f(t)( 1113857

+ α23eminus α1t

I0 iα3t( 11138571113872 1113873lowastf(t)

(24)


minus yPr(S+q)

radic

q + α4( 11138572

minus α261113872 1113873⎛⎝ ⎞⎠



+S minus α4( 1113857

α6e

minus α4tsinh α6t( 11138571113889

lowasterfcy

Pr

1113968

2t


(25)


minus y

λ q+α1( )

2+α23( 1113857

1113969

q + α4( 11138572

minus α261113872 1113873

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

B1(y t)lowastB3(t)

(26)



lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(27)

where

Alowast

α21 + α231113872 1113873

α24 minus α26

Blowast

2 α4 minus α6( 1113857A


2α6

Clowast

1 minus Alowast

minus Blowast

(28)


minus ySc Kc+q( )

1113968

q + α7( 11138572

minus α291113872 1113873⎛⎝ ⎞⎠



+Kc minus α7( 1113857

α9e

minus α7tsinh α9t( 11138571113889

lowasterfcy

Sc

1113968

2t


(29)

Similarly


minus y

λ q+α1( )

2+α23( 1113857

1113969

q + α7( 11138572

minus α291113872 1113873

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

B1(y t)lowastB4(t)

(30)



lowastClowast

+ Dlowaste


minus α7minus α9( )t1113874 1113875

lowaste


Dlowast

α21 + α231113872 1113873

α27 minus α29

Elowast

2 α7 minus α9( 1113857A


2α9

Flowast

1 minus Dlowast

minus Elowast

(31)


4 Applications




ySc

1113968

2t


(32)




yPr

1113968

2t


(33)



u(y t) II1 +

GT

λI

I2 minus

GT

λI

I3 +

Gc

λI

I4 minus

Gc

λI

I5 (34)

where

II1 B1(y t)

lowastB

I2(t) (35)


BI2(t) e


lowast δ(t) + α1H(t)( 1113857

+ α23eminus α1t


(36)


equation (25)


minus α4tcosh α6t( 11138571113872

+S minus α4( 1113857

α6e

minus α4tsinh α6t( 11138571113889

lowasterfcy

Pr

1113968

2t


(37)


II3 B1(y t)

lowastB

I3(t) (38)

where



lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(39)


tion (29)



α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


(40)


II5 B1(y t)

lowastB

I4(t) (41)

and



lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(42)




1 III2 III

3 III4 and III



ySc

1113968

2t



yPr

1113968

2t


u(y t) III1 +

GT

λI

II2 minus

GT

λI

II3 +

Gc

λI

II4 minus

Gc

λI

II5

(43)

where

III1 B1(y t)

lowastB

II2 (t) (44)


(A2))

III2 (t + λ)

lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


III3 B1(y t)

lowastB

II3 (t)

(45)


(A3))

III4 (t + λ)

lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


III5 B1(y t)

lowastB

II4 (t)

(46)



(A4))



ySc

1113968

2t


(47)



yPr

1113968

2t


(48)


u(y t) IIII1 +

GT

λI

III2 minus

GT

λI

III3 +

Gc

λI

III4 minus

Gc

λI

III5 (49)

where

IIII1 B1(y t)

lowastB

III2 (t) (50)


(A5))


lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


IIII3 B1(y t)

lowastB

III3 (t)

(51)


(A6))


lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


IIII5 B1(y t)

lowastB

III4 (t)

(52)


(A7))



C(y t) et

+ Kcet


ySc

1113968

2t


(53)


T(y t) et

+ Set


yPr

1113968

2t


(54)


u(y t) IIV1 +

GT

λI

IV2 minus

GT

λI

IV3 +

Gc

λI

IV4 minus

Gc

λI

IV5 (55)

where

IIV1 B1(y t)

lowastB

IV2 (t) (56)


(A8))


IIV2 e

t+ λe

t1113872 1113873

lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


IIV3 B1(y t)

lowastB

IV3 (t)

(57)


(A9))

IIV4 e

t+ λe

t1113872 1113873

lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


(58)

Similarly

IIV5 B1(y t)

lowastB

IV4 (t) (59)


(A10))


C(y t) et

+ Kc + 1( 1113857tet


ySc

1113968

2t


(60)


T(y t) et

+(S + 1)tet


yPr

1113968

2t


u(y t) IV1 +

GT

λI

V2 minus

GT

λI

V3 +

Gc

λI

V4 minus

Gc

λI

V5

(61)


(q minus 1)2)

IV1 B1(y t)

lowastB

V2 (t) (62)


(A11))

IV2 te

t+ λe

t(t + 1)1113872 1113873

lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


IV3 B1(y t)

lowastB

V3 (t)

(63)


(A12))

IV4 te

t+ λe

t(t + 1)1113872 1113873

lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


IV5 B1(y t)

lowastB

V4 (t)

(64)


(A13))





lowasterfcy

Sc

1113968

2t


T(y t) et(cos t +(1 + S)sin t)1113872 1113873

lowasterfcy

Pr

1113968

2t


u(y t) IVI1 +

GT

λI

VI2 minus

GT

λI

VI3 +

Gc

λI

VI4 minus

Gc

λI

VI5

IVI1 B1(y t)

lowastB

VI2 (t)

(65)


(A14))

IVI2 e

t sin t(1 + λ) + et cos t1113872 1113873

lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


IVI3 B1(y t)

lowastB

VI3 (t)

(66)


(A15))

IVI4 e

t sin t(1 + λ) + et cos t1113872 1113873

lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


(67)

Similarly

IVI5 B1(y t)

lowastB

VI4 (t) (68)





ySc

1113968

2t


(69)



yPr

1113968

2t


u(y t) IVII1 +

GT

λI

VII2 minus

GT

λI

VII3 +

Gc

λI

VII4 minus

Gc

λI

VII5

(70)

where

IVII1 B1(y t)

lowastB

VII2 (t) (71)


(A17))


lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


IVII3 B1(y t)

lowastB

VII3 (t)

(72)


tion (A18))


lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


IVII5 B1(y t)

lowastB

VII4 (t)

(73)





GC in equation (9))










f (t) = 1 g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

15

1

05

0

(a)

f (t) = t g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

2

15

1

05

0

(b)

f (t) = sint g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

15

1

05

0

(c)

f (t) = et g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

15

1

05

0

(d)

f (t) = tet g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

(e)

f (t) = et sint g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

5

4

3

2

1

0

Velo

city

(f )

Figure 1 Continued


f (t) = t sint g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

15

1

05

0

(g)


Conc

entr

atio

n

15

1

05

0

ndash050 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = H (t)

(a)

Conc

entr

atio

n

15

1

05

00 1 2 3 4 5

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = H (t)

(b)

005

004

003

002

001

0

ndash0010 1 2 3

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = t

(c)

02

015

01

005

00 1 2 3 4

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = t

(d)

Figure 2 Continued


005

004

003

002

001

0

ndash0010 1 2 3

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = sin t

(e)

02

015

01

005

00 1 2 3 4

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = sin t

(f )


04

03

02

01

0

ndash010 1 2 3

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = et

Conc

entr

atio

n

(a)

08

06

04

02

0

ndash020 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = etCo

ncen

trat

ion

(b)

004

0

006

002

ndash0020 1 2 3

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = tet

Conc

entr

atio

n

(c)

03

02

01

0

ndash010 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = tet

Conc

entr

atio

n

(d)

Figure 3 Continued


006

004

002

0

ndash0020 1 2 3

y

Sc = 14Sc = 096Sc = 022

Sc = 060

g (t) = et sintCo

ncen

trat

ion

(e)

03

02

01

00 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = et sint

Conc

entr

atio

n

(f )

001

8 times 10ndash3

6 times 10ndash3

4 times 10ndash3

2 times 10ndash3

0


0 1 2 3y

Sc = 14Sc = 096Sc = 022

Sc = 060

g (t) = t sint

Conc

entr

atio

n

(g)

0

008

006

004

002

ndash0020 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = t sintCo

ncen

trat

ion

(h)


Tem

pera

ture

15

1

05

0

ndash051 2 30

y

Pr = 70Pr = 50Pr = 071

Pr = 15

g (t) = H (t)

(a)

Tem

pera

ture

15

1

05

0

ndash051 2 30

y

Pr = 70Pr = 50Pr = 071

Pr = 15

g (t) = H (t)

(b)

Figure 4 Continued


0

008

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t

(c)

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t

(d)

0

008

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = sint

(e)

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = sint

(f )

05

04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et

(g)

Tem

pera

ture

1

08

06

04

02

00

y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et

(h)



0

008

01

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = tet

(a)

04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = tet

(b)

0

008

01

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et sint

(c)

04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et sint

(d)

Tem

pera

ture

0

002

0015

001

5 times 10ndash3


y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t sint

(e)

0

008

01

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t sint

(f )



Tem

pera

ture

15

1

05

00

y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = H (t)

(a)

Tem

pera

ture

12

1

08

06

04

02

00

y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = H (t)

(b)

0

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t

(c)

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t

(d)

0

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = sin t

(e)

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = sint

(f )

Figure 6 Continued


0y

S = 45S = 30S = 05

S = 15

05 1 15 2

05

04

03

02

01

0

Tem

pera

ture

g (t) = et

(g)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

Tem

pera

ture

1

08

06

04

02

0

g (t) = et

(h)


0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = tet

(a)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

04

03

02

01

0

ndash01

Tem

pera

ture

g (t) = tet

(b)

0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = et sint

(c)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

04

03

02

01

0

Tem

pera

ture

g (t) = et sint

(d)

Figure 7 Continued


Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

015

001

002

5 times 10ndash3

0

g (t) = t sint

(e)

0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t sint

(f )


0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = 1 g(t) = 1 h(t) = 1

(a)

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = t g(t) = 1 h(t) = 1

(b)

0 1 2 3 4 5y

Velo

city

2

15

1

05

0

M = 05M = 10

M = 15M = 25

f (t) = et g(t) = 1 h(t) = 1

(c)

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = sin t g(t) = 1 h(t) = 1

(d)

Figure 8 Continued






0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = tet g(t) = 1 h(t) = 1

(e)

0 1 2 3 4 5y

6

4

2

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = et sin t g(t) = 1 h(t) = 1

(f )

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = t sin t g(t) = 1 h(t) = 1

(g)



3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = sin t g (t) = 1 h (t) = 1

(c)

2

1

0

ndash1

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = et g (t) = 1 h (t) = 1

(d)

20 1 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

f (t) = tet g (t) = 1 h (t) = 1

(e)

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

5

4

3

2

1

0

Velo

city

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 9 Continued


2

1

0

ndash1

Velo

city

20 1 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


f (t) = 1 g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(a)

f (t) = t g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(b)

4

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

f (t) = sin t g (t) = 1 h (t) = 1

(c)

f (t) = et g (t) = 1 h (t) = 1

4

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(d)

Figure 10 Continued


f (t) = tet g (t) = 1 h (t) = 1

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

(e)

f (t) = et sin t g (t) = 1 h (t) = 1

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

5

4

3

2

1

0

Velo

city

(f )

f (t) = t sin t g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(g)


0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

4

3

2

1

0

Velo

city

f (t) = 1 g (t) = 1 h (t) = 1

(a)

GC = 80GC = 60GC = 20

GC = 40

0 1 2 3 54y

4

3

2

1

0

Velo

city

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 11 Continued


0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

4

3

2

1

0

Velo

city

f (t) = sin t g (t) = 1 h (t) = 1

(c)

GC = 80GC = 60GC = 20

GC = 40

0 1 2 3 54y

4

3

2

1

0

Velo

city

f (t) = et g (t) = 1 h (t) = 1

(d)

0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

f (t) = tet g (t) = 1 h (t) = 1

(e)

GC = 80GC = 60GC = 20

GC = 40

5

4

3

2

1

0

Velo


0 1 2 3 54y

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

f (t) = t sint g (t) = 1 h (t) = 1

(g)



3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = t g (t) = 1 h (t) = 1

(b)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = sin t g (t) = 1 h (t) = 1

(c)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = et g (t) = 1 h (t) = 1

(d)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

f (t) = tet g (t) = 1 h (t) = 1

(e)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

5

4

3

2

1

0

Velo

city

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 12 Continued


0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0f (t) = t sin t g (t) = 1 h (t) = 1

(g)


Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(a)

Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(b)

005

004

003

002

001

0

Conc

entr

atio

n

g (t) = t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(c)

Conc

entr

atio

n

02

015

01

005

0

g (t) = t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(d)

Figure 13 Continued


005

004

003

002

001

0

Conc

entr

atio

ng (t) = sin t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(e)

Conc

entr

atio

n

02

015

01

005

0

g (t) = sin t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(f )

03

04

02

01

0

Conc

entr

atio

n

g (t) = et

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(g)

Conc

entr

atio

n

08

06

04

02

0

g (t) = et

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(h)


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = tet

(a)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = tet

(b)

Figure 14 Continued


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = et sint

(c)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = et sint

(d)

001

8 times 10ndash3

6 times 10ndash3

4 times 10ndash3

2 times 10ndash3

0

Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = t sin t

(e)

0

008

006

004

002

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = t sin t

(f )


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = 1 g (t) = 1 h (t) = 1

(a)

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 15 Continued


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = sin t g (t) = 1 h (t) = 1

(c)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 105

0

ndash1 times 105

ndash2 times 105

ndash3 times 105

f (t) = et g (t) = 1 h (t) = 1

(d)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 106

0

ndash1 times 106

ndash2 times 106

ndash3 times 106

f (t) = tet g (t) = 1 h (t) = 1

(e)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

100

0

ndash100

ndash200

ndash300

ndash400

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t sin t g (t) = 1 h (t) = 1

(g)



3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = sin t g (t) = 1 h (t) = 1

(c)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et g (t) = 1 h (t) = 1

(d)

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = tet g (t) = 1 h (t) = 1

(e)

5

4

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 16 Continued


3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


3

2

1

00 1 2 3 54

y

M = 20M = 15M = 05

M = 10


Vel

ocity

(a)

1

0

ndash1

ndash2

ndash30 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

f (t) = 1 g (t) = sin t h (t) = tet

Vel

ocity

(b)

2

0

ndash2

ndash40 1 2 3 54

y

Gc = 80Gc = 60Gc = 20

Gc = 40


Vel

ocity

(c)

6

4

2

00 1 2 3 54

y

GT = 10GT = 60GT = 10

GT = 35

f (t) = 1 g (t) = sin t h (t) = t

Vel

ocity

(d)

Figure 17 Continued


2

1

0

ndash1

ndash2

ndash30 1 2 3 54

y

Sc = 078Sc = 060Sc = 022

Sc = 030


eloc

ity

(e)

3

2

1

0

ndash10 1 2 3 54

y

S = 40S = 30S = 10

S = 20

f (t) = t g (t) = et h (t) = 1

Vel

ocity

(f )


04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(a)

06

04

02

0

ndash02

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(b)



6 Conclusions






04

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(a)

06

04

02

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(b)


08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(a)

1

08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(b)







Appendix

B1(y t)

0 0lt tltyλ

eminus α1t

I0 iα3

t2

minus λy2

1113969

1113874 1113875 tgtyλ

⎧⎪⎪⎨

⎪⎪⎩(A1)

BII2 (t) e



I0 iα3t( 11138571113872 1113873lowastt (A2)

BII3 (t) (1 + λH(t))

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BII4 (t) (1 + λH(t)

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIII2 (t) e






Alowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste




Dlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIV2 (t) e


lowastH(t) + α1( 1113857

lowaste

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t (A8)

BIV3 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BIV4 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste

minus α1tI0 iα3t( 1113857 (A10)

BV2 (t) e


lowaste

t+ 1 + α1( 1113857te

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowastte

t (A11)

BV3 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A12)

BV4 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A13)

BVI2 (t) e


lowaste

t cos t + 1 + α1( 1113857et sin t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t sin t(A14)

BVI3 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A15)


BVI4 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A16)

BVII2 (t) e



minus α1tI0 iα3t( 11138571113872 1113873

lowastt sin t

(A17)



lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A18)



lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A19)

Lminus 1 e

minus bq2minus a2

radic

q2

minus a2

1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

0 0lt tlt b bgt 0

I0 a

t2

minus b2

1113969

1113888 1113889 tgt bRe(q)gtRe(a)

⎧⎪⎪⎨

⎪⎪⎩(A20)

Lminus 1


(A21)


q + a

radic)

q minus b1113890 1113891 e


Φ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A23)


q + a

radic)

q minus b1113890 1113891 e


Ψ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A25)

Data Availability




Acknowledgments


References































u(y t) II1 +

GT

λI

I2 minus

GT

λI

I3 +

Gc

λI

I4 minus

Gc

λI

I5 (34)

where

II1 B1(y t)

lowastB

I2(t) (35)


BI2(t) e


lowast δ(t) + α1H(t)( 1113857

+ α23eminus α1t


(36)


equation (25)


minus α4tcosh α6t( 11138571113872

+S minus α4( 1113857

α6e

minus α4tsinh α6t( 11138571113889

lowasterfcy

Pr

1113968

2t


(37)


II3 B1(y t)

lowastB

I3(t) (38)

where



lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(39)


tion (29)



α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


(40)


II5 B1(y t)

lowastB

I4(t) (41)

and



lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(42)




1 III2 III

3 III4 and III



ySc

1113968

2t



yPr

1113968

2t


u(y t) III1 +

GT

λI

II2 minus

GT

λI

II3 +

Gc

λI

II4 minus

Gc

λI

II5

(43)

where

III1 B1(y t)

lowastB

II2 (t) (44)


(A2))

III2 (t + λ)

lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


III3 B1(y t)

lowastB

II3 (t)

(45)


(A3))

III4 (t + λ)

lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


III5 B1(y t)

lowastB

II4 (t)

(46)



(A4))



ySc

1113968

2t


(47)



yPr

1113968

2t


(48)


u(y t) IIII1 +

GT

λI

III2 minus

GT

λI

III3 +

Gc

λI

III4 minus

Gc

λI

III5 (49)

where

IIII1 B1(y t)

lowastB

III2 (t) (50)


(A5))


lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


IIII3 B1(y t)

lowastB

III3 (t)

(51)


(A6))


lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


IIII5 B1(y t)

lowastB

III4 (t)

(52)


(A7))



C(y t) et

+ Kcet


ySc

1113968

2t


(53)


T(y t) et

+ Set


yPr

1113968

2t


(54)


u(y t) IIV1 +

GT

λI

IV2 minus

GT

λI

IV3 +

Gc

λI

IV4 minus

Gc

λI

IV5 (55)

where

IIV1 B1(y t)

lowastB

IV2 (t) (56)


(A8))


IIV2 e

t+ λe

t1113872 1113873

lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


IIV3 B1(y t)

lowastB

IV3 (t)

(57)


(A9))

IIV4 e

t+ λe

t1113872 1113873

lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


(58)

Similarly

IIV5 B1(y t)

lowastB

IV4 (t) (59)


(A10))


C(y t) et

+ Kc + 1( 1113857tet


ySc

1113968

2t


(60)


T(y t) et

+(S + 1)tet


yPr

1113968

2t


u(y t) IV1 +

GT

λI

V2 minus

GT

λI

V3 +

Gc

λI

V4 minus

Gc

λI

V5

(61)


(q minus 1)2)

IV1 B1(y t)

lowastB

V2 (t) (62)


(A11))

IV2 te

t+ λe

t(t + 1)1113872 1113873

lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


IV3 B1(y t)

lowastB

V3 (t)

(63)


(A12))

IV4 te

t+ λe

t(t + 1)1113872 1113873

lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


IV5 B1(y t)

lowastB

V4 (t)

(64)


(A13))





lowasterfcy

Sc

1113968

2t


T(y t) et(cos t +(1 + S)sin t)1113872 1113873

lowasterfcy

Pr

1113968

2t


u(y t) IVI1 +

GT

λI

VI2 minus

GT

λI

VI3 +

Gc

λI

VI4 minus

Gc

λI

VI5

IVI1 B1(y t)

lowastB

VI2 (t)

(65)


(A14))

IVI2 e

t sin t(1 + λ) + et cos t1113872 1113873

lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


IVI3 B1(y t)

lowastB

VI3 (t)

(66)


(A15))

IVI4 e

t sin t(1 + λ) + et cos t1113872 1113873

lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


(67)

Similarly

IVI5 B1(y t)

lowastB

VI4 (t) (68)





ySc

1113968

2t


(69)



yPr

1113968

2t


u(y t) IVII1 +

GT

λI

VII2 minus

GT

λI

VII3 +

Gc

λI

VII4 minus

Gc

λI

VII5

(70)

where

IVII1 B1(y t)

lowastB

VII2 (t) (71)


(A17))


lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


IVII3 B1(y t)

lowastB

VII3 (t)

(72)


tion (A18))


lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


IVII5 B1(y t)

lowastB

VII4 (t)

(73)





GC in equation (9))










f (t) = 1 g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

15

1

05

0

(a)

f (t) = t g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

2

15

1

05

0

(b)

f (t) = sint g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

15

1

05

0

(c)

f (t) = et g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

15

1

05

0

(d)

f (t) = tet g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

(e)

f (t) = et sint g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

5

4

3

2

1

0

Velo

city

(f )

Figure 1 Continued


f (t) = t sint g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

15

1

05

0

(g)


Conc

entr

atio

n

15

1

05

0

ndash050 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = H (t)

(a)

Conc

entr

atio

n

15

1

05

00 1 2 3 4 5

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = H (t)

(b)

005

004

003

002

001

0

ndash0010 1 2 3

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = t

(c)

02

015

01

005

00 1 2 3 4

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = t

(d)

Figure 2 Continued


005

004

003

002

001

0

ndash0010 1 2 3

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = sin t

(e)

02

015

01

005

00 1 2 3 4

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = sin t

(f )


04

03

02

01

0

ndash010 1 2 3

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = et

Conc

entr

atio

n

(a)

08

06

04

02

0

ndash020 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = etCo

ncen

trat

ion

(b)

004

0

006

002

ndash0020 1 2 3

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = tet

Conc

entr

atio

n

(c)

03

02

01

0

ndash010 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = tet

Conc

entr

atio

n

(d)

Figure 3 Continued


006

004

002

0

ndash0020 1 2 3

y

Sc = 14Sc = 096Sc = 022

Sc = 060

g (t) = et sintCo

ncen

trat

ion

(e)

03

02

01

00 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = et sint

Conc

entr

atio

n

(f )

001

8 times 10ndash3

6 times 10ndash3

4 times 10ndash3

2 times 10ndash3

0


0 1 2 3y

Sc = 14Sc = 096Sc = 022

Sc = 060

g (t) = t sint

Conc

entr

atio

n

(g)

0

008

006

004

002

ndash0020 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = t sintCo

ncen

trat

ion

(h)


Tem

pera

ture

15

1

05

0

ndash051 2 30

y

Pr = 70Pr = 50Pr = 071

Pr = 15

g (t) = H (t)

(a)

Tem

pera

ture

15

1

05

0

ndash051 2 30

y

Pr = 70Pr = 50Pr = 071

Pr = 15

g (t) = H (t)

(b)

Figure 4 Continued


0

008

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t

(c)

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t

(d)

0

008

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = sint

(e)

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = sint

(f )

05

04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et

(g)

Tem

pera

ture

1

08

06

04

02

00

y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et

(h)



0

008

01

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = tet

(a)

04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = tet

(b)

0

008

01

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et sint

(c)

04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et sint

(d)

Tem

pera

ture

0

002

0015

001

5 times 10ndash3


y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t sint

(e)

0

008

01

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t sint

(f )



Tem

pera

ture

15

1

05

00

y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = H (t)

(a)

Tem

pera

ture

12

1

08

06

04

02

00

y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = H (t)

(b)

0

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t

(c)

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t

(d)

0

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = sin t

(e)

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = sint

(f )

Figure 6 Continued


0y

S = 45S = 30S = 05

S = 15

05 1 15 2

05

04

03

02

01

0

Tem

pera

ture

g (t) = et

(g)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

Tem

pera

ture

1

08

06

04

02

0

g (t) = et

(h)


0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = tet

(a)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

04

03

02

01

0

ndash01

Tem

pera

ture

g (t) = tet

(b)

0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = et sint

(c)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

04

03

02

01

0

Tem

pera

ture

g (t) = et sint

(d)

Figure 7 Continued


Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

015

001

002

5 times 10ndash3

0

g (t) = t sint

(e)

0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t sint

(f )


0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = 1 g(t) = 1 h(t) = 1

(a)

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = t g(t) = 1 h(t) = 1

(b)

0 1 2 3 4 5y

Velo

city

2

15

1

05

0

M = 05M = 10

M = 15M = 25

f (t) = et g(t) = 1 h(t) = 1

(c)

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = sin t g(t) = 1 h(t) = 1

(d)

Figure 8 Continued






0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = tet g(t) = 1 h(t) = 1

(e)

0 1 2 3 4 5y

6

4

2

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = et sin t g(t) = 1 h(t) = 1

(f )

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = t sin t g(t) = 1 h(t) = 1

(g)



3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = sin t g (t) = 1 h (t) = 1

(c)

2

1

0

ndash1

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = et g (t) = 1 h (t) = 1

(d)

20 1 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

f (t) = tet g (t) = 1 h (t) = 1

(e)

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

5

4

3

2

1

0

Velo

city

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 9 Continued


2

1

0

ndash1

Velo

city

20 1 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


f (t) = 1 g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(a)

f (t) = t g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(b)

4

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

f (t) = sin t g (t) = 1 h (t) = 1

(c)

f (t) = et g (t) = 1 h (t) = 1

4

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(d)

Figure 10 Continued


f (t) = tet g (t) = 1 h (t) = 1

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

(e)

f (t) = et sin t g (t) = 1 h (t) = 1

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

5

4

3

2

1

0

Velo

city

(f )

f (t) = t sin t g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(g)


0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

4

3

2

1

0

Velo

city

f (t) = 1 g (t) = 1 h (t) = 1

(a)

GC = 80GC = 60GC = 20

GC = 40

0 1 2 3 54y

4

3

2

1

0

Velo

city

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 11 Continued


0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

4

3

2

1

0

Velo

city

f (t) = sin t g (t) = 1 h (t) = 1

(c)

GC = 80GC = 60GC = 20

GC = 40

0 1 2 3 54y

4

3

2

1

0

Velo

city

f (t) = et g (t) = 1 h (t) = 1

(d)

0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

f (t) = tet g (t) = 1 h (t) = 1

(e)

GC = 80GC = 60GC = 20

GC = 40

5

4

3

2

1

0

Velo


0 1 2 3 54y

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

f (t) = t sint g (t) = 1 h (t) = 1

(g)



3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = t g (t) = 1 h (t) = 1

(b)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = sin t g (t) = 1 h (t) = 1

(c)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = et g (t) = 1 h (t) = 1

(d)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

f (t) = tet g (t) = 1 h (t) = 1

(e)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

5

4

3

2

1

0

Velo

city

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 12 Continued


0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0f (t) = t sin t g (t) = 1 h (t) = 1

(g)


Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(a)

Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(b)

005

004

003

002

001

0

Conc

entr

atio

n

g (t) = t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(c)

Conc

entr

atio

n

02

015

01

005

0

g (t) = t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(d)

Figure 13 Continued


005

004

003

002

001

0

Conc

entr

atio

ng (t) = sin t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(e)

Conc

entr

atio

n

02

015

01

005

0

g (t) = sin t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(f )

03

04

02

01

0

Conc

entr

atio

n

g (t) = et

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(g)

Conc

entr

atio

n

08

06

04

02

0

g (t) = et

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(h)


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = tet

(a)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = tet

(b)

Figure 14 Continued


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = et sint

(c)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = et sint

(d)

001

8 times 10ndash3

6 times 10ndash3

4 times 10ndash3

2 times 10ndash3

0

Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = t sin t

(e)

0

008

006

004

002

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = t sin t

(f )


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = 1 g (t) = 1 h (t) = 1

(a)

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 15 Continued


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = sin t g (t) = 1 h (t) = 1

(c)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 105

0

ndash1 times 105

ndash2 times 105

ndash3 times 105

f (t) = et g (t) = 1 h (t) = 1

(d)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 106

0

ndash1 times 106

ndash2 times 106

ndash3 times 106

f (t) = tet g (t) = 1 h (t) = 1

(e)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

100

0

ndash100

ndash200

ndash300

ndash400

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t sin t g (t) = 1 h (t) = 1

(g)



3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = sin t g (t) = 1 h (t) = 1

(c)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et g (t) = 1 h (t) = 1

(d)

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = tet g (t) = 1 h (t) = 1

(e)

5

4

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 16 Continued


3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


3

2

1

00 1 2 3 54

y

M = 20M = 15M = 05

M = 10


Vel

ocity

(a)

1

0

ndash1

ndash2

ndash30 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

f (t) = 1 g (t) = sin t h (t) = tet

Vel

ocity

(b)

2

0

ndash2

ndash40 1 2 3 54

y

Gc = 80Gc = 60Gc = 20

Gc = 40


Vel

ocity

(c)

6

4

2

00 1 2 3 54

y

GT = 10GT = 60GT = 10

GT = 35

f (t) = 1 g (t) = sin t h (t) = t

Vel

ocity

(d)

Figure 17 Continued


2

1

0

ndash1

ndash2

ndash30 1 2 3 54

y

Sc = 078Sc = 060Sc = 022

Sc = 030


eloc

ity

(e)

3

2

1

0

ndash10 1 2 3 54

y

S = 40S = 30S = 10

S = 20

f (t) = t g (t) = et h (t) = 1

Vel

ocity

(f )


04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(a)

06

04

02

0

ndash02

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(b)



6 Conclusions






04

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(a)

06

04

02

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(b)


08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(a)

1

08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(b)







Appendix

B1(y t)

0 0lt tltyλ

eminus α1t

I0 iα3

t2

minus λy2

1113969

1113874 1113875 tgtyλ

⎧⎪⎪⎨

⎪⎪⎩(A1)

BII2 (t) e



I0 iα3t( 11138571113872 1113873lowastt (A2)

BII3 (t) (1 + λH(t))

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BII4 (t) (1 + λH(t)

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIII2 (t) e






Alowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste




Dlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIV2 (t) e


lowastH(t) + α1( 1113857

lowaste

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t (A8)

BIV3 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BIV4 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste

minus α1tI0 iα3t( 1113857 (A10)

BV2 (t) e


lowaste

t+ 1 + α1( 1113857te

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowastte

t (A11)

BV3 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A12)

BV4 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A13)

BVI2 (t) e


lowaste

t cos t + 1 + α1( 1113857et sin t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t sin t(A14)

BVI3 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A15)


BVI4 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A16)

BVII2 (t) e



minus α1tI0 iα3t( 11138571113872 1113873

lowastt sin t

(A17)



lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A18)



lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A19)

Lminus 1 e

minus bq2minus a2

radic

q2

minus a2

1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

0 0lt tlt b bgt 0

I0 a

t2

minus b2

1113969

1113888 1113889 tgt bRe(q)gtRe(a)

⎧⎪⎪⎨

⎪⎪⎩(A20)

Lminus 1


(A21)


q + a

radic)

q minus b1113890 1113891 e


Φ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A23)


q + a

radic)

q minus b1113890 1113891 e


Ψ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A25)

Data Availability




Acknowledgments


References
































(A4))



ySc

1113968

2t


(47)



yPr

1113968

2t


(48)


u(y t) IIII1 +

GT

λI

III2 minus

GT

λI

III3 +

Gc

λI

III4 minus

Gc

λI

III5 (49)

where

IIII1 B1(y t)

lowastB

III2 (t) (50)


(A5))


lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


IIII3 B1(y t)

lowastB

III3 (t)

(51)


(A6))


lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


IIII5 B1(y t)

lowastB

III4 (t)

(52)


(A7))



C(y t) et

+ Kcet


ySc

1113968

2t


(53)


T(y t) et

+ Set


yPr

1113968

2t


(54)


u(y t) IIV1 +

GT

λI

IV2 minus

GT

λI

IV3 +

Gc

λI

IV4 minus

Gc

λI

IV5 (55)

where

IIV1 B1(y t)

lowastB

IV2 (t) (56)


(A8))


IIV2 e

t+ λe

t1113872 1113873

lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


IIV3 B1(y t)

lowastB

IV3 (t)

(57)


(A9))

IIV4 e

t+ λe

t1113872 1113873

lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


(58)

Similarly

IIV5 B1(y t)

lowastB

IV4 (t) (59)


(A10))


C(y t) et

+ Kc + 1( 1113857tet


ySc

1113968

2t


(60)


T(y t) et

+(S + 1)tet


yPr

1113968

2t


u(y t) IV1 +

GT

λI

V2 minus

GT

λI

V3 +

Gc

λI

V4 minus

Gc

λI

V5

(61)


(q minus 1)2)

IV1 B1(y t)

lowastB

V2 (t) (62)


(A11))

IV2 te

t+ λe

t(t + 1)1113872 1113873

lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


IV3 B1(y t)

lowastB

V3 (t)

(63)


(A12))

IV4 te

t+ λe

t(t + 1)1113872 1113873

lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


IV5 B1(y t)

lowastB

V4 (t)

(64)


(A13))





lowasterfcy

Sc

1113968

2t


T(y t) et(cos t +(1 + S)sin t)1113872 1113873

lowasterfcy

Pr

1113968

2t


u(y t) IVI1 +

GT

λI

VI2 minus

GT

λI

VI3 +

Gc

λI

VI4 minus

Gc

λI

VI5

IVI1 B1(y t)

lowastB

VI2 (t)

(65)


(A14))

IVI2 e

t sin t(1 + λ) + et cos t1113872 1113873

lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


IVI3 B1(y t)

lowastB

VI3 (t)

(66)


(A15))

IVI4 e

t sin t(1 + λ) + et cos t1113872 1113873

lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


(67)

Similarly

IVI5 B1(y t)

lowastB

VI4 (t) (68)





ySc

1113968

2t


(69)



yPr

1113968

2t


u(y t) IVII1 +

GT

λI

VII2 minus

GT

λI

VII3 +

Gc

λI

VII4 minus

Gc

λI

VII5

(70)

where

IVII1 B1(y t)

lowastB

VII2 (t) (71)


(A17))


lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


IVII3 B1(y t)

lowastB

VII3 (t)

(72)


tion (A18))


lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


IVII5 B1(y t)

lowastB

VII4 (t)

(73)





GC in equation (9))










f (t) = 1 g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

15

1

05

0

(a)

f (t) = t g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

2

15

1

05

0

(b)

f (t) = sint g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

15

1

05

0

(c)

f (t) = et g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

15

1

05

0

(d)

f (t) = tet g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

(e)

f (t) = et sint g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

5

4

3

2

1

0

Velo

city

(f )

Figure 1 Continued


f (t) = t sint g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

15

1

05

0

(g)


Conc

entr

atio

n

15

1

05

0

ndash050 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = H (t)

(a)

Conc

entr

atio

n

15

1

05

00 1 2 3 4 5

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = H (t)

(b)

005

004

003

002

001

0

ndash0010 1 2 3

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = t

(c)

02

015

01

005

00 1 2 3 4

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = t

(d)

Figure 2 Continued


005

004

003

002

001

0

ndash0010 1 2 3

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = sin t

(e)

02

015

01

005

00 1 2 3 4

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = sin t

(f )


04

03

02

01

0

ndash010 1 2 3

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = et

Conc

entr

atio

n

(a)

08

06

04

02

0

ndash020 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = etCo

ncen

trat

ion

(b)

004

0

006

002

ndash0020 1 2 3

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = tet

Conc

entr

atio

n

(c)

03

02

01

0

ndash010 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = tet

Conc

entr

atio

n

(d)

Figure 3 Continued


006

004

002

0

ndash0020 1 2 3

y

Sc = 14Sc = 096Sc = 022

Sc = 060

g (t) = et sintCo

ncen

trat

ion

(e)

03

02

01

00 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = et sint

Conc

entr

atio

n

(f )

001

8 times 10ndash3

6 times 10ndash3

4 times 10ndash3

2 times 10ndash3

0


0 1 2 3y

Sc = 14Sc = 096Sc = 022

Sc = 060

g (t) = t sint

Conc

entr

atio

n

(g)

0

008

006

004

002

ndash0020 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = t sintCo

ncen

trat

ion

(h)


Tem

pera

ture

15

1

05

0

ndash051 2 30

y

Pr = 70Pr = 50Pr = 071

Pr = 15

g (t) = H (t)

(a)

Tem

pera

ture

15

1

05

0

ndash051 2 30

y

Pr = 70Pr = 50Pr = 071

Pr = 15

g (t) = H (t)

(b)

Figure 4 Continued


0

008

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t

(c)

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t

(d)

0

008

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = sint

(e)

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = sint

(f )

05

04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et

(g)

Tem

pera

ture

1

08

06

04

02

00

y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et

(h)



0

008

01

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = tet

(a)

04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = tet

(b)

0

008

01

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et sint

(c)

04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et sint

(d)

Tem

pera

ture

0

002

0015

001

5 times 10ndash3


y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t sint

(e)

0

008

01

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t sint

(f )



Tem

pera

ture

15

1

05

00

y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = H (t)

(a)

Tem

pera

ture

12

1

08

06

04

02

00

y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = H (t)

(b)

0

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t

(c)

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t

(d)

0

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = sin t

(e)

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = sint

(f )

Figure 6 Continued


0y

S = 45S = 30S = 05

S = 15

05 1 15 2

05

04

03

02

01

0

Tem

pera

ture

g (t) = et

(g)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

Tem

pera

ture

1

08

06

04

02

0

g (t) = et

(h)


0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = tet

(a)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

04

03

02

01

0

ndash01

Tem

pera

ture

g (t) = tet

(b)

0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = et sint

(c)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

04

03

02

01

0

Tem

pera

ture

g (t) = et sint

(d)

Figure 7 Continued


Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

015

001

002

5 times 10ndash3

0

g (t) = t sint

(e)

0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t sint

(f )


0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = 1 g(t) = 1 h(t) = 1

(a)

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = t g(t) = 1 h(t) = 1

(b)

0 1 2 3 4 5y

Velo

city

2

15

1

05

0

M = 05M = 10

M = 15M = 25

f (t) = et g(t) = 1 h(t) = 1

(c)

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = sin t g(t) = 1 h(t) = 1

(d)

Figure 8 Continued






0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = tet g(t) = 1 h(t) = 1

(e)

0 1 2 3 4 5y

6

4

2

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = et sin t g(t) = 1 h(t) = 1

(f )

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = t sin t g(t) = 1 h(t) = 1

(g)



3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = sin t g (t) = 1 h (t) = 1

(c)

2

1

0

ndash1

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = et g (t) = 1 h (t) = 1

(d)

20 1 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

f (t) = tet g (t) = 1 h (t) = 1

(e)

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

5

4

3

2

1

0

Velo

city

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 9 Continued


2

1

0

ndash1

Velo

city

20 1 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


f (t) = 1 g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(a)

f (t) = t g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(b)

4

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

f (t) = sin t g (t) = 1 h (t) = 1

(c)

f (t) = et g (t) = 1 h (t) = 1

4

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(d)

Figure 10 Continued


f (t) = tet g (t) = 1 h (t) = 1

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

(e)

f (t) = et sin t g (t) = 1 h (t) = 1

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

5

4

3

2

1

0

Velo

city

(f )

f (t) = t sin t g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(g)


0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

4

3

2

1

0

Velo

city

f (t) = 1 g (t) = 1 h (t) = 1

(a)

GC = 80GC = 60GC = 20

GC = 40

0 1 2 3 54y

4

3

2

1

0

Velo

city

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 11 Continued


0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

4

3

2

1

0

Velo

city

f (t) = sin t g (t) = 1 h (t) = 1

(c)

GC = 80GC = 60GC = 20

GC = 40

0 1 2 3 54y

4

3

2

1

0

Velo

city

f (t) = et g (t) = 1 h (t) = 1

(d)

0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

f (t) = tet g (t) = 1 h (t) = 1

(e)

GC = 80GC = 60GC = 20

GC = 40

5

4

3

2

1

0

Velo


0 1 2 3 54y

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

f (t) = t sint g (t) = 1 h (t) = 1

(g)



3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = t g (t) = 1 h (t) = 1

(b)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = sin t g (t) = 1 h (t) = 1

(c)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = et g (t) = 1 h (t) = 1

(d)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

f (t) = tet g (t) = 1 h (t) = 1

(e)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

5

4

3

2

1

0

Velo

city

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 12 Continued


0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0f (t) = t sin t g (t) = 1 h (t) = 1

(g)


Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(a)

Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(b)

005

004

003

002

001

0

Conc

entr

atio

n

g (t) = t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(c)

Conc

entr

atio

n

02

015

01

005

0

g (t) = t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(d)

Figure 13 Continued


005

004

003

002

001

0

Conc

entr

atio

ng (t) = sin t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(e)

Conc

entr

atio

n

02

015

01

005

0

g (t) = sin t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(f )

03

04

02

01

0

Conc

entr

atio

n

g (t) = et

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(g)

Conc

entr

atio

n

08

06

04

02

0

g (t) = et

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(h)


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = tet

(a)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = tet

(b)

Figure 14 Continued


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = et sint

(c)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = et sint

(d)

001

8 times 10ndash3

6 times 10ndash3

4 times 10ndash3

2 times 10ndash3

0

Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = t sin t

(e)

0

008

006

004

002

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = t sin t

(f )


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = 1 g (t) = 1 h (t) = 1

(a)

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 15 Continued


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = sin t g (t) = 1 h (t) = 1

(c)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 105

0

ndash1 times 105

ndash2 times 105

ndash3 times 105

f (t) = et g (t) = 1 h (t) = 1

(d)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 106

0

ndash1 times 106

ndash2 times 106

ndash3 times 106

f (t) = tet g (t) = 1 h (t) = 1

(e)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

100

0

ndash100

ndash200

ndash300

ndash400

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t sin t g (t) = 1 h (t) = 1

(g)



3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = sin t g (t) = 1 h (t) = 1

(c)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et g (t) = 1 h (t) = 1

(d)

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = tet g (t) = 1 h (t) = 1

(e)

5

4

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 16 Continued


3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


3

2

1

00 1 2 3 54

y

M = 20M = 15M = 05

M = 10


Vel

ocity

(a)

1

0

ndash1

ndash2

ndash30 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

f (t) = 1 g (t) = sin t h (t) = tet

Vel

ocity

(b)

2

0

ndash2

ndash40 1 2 3 54

y

Gc = 80Gc = 60Gc = 20

Gc = 40


Vel

ocity

(c)

6

4

2

00 1 2 3 54

y

GT = 10GT = 60GT = 10

GT = 35

f (t) = 1 g (t) = sin t h (t) = t

Vel

ocity

(d)

Figure 17 Continued


2

1

0

ndash1

ndash2

ndash30 1 2 3 54

y

Sc = 078Sc = 060Sc = 022

Sc = 030


eloc

ity

(e)

3

2

1

0

ndash10 1 2 3 54

y

S = 40S = 30S = 10

S = 20

f (t) = t g (t) = et h (t) = 1

Vel

ocity

(f )


04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(a)

06

04

02

0

ndash02

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(b)



6 Conclusions






04

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(a)

06

04

02

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(b)


08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(a)

1

08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(b)







Appendix

B1(y t)

0 0lt tltyλ

eminus α1t

I0 iα3

t2

minus λy2

1113969

1113874 1113875 tgtyλ

⎧⎪⎪⎨

⎪⎪⎩(A1)

BII2 (t) e



I0 iα3t( 11138571113872 1113873lowastt (A2)

BII3 (t) (1 + λH(t))

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BII4 (t) (1 + λH(t)

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIII2 (t) e






Alowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste




Dlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIV2 (t) e


lowastH(t) + α1( 1113857

lowaste

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t (A8)

BIV3 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BIV4 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste

minus α1tI0 iα3t( 1113857 (A10)

BV2 (t) e


lowaste

t+ 1 + α1( 1113857te

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowastte

t (A11)

BV3 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A12)

BV4 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A13)

BVI2 (t) e


lowaste

t cos t + 1 + α1( 1113857et sin t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t sin t(A14)

BVI3 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A15)


BVI4 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A16)

BVII2 (t) e



minus α1tI0 iα3t( 11138571113872 1113873

lowastt sin t

(A17)



lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A18)



lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A19)

Lminus 1 e

minus bq2minus a2

radic

q2

minus a2

1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

0 0lt tlt b bgt 0

I0 a

t2

minus b2

1113969

1113888 1113889 tgt bRe(q)gtRe(a)

⎧⎪⎪⎨

⎪⎪⎩(A20)

Lminus 1


(A21)


q + a

radic)

q minus b1113890 1113891 e


Φ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A23)


q + a

radic)

q minus b1113890 1113891 e


Ψ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A25)

Data Availability




Acknowledgments


References































IIV2 e

t+ λe

t1113872 1113873

lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


IIV3 B1(y t)

lowastB

IV3 (t)

(57)


(A9))

IIV4 e

t+ λe

t1113872 1113873

lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


(58)

Similarly

IIV5 B1(y t)

lowastB

IV4 (t) (59)


(A10))


C(y t) et

+ Kc + 1( 1113857tet


ySc

1113968

2t


(60)


T(y t) et

+(S + 1)tet


yPr

1113968

2t


u(y t) IV1 +

GT

λI

V2 minus

GT

λI

V3 +

Gc

λI

V4 minus

Gc

λI

V5

(61)


(q minus 1)2)

IV1 B1(y t)

lowastB

V2 (t) (62)


(A11))

IV2 te

t+ λe

t(t + 1)1113872 1113873

lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


IV3 B1(y t)

lowastB

V3 (t)

(63)


(A12))

IV4 te

t+ λe

t(t + 1)1113872 1113873

lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


IV5 B1(y t)

lowastB

V4 (t)

(64)


(A13))





lowasterfcy

Sc

1113968

2t


T(y t) et(cos t +(1 + S)sin t)1113872 1113873

lowasterfcy

Pr

1113968

2t


u(y t) IVI1 +

GT

λI

VI2 minus

GT

λI

VI3 +

Gc

λI

VI4 minus

Gc

λI

VI5

IVI1 B1(y t)

lowastB

VI2 (t)

(65)


(A14))

IVI2 e

t sin t(1 + λ) + et cos t1113872 1113873

lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


IVI3 B1(y t)

lowastB

VI3 (t)

(66)


(A15))

IVI4 e

t sin t(1 + λ) + et cos t1113872 1113873

lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


(67)

Similarly

IVI5 B1(y t)

lowastB

VI4 (t) (68)





ySc

1113968

2t


(69)



yPr

1113968

2t


u(y t) IVII1 +

GT

λI

VII2 minus

GT

λI

VII3 +

Gc

λI

VII4 minus

Gc

λI

VII5

(70)

where

IVII1 B1(y t)

lowastB

VII2 (t) (71)


(A17))


lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


IVII3 B1(y t)

lowastB

VII3 (t)

(72)


tion (A18))


lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


IVII5 B1(y t)

lowastB

VII4 (t)

(73)





GC in equation (9))










f (t) = 1 g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

15

1

05

0

(a)

f (t) = t g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

2

15

1

05

0

(b)

f (t) = sint g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

15

1

05

0

(c)

f (t) = et g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

15

1

05

0

(d)

f (t) = tet g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

(e)

f (t) = et sint g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

5

4

3

2

1

0

Velo

city

(f )

Figure 1 Continued


f (t) = t sint g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

15

1

05

0

(g)


Conc

entr

atio

n

15

1

05

0

ndash050 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = H (t)

(a)

Conc

entr

atio

n

15

1

05

00 1 2 3 4 5

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = H (t)

(b)

005

004

003

002

001

0

ndash0010 1 2 3

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = t

(c)

02

015

01

005

00 1 2 3 4

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = t

(d)

Figure 2 Continued


005

004

003

002

001

0

ndash0010 1 2 3

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = sin t

(e)

02

015

01

005

00 1 2 3 4

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = sin t

(f )


04

03

02

01

0

ndash010 1 2 3

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = et

Conc

entr

atio

n

(a)

08

06

04

02

0

ndash020 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = etCo

ncen

trat

ion

(b)

004

0

006

002

ndash0020 1 2 3

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = tet

Conc

entr

atio

n

(c)

03

02

01

0

ndash010 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = tet

Conc

entr

atio

n

(d)

Figure 3 Continued


006

004

002

0

ndash0020 1 2 3

y

Sc = 14Sc = 096Sc = 022

Sc = 060

g (t) = et sintCo

ncen

trat

ion

(e)

03

02

01

00 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = et sint

Conc

entr

atio

n

(f )

001

8 times 10ndash3

6 times 10ndash3

4 times 10ndash3

2 times 10ndash3

0


0 1 2 3y

Sc = 14Sc = 096Sc = 022

Sc = 060

g (t) = t sint

Conc

entr

atio

n

(g)

0

008

006

004

002

ndash0020 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = t sintCo

ncen

trat

ion

(h)


Tem

pera

ture

15

1

05

0

ndash051 2 30

y

Pr = 70Pr = 50Pr = 071

Pr = 15

g (t) = H (t)

(a)

Tem

pera

ture

15

1

05

0

ndash051 2 30

y

Pr = 70Pr = 50Pr = 071

Pr = 15

g (t) = H (t)

(b)

Figure 4 Continued


0

008

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t

(c)

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t

(d)

0

008

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = sint

(e)

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = sint

(f )

05

04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et

(g)

Tem

pera

ture

1

08

06

04

02

00

y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et

(h)



0

008

01

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = tet

(a)

04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = tet

(b)

0

008

01

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et sint

(c)

04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et sint

(d)

Tem

pera

ture

0

002

0015

001

5 times 10ndash3


y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t sint

(e)

0

008

01

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t sint

(f )



Tem

pera

ture

15

1

05

00

y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = H (t)

(a)

Tem

pera

ture

12

1

08

06

04

02

00

y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = H (t)

(b)

0

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t

(c)

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t

(d)

0

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = sin t

(e)

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = sint

(f )

Figure 6 Continued


0y

S = 45S = 30S = 05

S = 15

05 1 15 2

05

04

03

02

01

0

Tem

pera

ture

g (t) = et

(g)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

Tem

pera

ture

1

08

06

04

02

0

g (t) = et

(h)


0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = tet

(a)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

04

03

02

01

0

ndash01

Tem

pera

ture

g (t) = tet

(b)

0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = et sint

(c)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

04

03

02

01

0

Tem

pera

ture

g (t) = et sint

(d)

Figure 7 Continued


Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

015

001

002

5 times 10ndash3

0

g (t) = t sint

(e)

0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t sint

(f )


0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = 1 g(t) = 1 h(t) = 1

(a)

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = t g(t) = 1 h(t) = 1

(b)

0 1 2 3 4 5y

Velo

city

2

15

1

05

0

M = 05M = 10

M = 15M = 25

f (t) = et g(t) = 1 h(t) = 1

(c)

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = sin t g(t) = 1 h(t) = 1

(d)

Figure 8 Continued






0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = tet g(t) = 1 h(t) = 1

(e)

0 1 2 3 4 5y

6

4

2

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = et sin t g(t) = 1 h(t) = 1

(f )

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = t sin t g(t) = 1 h(t) = 1

(g)



3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = sin t g (t) = 1 h (t) = 1

(c)

2

1

0

ndash1

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = et g (t) = 1 h (t) = 1

(d)

20 1 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

f (t) = tet g (t) = 1 h (t) = 1

(e)

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

5

4

3

2

1

0

Velo

city

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 9 Continued


2

1

0

ndash1

Velo

city

20 1 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


f (t) = 1 g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(a)

f (t) = t g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(b)

4

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

f (t) = sin t g (t) = 1 h (t) = 1

(c)

f (t) = et g (t) = 1 h (t) = 1

4

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(d)

Figure 10 Continued


f (t) = tet g (t) = 1 h (t) = 1

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

(e)

f (t) = et sin t g (t) = 1 h (t) = 1

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

5

4

3

2

1

0

Velo

city

(f )

f (t) = t sin t g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(g)


0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

4

3

2

1

0

Velo

city

f (t) = 1 g (t) = 1 h (t) = 1

(a)

GC = 80GC = 60GC = 20

GC = 40

0 1 2 3 54y

4

3

2

1

0

Velo

city

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 11 Continued


0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

4

3

2

1

0

Velo

city

f (t) = sin t g (t) = 1 h (t) = 1

(c)

GC = 80GC = 60GC = 20

GC = 40

0 1 2 3 54y

4

3

2

1

0

Velo

city

f (t) = et g (t) = 1 h (t) = 1

(d)

0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

f (t) = tet g (t) = 1 h (t) = 1

(e)

GC = 80GC = 60GC = 20

GC = 40

5

4

3

2

1

0

Velo


0 1 2 3 54y

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

f (t) = t sint g (t) = 1 h (t) = 1

(g)



3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = t g (t) = 1 h (t) = 1

(b)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = sin t g (t) = 1 h (t) = 1

(c)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = et g (t) = 1 h (t) = 1

(d)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

f (t) = tet g (t) = 1 h (t) = 1

(e)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

5

4

3

2

1

0

Velo

city

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 12 Continued


0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0f (t) = t sin t g (t) = 1 h (t) = 1

(g)


Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(a)

Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(b)

005

004

003

002

001

0

Conc

entr

atio

n

g (t) = t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(c)

Conc

entr

atio

n

02

015

01

005

0

g (t) = t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(d)

Figure 13 Continued


005

004

003

002

001

0

Conc

entr

atio

ng (t) = sin t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(e)

Conc

entr

atio

n

02

015

01

005

0

g (t) = sin t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(f )

03

04

02

01

0

Conc

entr

atio

n

g (t) = et

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(g)

Conc

entr

atio

n

08

06

04

02

0

g (t) = et

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(h)


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = tet

(a)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = tet

(b)

Figure 14 Continued


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = et sint

(c)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = et sint

(d)

001

8 times 10ndash3

6 times 10ndash3

4 times 10ndash3

2 times 10ndash3

0

Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = t sin t

(e)

0

008

006

004

002

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = t sin t

(f )


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = 1 g (t) = 1 h (t) = 1

(a)

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 15 Continued


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = sin t g (t) = 1 h (t) = 1

(c)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 105

0

ndash1 times 105

ndash2 times 105

ndash3 times 105

f (t) = et g (t) = 1 h (t) = 1

(d)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 106

0

ndash1 times 106

ndash2 times 106

ndash3 times 106

f (t) = tet g (t) = 1 h (t) = 1

(e)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

100

0

ndash100

ndash200

ndash300

ndash400

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t sin t g (t) = 1 h (t) = 1

(g)



3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = sin t g (t) = 1 h (t) = 1

(c)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et g (t) = 1 h (t) = 1

(d)

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = tet g (t) = 1 h (t) = 1

(e)

5

4

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 16 Continued


3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


3

2

1

00 1 2 3 54

y

M = 20M = 15M = 05

M = 10


Vel

ocity

(a)

1

0

ndash1

ndash2

ndash30 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

f (t) = 1 g (t) = sin t h (t) = tet

Vel

ocity

(b)

2

0

ndash2

ndash40 1 2 3 54

y

Gc = 80Gc = 60Gc = 20

Gc = 40


Vel

ocity

(c)

6

4

2

00 1 2 3 54

y

GT = 10GT = 60GT = 10

GT = 35

f (t) = 1 g (t) = sin t h (t) = t

Vel

ocity

(d)

Figure 17 Continued


2

1

0

ndash1

ndash2

ndash30 1 2 3 54

y

Sc = 078Sc = 060Sc = 022

Sc = 030


eloc

ity

(e)

3

2

1

0

ndash10 1 2 3 54

y

S = 40S = 30S = 10

S = 20

f (t) = t g (t) = et h (t) = 1

Vel

ocity

(f )


04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(a)

06

04

02

0

ndash02

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(b)



6 Conclusions






04

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(a)

06

04

02

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(b)


08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(a)

1

08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(b)







Appendix

B1(y t)

0 0lt tltyλ

eminus α1t

I0 iα3

t2

minus λy2

1113969

1113874 1113875 tgtyλ

⎧⎪⎪⎨

⎪⎪⎩(A1)

BII2 (t) e



I0 iα3t( 11138571113872 1113873lowastt (A2)

BII3 (t) (1 + λH(t))

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BII4 (t) (1 + λH(t)

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIII2 (t) e






Alowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste




Dlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIV2 (t) e


lowastH(t) + α1( 1113857

lowaste

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t (A8)

BIV3 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BIV4 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste

minus α1tI0 iα3t( 1113857 (A10)

BV2 (t) e


lowaste

t+ 1 + α1( 1113857te

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowastte

t (A11)

BV3 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A12)

BV4 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A13)

BVI2 (t) e


lowaste

t cos t + 1 + α1( 1113857et sin t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t sin t(A14)

BVI3 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A15)


BVI4 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A16)

BVII2 (t) e



minus α1tI0 iα3t( 11138571113872 1113873

lowastt sin t

(A17)



lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A18)



lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A19)

Lminus 1 e

minus bq2minus a2

radic

q2

minus a2

1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

0 0lt tlt b bgt 0

I0 a

t2

minus b2

1113969

1113888 1113889 tgt bRe(q)gtRe(a)

⎧⎪⎪⎨

⎪⎪⎩(A20)

Lminus 1


(A21)


q + a

radic)

q minus b1113890 1113891 e


Φ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A23)


q + a

radic)

q minus b1113890 1113891 e


Ψ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A25)

Data Availability




Acknowledgments


References

































ySc

1113968

2t


(69)



yPr

1113968

2t


u(y t) IVII1 +

GT

λI

VII2 minus

GT

λI

VII3 +

Gc

λI

VII4 minus

Gc

λI

VII5

(70)

where

IVII1 B1(y t)

lowastB

VII2 (t) (71)


(A17))


lowaste


α6e

minus α4tsinh α6t( 11138571113888 1113889

lowasterfcy

Pr

1113968

2t


IVII3 B1(y t)

lowastB

VII3 (t)

(72)


tion (A18))


lowaste


α9e

minus α7tsinh α9t( 11138571113888 1113889

lowasterfcy

Sc

1113968

2t


IVII5 B1(y t)

lowastB

VII4 (t)

(73)





GC in equation (9))










f (t) = 1 g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

15

1

05

0

(a)

f (t) = t g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

2

15

1

05

0

(b)

f (t) = sint g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

15

1

05

0

(c)

f (t) = et g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

15

1

05

0

(d)

f (t) = tet g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

(e)

f (t) = et sint g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

5

4

3

2

1

0

Velo

city

(f )

Figure 1 Continued


f (t) = t sint g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

15

1

05

0

(g)


Conc

entr

atio

n

15

1

05

0

ndash050 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = H (t)

(a)

Conc

entr

atio

n

15

1

05

00 1 2 3 4 5

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = H (t)

(b)

005

004

003

002

001

0

ndash0010 1 2 3

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = t

(c)

02

015

01

005

00 1 2 3 4

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = t

(d)

Figure 2 Continued


005

004

003

002

001

0

ndash0010 1 2 3

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = sin t

(e)

02

015

01

005

00 1 2 3 4

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = sin t

(f )


04

03

02

01

0

ndash010 1 2 3

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = et

Conc

entr

atio

n

(a)

08

06

04

02

0

ndash020 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = etCo

ncen

trat

ion

(b)

004

0

006

002

ndash0020 1 2 3

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = tet

Conc

entr

atio

n

(c)

03

02

01

0

ndash010 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = tet

Conc

entr

atio

n

(d)

Figure 3 Continued


006

004

002

0

ndash0020 1 2 3

y

Sc = 14Sc = 096Sc = 022

Sc = 060

g (t) = et sintCo

ncen

trat

ion

(e)

03

02

01

00 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = et sint

Conc

entr

atio

n

(f )

001

8 times 10ndash3

6 times 10ndash3

4 times 10ndash3

2 times 10ndash3

0


0 1 2 3y

Sc = 14Sc = 096Sc = 022

Sc = 060

g (t) = t sint

Conc

entr

atio

n

(g)

0

008

006

004

002

ndash0020 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = t sintCo

ncen

trat

ion

(h)


Tem

pera

ture

15

1

05

0

ndash051 2 30

y

Pr = 70Pr = 50Pr = 071

Pr = 15

g (t) = H (t)

(a)

Tem

pera

ture

15

1

05

0

ndash051 2 30

y

Pr = 70Pr = 50Pr = 071

Pr = 15

g (t) = H (t)

(b)

Figure 4 Continued


0

008

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t

(c)

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t

(d)

0

008

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = sint

(e)

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = sint

(f )

05

04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et

(g)

Tem

pera

ture

1

08

06

04

02

00

y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et

(h)



0

008

01

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = tet

(a)

04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = tet

(b)

0

008

01

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et sint

(c)

04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et sint

(d)

Tem

pera

ture

0

002

0015

001

5 times 10ndash3


y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t sint

(e)

0

008

01

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t sint

(f )



Tem

pera

ture

15

1

05

00

y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = H (t)

(a)

Tem

pera

ture

12

1

08

06

04

02

00

y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = H (t)

(b)

0

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t

(c)

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t

(d)

0

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = sin t

(e)

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = sint

(f )

Figure 6 Continued


0y

S = 45S = 30S = 05

S = 15

05 1 15 2

05

04

03

02

01

0

Tem

pera

ture

g (t) = et

(g)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

Tem

pera

ture

1

08

06

04

02

0

g (t) = et

(h)


0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = tet

(a)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

04

03

02

01

0

ndash01

Tem

pera

ture

g (t) = tet

(b)

0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = et sint

(c)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

04

03

02

01

0

Tem

pera

ture

g (t) = et sint

(d)

Figure 7 Continued


Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

015

001

002

5 times 10ndash3

0

g (t) = t sint

(e)

0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t sint

(f )


0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = 1 g(t) = 1 h(t) = 1

(a)

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = t g(t) = 1 h(t) = 1

(b)

0 1 2 3 4 5y

Velo

city

2

15

1

05

0

M = 05M = 10

M = 15M = 25

f (t) = et g(t) = 1 h(t) = 1

(c)

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = sin t g(t) = 1 h(t) = 1

(d)

Figure 8 Continued






0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = tet g(t) = 1 h(t) = 1

(e)

0 1 2 3 4 5y

6

4

2

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = et sin t g(t) = 1 h(t) = 1

(f )

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = t sin t g(t) = 1 h(t) = 1

(g)



3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = sin t g (t) = 1 h (t) = 1

(c)

2

1

0

ndash1

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = et g (t) = 1 h (t) = 1

(d)

20 1 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

f (t) = tet g (t) = 1 h (t) = 1

(e)

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

5

4

3

2

1

0

Velo

city

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 9 Continued


2

1

0

ndash1

Velo

city

20 1 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


f (t) = 1 g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(a)

f (t) = t g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(b)

4

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

f (t) = sin t g (t) = 1 h (t) = 1

(c)

f (t) = et g (t) = 1 h (t) = 1

4

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(d)

Figure 10 Continued


f (t) = tet g (t) = 1 h (t) = 1

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

(e)

f (t) = et sin t g (t) = 1 h (t) = 1

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

5

4

3

2

1

0

Velo

city

(f )

f (t) = t sin t g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(g)


0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

4

3

2

1

0

Velo

city

f (t) = 1 g (t) = 1 h (t) = 1

(a)

GC = 80GC = 60GC = 20

GC = 40

0 1 2 3 54y

4

3

2

1

0

Velo

city

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 11 Continued


0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

4

3

2

1

0

Velo

city

f (t) = sin t g (t) = 1 h (t) = 1

(c)

GC = 80GC = 60GC = 20

GC = 40

0 1 2 3 54y

4

3

2

1

0

Velo

city

f (t) = et g (t) = 1 h (t) = 1

(d)

0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

f (t) = tet g (t) = 1 h (t) = 1

(e)

GC = 80GC = 60GC = 20

GC = 40

5

4

3

2

1

0

Velo


0 1 2 3 54y

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

f (t) = t sint g (t) = 1 h (t) = 1

(g)



3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = t g (t) = 1 h (t) = 1

(b)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = sin t g (t) = 1 h (t) = 1

(c)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = et g (t) = 1 h (t) = 1

(d)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

f (t) = tet g (t) = 1 h (t) = 1

(e)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

5

4

3

2

1

0

Velo

city

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 12 Continued


0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0f (t) = t sin t g (t) = 1 h (t) = 1

(g)


Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(a)

Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(b)

005

004

003

002

001

0

Conc

entr

atio

n

g (t) = t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(c)

Conc

entr

atio

n

02

015

01

005

0

g (t) = t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(d)

Figure 13 Continued


005

004

003

002

001

0

Conc

entr

atio

ng (t) = sin t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(e)

Conc

entr

atio

n

02

015

01

005

0

g (t) = sin t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(f )

03

04

02

01

0

Conc

entr

atio

n

g (t) = et

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(g)

Conc

entr

atio

n

08

06

04

02

0

g (t) = et

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(h)


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = tet

(a)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = tet

(b)

Figure 14 Continued


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = et sint

(c)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = et sint

(d)

001

8 times 10ndash3

6 times 10ndash3

4 times 10ndash3

2 times 10ndash3

0

Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = t sin t

(e)

0

008

006

004

002

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = t sin t

(f )


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = 1 g (t) = 1 h (t) = 1

(a)

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 15 Continued


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = sin t g (t) = 1 h (t) = 1

(c)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 105

0

ndash1 times 105

ndash2 times 105

ndash3 times 105

f (t) = et g (t) = 1 h (t) = 1

(d)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 106

0

ndash1 times 106

ndash2 times 106

ndash3 times 106

f (t) = tet g (t) = 1 h (t) = 1

(e)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

100

0

ndash100

ndash200

ndash300

ndash400

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t sin t g (t) = 1 h (t) = 1

(g)



3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = sin t g (t) = 1 h (t) = 1

(c)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et g (t) = 1 h (t) = 1

(d)

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = tet g (t) = 1 h (t) = 1

(e)

5

4

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 16 Continued


3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


3

2

1

00 1 2 3 54

y

M = 20M = 15M = 05

M = 10


Vel

ocity

(a)

1

0

ndash1

ndash2

ndash30 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

f (t) = 1 g (t) = sin t h (t) = tet

Vel

ocity

(b)

2

0

ndash2

ndash40 1 2 3 54

y

Gc = 80Gc = 60Gc = 20

Gc = 40


Vel

ocity

(c)

6

4

2

00 1 2 3 54

y

GT = 10GT = 60GT = 10

GT = 35

f (t) = 1 g (t) = sin t h (t) = t

Vel

ocity

(d)

Figure 17 Continued


2

1

0

ndash1

ndash2

ndash30 1 2 3 54

y

Sc = 078Sc = 060Sc = 022

Sc = 030


eloc

ity

(e)

3

2

1

0

ndash10 1 2 3 54

y

S = 40S = 30S = 10

S = 20

f (t) = t g (t) = et h (t) = 1

Vel

ocity

(f )


04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(a)

06

04

02

0

ndash02

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(b)



6 Conclusions






04

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(a)

06

04

02

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(b)


08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(a)

1

08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(b)







Appendix

B1(y t)

0 0lt tltyλ

eminus α1t

I0 iα3

t2

minus λy2

1113969

1113874 1113875 tgtyλ

⎧⎪⎪⎨

⎪⎪⎩(A1)

BII2 (t) e



I0 iα3t( 11138571113872 1113873lowastt (A2)

BII3 (t) (1 + λH(t))

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BII4 (t) (1 + λH(t)

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIII2 (t) e






Alowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste




Dlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIV2 (t) e


lowastH(t) + α1( 1113857

lowaste

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t (A8)

BIV3 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BIV4 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste

minus α1tI0 iα3t( 1113857 (A10)

BV2 (t) e


lowaste

t+ 1 + α1( 1113857te

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowastte

t (A11)

BV3 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A12)

BV4 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A13)

BVI2 (t) e


lowaste

t cos t + 1 + α1( 1113857et sin t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t sin t(A14)

BVI3 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A15)


BVI4 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A16)

BVII2 (t) e



minus α1tI0 iα3t( 11138571113872 1113873

lowastt sin t

(A17)



lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A18)



lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A19)

Lminus 1 e

minus bq2minus a2

radic

q2

minus a2

1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

0 0lt tlt b bgt 0

I0 a

t2

minus b2

1113969

1113888 1113889 tgt bRe(q)gtRe(a)

⎧⎪⎪⎨

⎪⎪⎩(A20)

Lminus 1


(A21)


q + a

radic)

q minus b1113890 1113891 e


Φ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A23)


q + a

radic)

q minus b1113890 1113891 e


Ψ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A25)

Data Availability




Acknowledgments


References































f (t) = 1 g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

15

1

05

0

(a)

f (t) = t g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

2

15

1

05

0

(b)

f (t) = sint g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

15

1

05

0

(c)

f (t) = et g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

15

1

05

0

(d)

f (t) = tet g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

(e)

f (t) = et sint g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

5

4

3

2

1

0

Velo

city

(f )

Figure 1 Continued


f (t) = t sint g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

15

1

05

0

(g)


Conc

entr

atio

n

15

1

05

0

ndash050 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = H (t)

(a)

Conc

entr

atio

n

15

1

05

00 1 2 3 4 5

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = H (t)

(b)

005

004

003

002

001

0

ndash0010 1 2 3

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = t

(c)

02

015

01

005

00 1 2 3 4

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = t

(d)

Figure 2 Continued


005

004

003

002

001

0

ndash0010 1 2 3

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = sin t

(e)

02

015

01

005

00 1 2 3 4

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = sin t

(f )


04

03

02

01

0

ndash010 1 2 3

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = et

Conc

entr

atio

n

(a)

08

06

04

02

0

ndash020 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = etCo

ncen

trat

ion

(b)

004

0

006

002

ndash0020 1 2 3

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = tet

Conc

entr

atio

n

(c)

03

02

01

0

ndash010 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = tet

Conc

entr

atio

n

(d)

Figure 3 Continued


006

004

002

0

ndash0020 1 2 3

y

Sc = 14Sc = 096Sc = 022

Sc = 060

g (t) = et sintCo

ncen

trat

ion

(e)

03

02

01

00 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = et sint

Conc

entr

atio

n

(f )

001

8 times 10ndash3

6 times 10ndash3

4 times 10ndash3

2 times 10ndash3

0


0 1 2 3y

Sc = 14Sc = 096Sc = 022

Sc = 060

g (t) = t sint

Conc

entr

atio

n

(g)

0

008

006

004

002

ndash0020 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = t sintCo

ncen

trat

ion

(h)


Tem

pera

ture

15

1

05

0

ndash051 2 30

y

Pr = 70Pr = 50Pr = 071

Pr = 15

g (t) = H (t)

(a)

Tem

pera

ture

15

1

05

0

ndash051 2 30

y

Pr = 70Pr = 50Pr = 071

Pr = 15

g (t) = H (t)

(b)

Figure 4 Continued


0

008

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t

(c)

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t

(d)

0

008

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = sint

(e)

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = sint

(f )

05

04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et

(g)

Tem

pera

ture

1

08

06

04

02

00

y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et

(h)



0

008

01

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = tet

(a)

04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = tet

(b)

0

008

01

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et sint

(c)

04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et sint

(d)

Tem

pera

ture

0

002

0015

001

5 times 10ndash3


y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t sint

(e)

0

008

01

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t sint

(f )



Tem

pera

ture

15

1

05

00

y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = H (t)

(a)

Tem

pera

ture

12

1

08

06

04

02

00

y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = H (t)

(b)

0

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t

(c)

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t

(d)

0

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = sin t

(e)

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = sint

(f )

Figure 6 Continued


0y

S = 45S = 30S = 05

S = 15

05 1 15 2

05

04

03

02

01

0

Tem

pera

ture

g (t) = et

(g)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

Tem

pera

ture

1

08

06

04

02

0

g (t) = et

(h)


0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = tet

(a)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

04

03

02

01

0

ndash01

Tem

pera

ture

g (t) = tet

(b)

0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = et sint

(c)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

04

03

02

01

0

Tem

pera

ture

g (t) = et sint

(d)

Figure 7 Continued


Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

015

001

002

5 times 10ndash3

0

g (t) = t sint

(e)

0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t sint

(f )


0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = 1 g(t) = 1 h(t) = 1

(a)

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = t g(t) = 1 h(t) = 1

(b)

0 1 2 3 4 5y

Velo

city

2

15

1

05

0

M = 05M = 10

M = 15M = 25

f (t) = et g(t) = 1 h(t) = 1

(c)

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = sin t g(t) = 1 h(t) = 1

(d)

Figure 8 Continued






0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = tet g(t) = 1 h(t) = 1

(e)

0 1 2 3 4 5y

6

4

2

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = et sin t g(t) = 1 h(t) = 1

(f )

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = t sin t g(t) = 1 h(t) = 1

(g)



3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = sin t g (t) = 1 h (t) = 1

(c)

2

1

0

ndash1

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = et g (t) = 1 h (t) = 1

(d)

20 1 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

f (t) = tet g (t) = 1 h (t) = 1

(e)

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

5

4

3

2

1

0

Velo

city

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 9 Continued


2

1

0

ndash1

Velo

city

20 1 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


f (t) = 1 g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(a)

f (t) = t g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(b)

4

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

f (t) = sin t g (t) = 1 h (t) = 1

(c)

f (t) = et g (t) = 1 h (t) = 1

4

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(d)

Figure 10 Continued


f (t) = tet g (t) = 1 h (t) = 1

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

(e)

f (t) = et sin t g (t) = 1 h (t) = 1

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

5

4

3

2

1

0

Velo

city

(f )

f (t) = t sin t g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(g)


0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

4

3

2

1

0

Velo

city

f (t) = 1 g (t) = 1 h (t) = 1

(a)

GC = 80GC = 60GC = 20

GC = 40

0 1 2 3 54y

4

3

2

1

0

Velo

city

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 11 Continued


0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

4

3

2

1

0

Velo

city

f (t) = sin t g (t) = 1 h (t) = 1

(c)

GC = 80GC = 60GC = 20

GC = 40

0 1 2 3 54y

4

3

2

1

0

Velo

city

f (t) = et g (t) = 1 h (t) = 1

(d)

0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

f (t) = tet g (t) = 1 h (t) = 1

(e)

GC = 80GC = 60GC = 20

GC = 40

5

4

3

2

1

0

Velo


0 1 2 3 54y

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

f (t) = t sint g (t) = 1 h (t) = 1

(g)



3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = t g (t) = 1 h (t) = 1

(b)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = sin t g (t) = 1 h (t) = 1

(c)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = et g (t) = 1 h (t) = 1

(d)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

f (t) = tet g (t) = 1 h (t) = 1

(e)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

5

4

3

2

1

0

Velo

city

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 12 Continued


0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0f (t) = t sin t g (t) = 1 h (t) = 1

(g)


Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(a)

Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(b)

005

004

003

002

001

0

Conc

entr

atio

n

g (t) = t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(c)

Conc

entr

atio

n

02

015

01

005

0

g (t) = t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(d)

Figure 13 Continued


005

004

003

002

001

0

Conc

entr

atio

ng (t) = sin t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(e)

Conc

entr

atio

n

02

015

01

005

0

g (t) = sin t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(f )

03

04

02

01

0

Conc

entr

atio

n

g (t) = et

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(g)

Conc

entr

atio

n

08

06

04

02

0

g (t) = et

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(h)


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = tet

(a)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = tet

(b)

Figure 14 Continued


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = et sint

(c)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = et sint

(d)

001

8 times 10ndash3

6 times 10ndash3

4 times 10ndash3

2 times 10ndash3

0

Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = t sin t

(e)

0

008

006

004

002

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = t sin t

(f )


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = 1 g (t) = 1 h (t) = 1

(a)

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 15 Continued


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = sin t g (t) = 1 h (t) = 1

(c)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 105

0

ndash1 times 105

ndash2 times 105

ndash3 times 105

f (t) = et g (t) = 1 h (t) = 1

(d)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 106

0

ndash1 times 106

ndash2 times 106

ndash3 times 106

f (t) = tet g (t) = 1 h (t) = 1

(e)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

100

0

ndash100

ndash200

ndash300

ndash400

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t sin t g (t) = 1 h (t) = 1

(g)



3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = sin t g (t) = 1 h (t) = 1

(c)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et g (t) = 1 h (t) = 1

(d)

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = tet g (t) = 1 h (t) = 1

(e)

5

4

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 16 Continued


3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


3

2

1

00 1 2 3 54

y

M = 20M = 15M = 05

M = 10


Vel

ocity

(a)

1

0

ndash1

ndash2

ndash30 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

f (t) = 1 g (t) = sin t h (t) = tet

Vel

ocity

(b)

2

0

ndash2

ndash40 1 2 3 54

y

Gc = 80Gc = 60Gc = 20

Gc = 40


Vel

ocity

(c)

6

4

2

00 1 2 3 54

y

GT = 10GT = 60GT = 10

GT = 35

f (t) = 1 g (t) = sin t h (t) = t

Vel

ocity

(d)

Figure 17 Continued


2

1

0

ndash1

ndash2

ndash30 1 2 3 54

y

Sc = 078Sc = 060Sc = 022

Sc = 030


eloc

ity

(e)

3

2

1

0

ndash10 1 2 3 54

y

S = 40S = 30S = 10

S = 20

f (t) = t g (t) = et h (t) = 1

Vel

ocity

(f )


04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(a)

06

04

02

0

ndash02

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(b)



6 Conclusions






04

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(a)

06

04

02

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(b)


08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(a)

1

08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(b)







Appendix

B1(y t)

0 0lt tltyλ

eminus α1t

I0 iα3

t2

minus λy2

1113969

1113874 1113875 tgtyλ

⎧⎪⎪⎨

⎪⎪⎩(A1)

BII2 (t) e



I0 iα3t( 11138571113872 1113873lowastt (A2)

BII3 (t) (1 + λH(t))

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BII4 (t) (1 + λH(t)

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIII2 (t) e






Alowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste




Dlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIV2 (t) e


lowastH(t) + α1( 1113857

lowaste

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t (A8)

BIV3 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BIV4 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste

minus α1tI0 iα3t( 1113857 (A10)

BV2 (t) e


lowaste

t+ 1 + α1( 1113857te

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowastte

t (A11)

BV3 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A12)

BV4 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A13)

BVI2 (t) e


lowaste

t cos t + 1 + α1( 1113857et sin t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t sin t(A14)

BVI3 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A15)


BVI4 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A16)

BVII2 (t) e



minus α1tI0 iα3t( 11138571113872 1113873

lowastt sin t

(A17)



lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A18)



lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A19)

Lminus 1 e

minus bq2minus a2

radic

q2

minus a2

1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

0 0lt tlt b bgt 0

I0 a

t2

minus b2

1113969

1113888 1113889 tgt bRe(q)gtRe(a)

⎧⎪⎪⎨

⎪⎪⎩(A20)

Lminus 1


(A21)


q + a

radic)

q minus b1113890 1113891 e


Φ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A23)


q + a

radic)

q minus b1113890 1113891 e


Ψ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A25)

Data Availability




Acknowledgments


References































f (t) = t sint g (t) = 1 h (t) = 1

0 1 2 3 54y

Sc = 250Sc = 196Sc = 022

Sc = 085

Velo

city

15

1

05

0

(g)


Conc

entr

atio

n

15

1

05

0

ndash050 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = H (t)

(a)

Conc

entr

atio

n

15

1

05

00 1 2 3 4 5

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = H (t)

(b)

005

004

003

002

001

0

ndash0010 1 2 3

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = t

(c)

02

015

01

005

00 1 2 3 4

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = t

(d)

Figure 2 Continued


005

004

003

002

001

0

ndash0010 1 2 3

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = sin t

(e)

02

015

01

005

00 1 2 3 4

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = sin t

(f )


04

03

02

01

0

ndash010 1 2 3

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = et

Conc

entr

atio

n

(a)

08

06

04

02

0

ndash020 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = etCo

ncen

trat

ion

(b)

004

0

006

002

ndash0020 1 2 3

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = tet

Conc

entr

atio

n

(c)

03

02

01

0

ndash010 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = tet

Conc

entr

atio

n

(d)

Figure 3 Continued


006

004

002

0

ndash0020 1 2 3

y

Sc = 14Sc = 096Sc = 022

Sc = 060

g (t) = et sintCo

ncen

trat

ion

(e)

03

02

01

00 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = et sint

Conc

entr

atio

n

(f )

001

8 times 10ndash3

6 times 10ndash3

4 times 10ndash3

2 times 10ndash3

0


0 1 2 3y

Sc = 14Sc = 096Sc = 022

Sc = 060

g (t) = t sint

Conc

entr

atio

n

(g)

0

008

006

004

002

ndash0020 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = t sintCo

ncen

trat

ion

(h)


Tem

pera

ture

15

1

05

0

ndash051 2 30

y

Pr = 70Pr = 50Pr = 071

Pr = 15

g (t) = H (t)

(a)

Tem

pera

ture

15

1

05

0

ndash051 2 30

y

Pr = 70Pr = 50Pr = 071

Pr = 15

g (t) = H (t)

(b)

Figure 4 Continued


0

008

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t

(c)

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t

(d)

0

008

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = sint

(e)

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = sint

(f )

05

04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et

(g)

Tem

pera

ture

1

08

06

04

02

00

y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et

(h)



0

008

01

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = tet

(a)

04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = tet

(b)

0

008

01

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et sint

(c)

04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et sint

(d)

Tem

pera

ture

0

002

0015

001

5 times 10ndash3


y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t sint

(e)

0

008

01

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t sint

(f )



Tem

pera

ture

15

1

05

00

y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = H (t)

(a)

Tem

pera

ture

12

1

08

06

04

02

00

y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = H (t)

(b)

0

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t

(c)

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t

(d)

0

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = sin t

(e)

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = sint

(f )

Figure 6 Continued


0y

S = 45S = 30S = 05

S = 15

05 1 15 2

05

04

03

02

01

0

Tem

pera

ture

g (t) = et

(g)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

Tem

pera

ture

1

08

06

04

02

0

g (t) = et

(h)


0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = tet

(a)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

04

03

02

01

0

ndash01

Tem

pera

ture

g (t) = tet

(b)

0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = et sint

(c)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

04

03

02

01

0

Tem

pera

ture

g (t) = et sint

(d)

Figure 7 Continued


Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

015

001

002

5 times 10ndash3

0

g (t) = t sint

(e)

0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t sint

(f )


0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = 1 g(t) = 1 h(t) = 1

(a)

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = t g(t) = 1 h(t) = 1

(b)

0 1 2 3 4 5y

Velo

city

2

15

1

05

0

M = 05M = 10

M = 15M = 25

f (t) = et g(t) = 1 h(t) = 1

(c)

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = sin t g(t) = 1 h(t) = 1

(d)

Figure 8 Continued






0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = tet g(t) = 1 h(t) = 1

(e)

0 1 2 3 4 5y

6

4

2

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = et sin t g(t) = 1 h(t) = 1

(f )

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = t sin t g(t) = 1 h(t) = 1

(g)



3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = sin t g (t) = 1 h (t) = 1

(c)

2

1

0

ndash1

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = et g (t) = 1 h (t) = 1

(d)

20 1 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

f (t) = tet g (t) = 1 h (t) = 1

(e)

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

5

4

3

2

1

0

Velo

city

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 9 Continued


2

1

0

ndash1

Velo

city

20 1 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


f (t) = 1 g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(a)

f (t) = t g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(b)

4

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

f (t) = sin t g (t) = 1 h (t) = 1

(c)

f (t) = et g (t) = 1 h (t) = 1

4

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(d)

Figure 10 Continued


f (t) = tet g (t) = 1 h (t) = 1

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

(e)

f (t) = et sin t g (t) = 1 h (t) = 1

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

5

4

3

2

1

0

Velo

city

(f )

f (t) = t sin t g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(g)


0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

4

3

2

1

0

Velo

city

f (t) = 1 g (t) = 1 h (t) = 1

(a)

GC = 80GC = 60GC = 20

GC = 40

0 1 2 3 54y

4

3

2

1

0

Velo

city

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 11 Continued


0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

4

3

2

1

0

Velo

city

f (t) = sin t g (t) = 1 h (t) = 1

(c)

GC = 80GC = 60GC = 20

GC = 40

0 1 2 3 54y

4

3

2

1

0

Velo

city

f (t) = et g (t) = 1 h (t) = 1

(d)

0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

f (t) = tet g (t) = 1 h (t) = 1

(e)

GC = 80GC = 60GC = 20

GC = 40

5

4

3

2

1

0

Velo


0 1 2 3 54y

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

f (t) = t sint g (t) = 1 h (t) = 1

(g)



3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = t g (t) = 1 h (t) = 1

(b)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = sin t g (t) = 1 h (t) = 1

(c)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = et g (t) = 1 h (t) = 1

(d)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

f (t) = tet g (t) = 1 h (t) = 1

(e)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

5

4

3

2

1

0

Velo

city

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 12 Continued


0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0f (t) = t sin t g (t) = 1 h (t) = 1

(g)


Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(a)

Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(b)

005

004

003

002

001

0

Conc

entr

atio

n

g (t) = t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(c)

Conc

entr

atio

n

02

015

01

005

0

g (t) = t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(d)

Figure 13 Continued


005

004

003

002

001

0

Conc

entr

atio

ng (t) = sin t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(e)

Conc

entr

atio

n

02

015

01

005

0

g (t) = sin t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(f )

03

04

02

01

0

Conc

entr

atio

n

g (t) = et

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(g)

Conc

entr

atio

n

08

06

04

02

0

g (t) = et

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(h)


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = tet

(a)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = tet

(b)

Figure 14 Continued


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = et sint

(c)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = et sint

(d)

001

8 times 10ndash3

6 times 10ndash3

4 times 10ndash3

2 times 10ndash3

0

Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = t sin t

(e)

0

008

006

004

002

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = t sin t

(f )


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = 1 g (t) = 1 h (t) = 1

(a)

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 15 Continued


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = sin t g (t) = 1 h (t) = 1

(c)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 105

0

ndash1 times 105

ndash2 times 105

ndash3 times 105

f (t) = et g (t) = 1 h (t) = 1

(d)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 106

0

ndash1 times 106

ndash2 times 106

ndash3 times 106

f (t) = tet g (t) = 1 h (t) = 1

(e)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

100

0

ndash100

ndash200

ndash300

ndash400

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t sin t g (t) = 1 h (t) = 1

(g)



3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = sin t g (t) = 1 h (t) = 1

(c)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et g (t) = 1 h (t) = 1

(d)

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = tet g (t) = 1 h (t) = 1

(e)

5

4

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 16 Continued


3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


3

2

1

00 1 2 3 54

y

M = 20M = 15M = 05

M = 10


Vel

ocity

(a)

1

0

ndash1

ndash2

ndash30 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

f (t) = 1 g (t) = sin t h (t) = tet

Vel

ocity

(b)

2

0

ndash2

ndash40 1 2 3 54

y

Gc = 80Gc = 60Gc = 20

Gc = 40


Vel

ocity

(c)

6

4

2

00 1 2 3 54

y

GT = 10GT = 60GT = 10

GT = 35

f (t) = 1 g (t) = sin t h (t) = t

Vel

ocity

(d)

Figure 17 Continued


2

1

0

ndash1

ndash2

ndash30 1 2 3 54

y

Sc = 078Sc = 060Sc = 022

Sc = 030


eloc

ity

(e)

3

2

1

0

ndash10 1 2 3 54

y

S = 40S = 30S = 10

S = 20

f (t) = t g (t) = et h (t) = 1

Vel

ocity

(f )


04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(a)

06

04

02

0

ndash02

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(b)



6 Conclusions






04

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(a)

06

04

02

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(b)


08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(a)

1

08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(b)







Appendix

B1(y t)

0 0lt tltyλ

eminus α1t

I0 iα3

t2

minus λy2

1113969

1113874 1113875 tgtyλ

⎧⎪⎪⎨

⎪⎪⎩(A1)

BII2 (t) e



I0 iα3t( 11138571113872 1113873lowastt (A2)

BII3 (t) (1 + λH(t))

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BII4 (t) (1 + λH(t)

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIII2 (t) e






Alowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste




Dlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIV2 (t) e


lowastH(t) + α1( 1113857

lowaste

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t (A8)

BIV3 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BIV4 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste

minus α1tI0 iα3t( 1113857 (A10)

BV2 (t) e


lowaste

t+ 1 + α1( 1113857te

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowastte

t (A11)

BV3 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A12)

BV4 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A13)

BVI2 (t) e


lowaste

t cos t + 1 + α1( 1113857et sin t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t sin t(A14)

BVI3 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A15)


BVI4 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A16)

BVII2 (t) e



minus α1tI0 iα3t( 11138571113872 1113873

lowastt sin t

(A17)



lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A18)



lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A19)

Lminus 1 e

minus bq2minus a2

radic

q2

minus a2

1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

0 0lt tlt b bgt 0

I0 a

t2

minus b2

1113969

1113888 1113889 tgt bRe(q)gtRe(a)

⎧⎪⎪⎨

⎪⎪⎩(A20)

Lminus 1


(A21)


q + a

radic)

q minus b1113890 1113891 e


Φ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A23)


q + a

radic)

q minus b1113890 1113891 e


Ψ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A25)

Data Availability




Acknowledgments


References































005

004

003

002

001

0

ndash0010 1 2 3

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = sin t

(e)

02

015

01

005

00 1 2 3 4

Conc

entr

atio

n

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = sin t

(f )


04

03

02

01

0

ndash010 1 2 3

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = et

Conc

entr

atio

n

(a)

08

06

04

02

0

ndash020 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = etCo

ncen

trat

ion

(b)

004

0

006

002

ndash0020 1 2 3

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = tet

Conc

entr

atio

n

(c)

03

02

01

0

ndash010 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = tet

Conc

entr

atio

n

(d)

Figure 3 Continued


006

004

002

0

ndash0020 1 2 3

y

Sc = 14Sc = 096Sc = 022

Sc = 060

g (t) = et sintCo

ncen

trat

ion

(e)

03

02

01

00 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = et sint

Conc

entr

atio

n

(f )

001

8 times 10ndash3

6 times 10ndash3

4 times 10ndash3

2 times 10ndash3

0


0 1 2 3y

Sc = 14Sc = 096Sc = 022

Sc = 060

g (t) = t sint

Conc

entr

atio

n

(g)

0

008

006

004

002

ndash0020 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = t sintCo

ncen

trat

ion

(h)


Tem

pera

ture

15

1

05

0

ndash051 2 30

y

Pr = 70Pr = 50Pr = 071

Pr = 15

g (t) = H (t)

(a)

Tem

pera

ture

15

1

05

0

ndash051 2 30

y

Pr = 70Pr = 50Pr = 071

Pr = 15

g (t) = H (t)

(b)

Figure 4 Continued


0

008

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t

(c)

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t

(d)

0

008

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = sint

(e)

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = sint

(f )

05

04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et

(g)

Tem

pera

ture

1

08

06

04

02

00

y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et

(h)



0

008

01

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = tet

(a)

04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = tet

(b)

0

008

01

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et sint

(c)

04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et sint

(d)

Tem

pera

ture

0

002

0015

001

5 times 10ndash3


y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t sint

(e)

0

008

01

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t sint

(f )



Tem

pera

ture

15

1

05

00

y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = H (t)

(a)

Tem

pera

ture

12

1

08

06

04

02

00

y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = H (t)

(b)

0

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t

(c)

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t

(d)

0

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = sin t

(e)

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = sint

(f )

Figure 6 Continued


0y

S = 45S = 30S = 05

S = 15

05 1 15 2

05

04

03

02

01

0

Tem

pera

ture

g (t) = et

(g)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

Tem

pera

ture

1

08

06

04

02

0

g (t) = et

(h)


0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = tet

(a)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

04

03

02

01

0

ndash01

Tem

pera

ture

g (t) = tet

(b)

0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = et sint

(c)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

04

03

02

01

0

Tem

pera

ture

g (t) = et sint

(d)

Figure 7 Continued


Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

015

001

002

5 times 10ndash3

0

g (t) = t sint

(e)

0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t sint

(f )


0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = 1 g(t) = 1 h(t) = 1

(a)

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = t g(t) = 1 h(t) = 1

(b)

0 1 2 3 4 5y

Velo

city

2

15

1

05

0

M = 05M = 10

M = 15M = 25

f (t) = et g(t) = 1 h(t) = 1

(c)

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = sin t g(t) = 1 h(t) = 1

(d)

Figure 8 Continued






0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = tet g(t) = 1 h(t) = 1

(e)

0 1 2 3 4 5y

6

4

2

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = et sin t g(t) = 1 h(t) = 1

(f )

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = t sin t g(t) = 1 h(t) = 1

(g)



3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = sin t g (t) = 1 h (t) = 1

(c)

2

1

0

ndash1

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = et g (t) = 1 h (t) = 1

(d)

20 1 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

f (t) = tet g (t) = 1 h (t) = 1

(e)

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

5

4

3

2

1

0

Velo

city

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 9 Continued


2

1

0

ndash1

Velo

city

20 1 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


f (t) = 1 g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(a)

f (t) = t g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(b)

4

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

f (t) = sin t g (t) = 1 h (t) = 1

(c)

f (t) = et g (t) = 1 h (t) = 1

4

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(d)

Figure 10 Continued


f (t) = tet g (t) = 1 h (t) = 1

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

(e)

f (t) = et sin t g (t) = 1 h (t) = 1

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

5

4

3

2

1

0

Velo

city

(f )

f (t) = t sin t g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(g)


0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

4

3

2

1

0

Velo

city

f (t) = 1 g (t) = 1 h (t) = 1

(a)

GC = 80GC = 60GC = 20

GC = 40

0 1 2 3 54y

4

3

2

1

0

Velo

city

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 11 Continued


0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

4

3

2

1

0

Velo

city

f (t) = sin t g (t) = 1 h (t) = 1

(c)

GC = 80GC = 60GC = 20

GC = 40

0 1 2 3 54y

4

3

2

1

0

Velo

city

f (t) = et g (t) = 1 h (t) = 1

(d)

0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

f (t) = tet g (t) = 1 h (t) = 1

(e)

GC = 80GC = 60GC = 20

GC = 40

5

4

3

2

1

0

Velo


0 1 2 3 54y

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

f (t) = t sint g (t) = 1 h (t) = 1

(g)



3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = t g (t) = 1 h (t) = 1

(b)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = sin t g (t) = 1 h (t) = 1

(c)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = et g (t) = 1 h (t) = 1

(d)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

f (t) = tet g (t) = 1 h (t) = 1

(e)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

5

4

3

2

1

0

Velo

city

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 12 Continued


0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0f (t) = t sin t g (t) = 1 h (t) = 1

(g)


Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(a)

Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(b)

005

004

003

002

001

0

Conc

entr

atio

n

g (t) = t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(c)

Conc

entr

atio

n

02

015

01

005

0

g (t) = t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(d)

Figure 13 Continued


005

004

003

002

001

0

Conc

entr

atio

ng (t) = sin t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(e)

Conc

entr

atio

n

02

015

01

005

0

g (t) = sin t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(f )

03

04

02

01

0

Conc

entr

atio

n

g (t) = et

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(g)

Conc

entr

atio

n

08

06

04

02

0

g (t) = et

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(h)


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = tet

(a)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = tet

(b)

Figure 14 Continued


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = et sint

(c)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = et sint

(d)

001

8 times 10ndash3

6 times 10ndash3

4 times 10ndash3

2 times 10ndash3

0

Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = t sin t

(e)

0

008

006

004

002

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = t sin t

(f )


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = 1 g (t) = 1 h (t) = 1

(a)

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 15 Continued


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = sin t g (t) = 1 h (t) = 1

(c)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 105

0

ndash1 times 105

ndash2 times 105

ndash3 times 105

f (t) = et g (t) = 1 h (t) = 1

(d)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 106

0

ndash1 times 106

ndash2 times 106

ndash3 times 106

f (t) = tet g (t) = 1 h (t) = 1

(e)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

100

0

ndash100

ndash200

ndash300

ndash400

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t sin t g (t) = 1 h (t) = 1

(g)



3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = sin t g (t) = 1 h (t) = 1

(c)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et g (t) = 1 h (t) = 1

(d)

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = tet g (t) = 1 h (t) = 1

(e)

5

4

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 16 Continued


3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


3

2

1

00 1 2 3 54

y

M = 20M = 15M = 05

M = 10


Vel

ocity

(a)

1

0

ndash1

ndash2

ndash30 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

f (t) = 1 g (t) = sin t h (t) = tet

Vel

ocity

(b)

2

0

ndash2

ndash40 1 2 3 54

y

Gc = 80Gc = 60Gc = 20

Gc = 40


Vel

ocity

(c)

6

4

2

00 1 2 3 54

y

GT = 10GT = 60GT = 10

GT = 35

f (t) = 1 g (t) = sin t h (t) = t

Vel

ocity

(d)

Figure 17 Continued


2

1

0

ndash1

ndash2

ndash30 1 2 3 54

y

Sc = 078Sc = 060Sc = 022

Sc = 030


eloc

ity

(e)

3

2

1

0

ndash10 1 2 3 54

y

S = 40S = 30S = 10

S = 20

f (t) = t g (t) = et h (t) = 1

Vel

ocity

(f )


04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(a)

06

04

02

0

ndash02

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(b)



6 Conclusions






04

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(a)

06

04

02

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(b)


08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(a)

1

08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(b)







Appendix

B1(y t)

0 0lt tltyλ

eminus α1t

I0 iα3

t2

minus λy2

1113969

1113874 1113875 tgtyλ

⎧⎪⎪⎨

⎪⎪⎩(A1)

BII2 (t) e



I0 iα3t( 11138571113872 1113873lowastt (A2)

BII3 (t) (1 + λH(t))

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BII4 (t) (1 + λH(t)

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIII2 (t) e






Alowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste




Dlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIV2 (t) e


lowastH(t) + α1( 1113857

lowaste

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t (A8)

BIV3 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BIV4 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste

minus α1tI0 iα3t( 1113857 (A10)

BV2 (t) e


lowaste

t+ 1 + α1( 1113857te

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowastte

t (A11)

BV3 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A12)

BV4 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A13)

BVI2 (t) e


lowaste

t cos t + 1 + α1( 1113857et sin t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t sin t(A14)

BVI3 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A15)


BVI4 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A16)

BVII2 (t) e



minus α1tI0 iα3t( 11138571113872 1113873

lowastt sin t

(A17)



lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A18)



lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A19)

Lminus 1 e

minus bq2minus a2

radic

q2

minus a2

1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

0 0lt tlt b bgt 0

I0 a

t2

minus b2

1113969

1113888 1113889 tgt bRe(q)gtRe(a)

⎧⎪⎪⎨

⎪⎪⎩(A20)

Lminus 1


(A21)


q + a

radic)

q minus b1113890 1113891 e


Φ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A23)


q + a

radic)

q minus b1113890 1113891 e


Ψ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A25)

Data Availability




Acknowledgments


References































006

004

002

0

ndash0020 1 2 3

y

Sc = 14Sc = 096Sc = 022

Sc = 060

g (t) = et sintCo

ncen

trat

ion

(e)

03

02

01

00 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = et sint

Conc

entr

atio

n

(f )

001

8 times 10ndash3

6 times 10ndash3

4 times 10ndash3

2 times 10ndash3

0


0 1 2 3y

Sc = 14Sc = 096Sc = 022

Sc = 060

g (t) = t sint

Conc

entr

atio

n

(g)

0

008

006

004

002

ndash0020 1 2 3 4

y

Sc = 022Sc = 060 Sc = 14

Sc = 096

g (t) = t sintCo

ncen

trat

ion

(h)


Tem

pera

ture

15

1

05

0

ndash051 2 30

y

Pr = 70Pr = 50Pr = 071

Pr = 15

g (t) = H (t)

(a)

Tem

pera

ture

15

1

05

0

ndash051 2 30

y

Pr = 70Pr = 50Pr = 071

Pr = 15

g (t) = H (t)

(b)

Figure 4 Continued


0

008

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t

(c)

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t

(d)

0

008

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = sint

(e)

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = sint

(f )

05

04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et

(g)

Tem

pera

ture

1

08

06

04

02

00

y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et

(h)



0

008

01

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = tet

(a)

04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = tet

(b)

0

008

01

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et sint

(c)

04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et sint

(d)

Tem

pera

ture

0

002

0015

001

5 times 10ndash3


y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t sint

(e)

0

008

01

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t sint

(f )



Tem

pera

ture

15

1

05

00

y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = H (t)

(a)

Tem

pera

ture

12

1

08

06

04

02

00

y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = H (t)

(b)

0

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t

(c)

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t

(d)

0

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = sin t

(e)

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = sint

(f )

Figure 6 Continued


0y

S = 45S = 30S = 05

S = 15

05 1 15 2

05

04

03

02

01

0

Tem

pera

ture

g (t) = et

(g)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

Tem

pera

ture

1

08

06

04

02

0

g (t) = et

(h)


0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = tet

(a)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

04

03

02

01

0

ndash01

Tem

pera

ture

g (t) = tet

(b)

0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = et sint

(c)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

04

03

02

01

0

Tem

pera

ture

g (t) = et sint

(d)

Figure 7 Continued


Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

015

001

002

5 times 10ndash3

0

g (t) = t sint

(e)

0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t sint

(f )


0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = 1 g(t) = 1 h(t) = 1

(a)

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = t g(t) = 1 h(t) = 1

(b)

0 1 2 3 4 5y

Velo

city

2

15

1

05

0

M = 05M = 10

M = 15M = 25

f (t) = et g(t) = 1 h(t) = 1

(c)

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = sin t g(t) = 1 h(t) = 1

(d)

Figure 8 Continued






0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = tet g(t) = 1 h(t) = 1

(e)

0 1 2 3 4 5y

6

4

2

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = et sin t g(t) = 1 h(t) = 1

(f )

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = t sin t g(t) = 1 h(t) = 1

(g)



3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = sin t g (t) = 1 h (t) = 1

(c)

2

1

0

ndash1

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = et g (t) = 1 h (t) = 1

(d)

20 1 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

f (t) = tet g (t) = 1 h (t) = 1

(e)

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

5

4

3

2

1

0

Velo

city

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 9 Continued


2

1

0

ndash1

Velo

city

20 1 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


f (t) = 1 g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(a)

f (t) = t g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(b)

4

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

f (t) = sin t g (t) = 1 h (t) = 1

(c)

f (t) = et g (t) = 1 h (t) = 1

4

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(d)

Figure 10 Continued


f (t) = tet g (t) = 1 h (t) = 1

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

(e)

f (t) = et sin t g (t) = 1 h (t) = 1

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

5

4

3

2

1

0

Velo

city

(f )

f (t) = t sin t g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(g)


0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

4

3

2

1

0

Velo

city

f (t) = 1 g (t) = 1 h (t) = 1

(a)

GC = 80GC = 60GC = 20

GC = 40

0 1 2 3 54y

4

3

2

1

0

Velo

city

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 11 Continued


0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

4

3

2

1

0

Velo

city

f (t) = sin t g (t) = 1 h (t) = 1

(c)

GC = 80GC = 60GC = 20

GC = 40

0 1 2 3 54y

4

3

2

1

0

Velo

city

f (t) = et g (t) = 1 h (t) = 1

(d)

0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

f (t) = tet g (t) = 1 h (t) = 1

(e)

GC = 80GC = 60GC = 20

GC = 40

5

4

3

2

1

0

Velo


0 1 2 3 54y

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

f (t) = t sint g (t) = 1 h (t) = 1

(g)



3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = t g (t) = 1 h (t) = 1

(b)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = sin t g (t) = 1 h (t) = 1

(c)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = et g (t) = 1 h (t) = 1

(d)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

f (t) = tet g (t) = 1 h (t) = 1

(e)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

5

4

3

2

1

0

Velo

city

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 12 Continued


0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0f (t) = t sin t g (t) = 1 h (t) = 1

(g)


Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(a)

Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(b)

005

004

003

002

001

0

Conc

entr

atio

n

g (t) = t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(c)

Conc

entr

atio

n

02

015

01

005

0

g (t) = t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(d)

Figure 13 Continued


005

004

003

002

001

0

Conc

entr

atio

ng (t) = sin t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(e)

Conc

entr

atio

n

02

015

01

005

0

g (t) = sin t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(f )

03

04

02

01

0

Conc

entr

atio

n

g (t) = et

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(g)

Conc

entr

atio

n

08

06

04

02

0

g (t) = et

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(h)


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = tet

(a)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = tet

(b)

Figure 14 Continued


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = et sint

(c)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = et sint

(d)

001

8 times 10ndash3

6 times 10ndash3

4 times 10ndash3

2 times 10ndash3

0

Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = t sin t

(e)

0

008

006

004

002

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = t sin t

(f )


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = 1 g (t) = 1 h (t) = 1

(a)

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 15 Continued


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = sin t g (t) = 1 h (t) = 1

(c)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 105

0

ndash1 times 105

ndash2 times 105

ndash3 times 105

f (t) = et g (t) = 1 h (t) = 1

(d)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 106

0

ndash1 times 106

ndash2 times 106

ndash3 times 106

f (t) = tet g (t) = 1 h (t) = 1

(e)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

100

0

ndash100

ndash200

ndash300

ndash400

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t sin t g (t) = 1 h (t) = 1

(g)



3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = sin t g (t) = 1 h (t) = 1

(c)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et g (t) = 1 h (t) = 1

(d)

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = tet g (t) = 1 h (t) = 1

(e)

5

4

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 16 Continued


3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


3

2

1

00 1 2 3 54

y

M = 20M = 15M = 05

M = 10


Vel

ocity

(a)

1

0

ndash1

ndash2

ndash30 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

f (t) = 1 g (t) = sin t h (t) = tet

Vel

ocity

(b)

2

0

ndash2

ndash40 1 2 3 54

y

Gc = 80Gc = 60Gc = 20

Gc = 40


Vel

ocity

(c)

6

4

2

00 1 2 3 54

y

GT = 10GT = 60GT = 10

GT = 35

f (t) = 1 g (t) = sin t h (t) = t

Vel

ocity

(d)

Figure 17 Continued


2

1

0

ndash1

ndash2

ndash30 1 2 3 54

y

Sc = 078Sc = 060Sc = 022

Sc = 030


eloc

ity

(e)

3

2

1

0

ndash10 1 2 3 54

y

S = 40S = 30S = 10

S = 20

f (t) = t g (t) = et h (t) = 1

Vel

ocity

(f )


04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(a)

06

04

02

0

ndash02

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(b)



6 Conclusions






04

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(a)

06

04

02

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(b)


08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(a)

1

08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(b)







Appendix

B1(y t)

0 0lt tltyλ

eminus α1t

I0 iα3

t2

minus λy2

1113969

1113874 1113875 tgtyλ

⎧⎪⎪⎨

⎪⎪⎩(A1)

BII2 (t) e



I0 iα3t( 11138571113872 1113873lowastt (A2)

BII3 (t) (1 + λH(t))

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BII4 (t) (1 + λH(t)

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIII2 (t) e






Alowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste




Dlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIV2 (t) e


lowastH(t) + α1( 1113857

lowaste

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t (A8)

BIV3 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BIV4 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste

minus α1tI0 iα3t( 1113857 (A10)

BV2 (t) e


lowaste

t+ 1 + α1( 1113857te

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowastte

t (A11)

BV3 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A12)

BV4 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A13)

BVI2 (t) e


lowaste

t cos t + 1 + α1( 1113857et sin t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t sin t(A14)

BVI3 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A15)


BVI4 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A16)

BVII2 (t) e



minus α1tI0 iα3t( 11138571113872 1113873

lowastt sin t

(A17)



lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A18)



lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A19)

Lminus 1 e

minus bq2minus a2

radic

q2

minus a2

1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

0 0lt tlt b bgt 0

I0 a

t2

minus b2

1113969

1113888 1113889 tgt bRe(q)gtRe(a)

⎧⎪⎪⎨

⎪⎪⎩(A20)

Lminus 1


(A21)


q + a

radic)

q minus b1113890 1113891 e


Φ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A23)


q + a

radic)

q minus b1113890 1113891 e


Ψ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A25)

Data Availability




Acknowledgments


References































0

008

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t

(c)

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t

(d)

0

008

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = sint

(e)

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = sint

(f )

05

04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et

(g)

Tem

pera

ture

1

08

06

04

02

00

y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et

(h)



0

008

01

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = tet

(a)

04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = tet

(b)

0

008

01

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et sint

(c)

04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et sint

(d)

Tem

pera

ture

0

002

0015

001

5 times 10ndash3


y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t sint

(e)

0

008

01

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t sint

(f )



Tem

pera

ture

15

1

05

00

y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = H (t)

(a)

Tem

pera

ture

12

1

08

06

04

02

00

y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = H (t)

(b)

0

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t

(c)

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t

(d)

0

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = sin t

(e)

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = sint

(f )

Figure 6 Continued


0y

S = 45S = 30S = 05

S = 15

05 1 15 2

05

04

03

02

01

0

Tem

pera

ture

g (t) = et

(g)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

Tem

pera

ture

1

08

06

04

02

0

g (t) = et

(h)


0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = tet

(a)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

04

03

02

01

0

ndash01

Tem

pera

ture

g (t) = tet

(b)

0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = et sint

(c)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

04

03

02

01

0

Tem

pera

ture

g (t) = et sint

(d)

Figure 7 Continued


Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

015

001

002

5 times 10ndash3

0

g (t) = t sint

(e)

0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t sint

(f )


0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = 1 g(t) = 1 h(t) = 1

(a)

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = t g(t) = 1 h(t) = 1

(b)

0 1 2 3 4 5y

Velo

city

2

15

1

05

0

M = 05M = 10

M = 15M = 25

f (t) = et g(t) = 1 h(t) = 1

(c)

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = sin t g(t) = 1 h(t) = 1

(d)

Figure 8 Continued






0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = tet g(t) = 1 h(t) = 1

(e)

0 1 2 3 4 5y

6

4

2

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = et sin t g(t) = 1 h(t) = 1

(f )

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = t sin t g(t) = 1 h(t) = 1

(g)



3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = sin t g (t) = 1 h (t) = 1

(c)

2

1

0

ndash1

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = et g (t) = 1 h (t) = 1

(d)

20 1 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

f (t) = tet g (t) = 1 h (t) = 1

(e)

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

5

4

3

2

1

0

Velo

city

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 9 Continued


2

1

0

ndash1

Velo

city

20 1 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


f (t) = 1 g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(a)

f (t) = t g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(b)

4

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

f (t) = sin t g (t) = 1 h (t) = 1

(c)

f (t) = et g (t) = 1 h (t) = 1

4

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(d)

Figure 10 Continued


f (t) = tet g (t) = 1 h (t) = 1

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

(e)

f (t) = et sin t g (t) = 1 h (t) = 1

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

5

4

3

2

1

0

Velo

city

(f )

f (t) = t sin t g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(g)


0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

4

3

2

1

0

Velo

city

f (t) = 1 g (t) = 1 h (t) = 1

(a)

GC = 80GC = 60GC = 20

GC = 40

0 1 2 3 54y

4

3

2

1

0

Velo

city

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 11 Continued


0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

4

3

2

1

0

Velo

city

f (t) = sin t g (t) = 1 h (t) = 1

(c)

GC = 80GC = 60GC = 20

GC = 40

0 1 2 3 54y

4

3

2

1

0

Velo

city

f (t) = et g (t) = 1 h (t) = 1

(d)

0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

f (t) = tet g (t) = 1 h (t) = 1

(e)

GC = 80GC = 60GC = 20

GC = 40

5

4

3

2

1

0

Velo


0 1 2 3 54y

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

f (t) = t sint g (t) = 1 h (t) = 1

(g)



3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = t g (t) = 1 h (t) = 1

(b)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = sin t g (t) = 1 h (t) = 1

(c)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = et g (t) = 1 h (t) = 1

(d)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

f (t) = tet g (t) = 1 h (t) = 1

(e)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

5

4

3

2

1

0

Velo

city

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 12 Continued


0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0f (t) = t sin t g (t) = 1 h (t) = 1

(g)


Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(a)

Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(b)

005

004

003

002

001

0

Conc

entr

atio

n

g (t) = t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(c)

Conc

entr

atio

n

02

015

01

005

0

g (t) = t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(d)

Figure 13 Continued


005

004

003

002

001

0

Conc

entr

atio

ng (t) = sin t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(e)

Conc

entr

atio

n

02

015

01

005

0

g (t) = sin t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(f )

03

04

02

01

0

Conc

entr

atio

n

g (t) = et

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(g)

Conc

entr

atio

n

08

06

04

02

0

g (t) = et

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(h)


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = tet

(a)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = tet

(b)

Figure 14 Continued


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = et sint

(c)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = et sint

(d)

001

8 times 10ndash3

6 times 10ndash3

4 times 10ndash3

2 times 10ndash3

0

Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = t sin t

(e)

0

008

006

004

002

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = t sin t

(f )


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = 1 g (t) = 1 h (t) = 1

(a)

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 15 Continued


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = sin t g (t) = 1 h (t) = 1

(c)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 105

0

ndash1 times 105

ndash2 times 105

ndash3 times 105

f (t) = et g (t) = 1 h (t) = 1

(d)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 106

0

ndash1 times 106

ndash2 times 106

ndash3 times 106

f (t) = tet g (t) = 1 h (t) = 1

(e)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

100

0

ndash100

ndash200

ndash300

ndash400

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t sin t g (t) = 1 h (t) = 1

(g)



3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = sin t g (t) = 1 h (t) = 1

(c)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et g (t) = 1 h (t) = 1

(d)

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = tet g (t) = 1 h (t) = 1

(e)

5

4

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 16 Continued


3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


3

2

1

00 1 2 3 54

y

M = 20M = 15M = 05

M = 10


Vel

ocity

(a)

1

0

ndash1

ndash2

ndash30 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

f (t) = 1 g (t) = sin t h (t) = tet

Vel

ocity

(b)

2

0

ndash2

ndash40 1 2 3 54

y

Gc = 80Gc = 60Gc = 20

Gc = 40


Vel

ocity

(c)

6

4

2

00 1 2 3 54

y

GT = 10GT = 60GT = 10

GT = 35

f (t) = 1 g (t) = sin t h (t) = t

Vel

ocity

(d)

Figure 17 Continued


2

1

0

ndash1

ndash2

ndash30 1 2 3 54

y

Sc = 078Sc = 060Sc = 022

Sc = 030


eloc

ity

(e)

3

2

1

0

ndash10 1 2 3 54

y

S = 40S = 30S = 10

S = 20

f (t) = t g (t) = et h (t) = 1

Vel

ocity

(f )


04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(a)

06

04

02

0

ndash02

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(b)



6 Conclusions






04

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(a)

06

04

02

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(b)


08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(a)

1

08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(b)







Appendix

B1(y t)

0 0lt tltyλ

eminus α1t

I0 iα3

t2

minus λy2

1113969

1113874 1113875 tgtyλ

⎧⎪⎪⎨

⎪⎪⎩(A1)

BII2 (t) e



I0 iα3t( 11138571113872 1113873lowastt (A2)

BII3 (t) (1 + λH(t))

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BII4 (t) (1 + λH(t)

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIII2 (t) e






Alowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste




Dlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIV2 (t) e


lowastH(t) + α1( 1113857

lowaste

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t (A8)

BIV3 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BIV4 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste

minus α1tI0 iα3t( 1113857 (A10)

BV2 (t) e


lowaste

t+ 1 + α1( 1113857te

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowastte

t (A11)

BV3 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A12)

BV4 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A13)

BVI2 (t) e


lowaste

t cos t + 1 + α1( 1113857et sin t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t sin t(A14)

BVI3 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A15)


BVI4 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A16)

BVII2 (t) e



minus α1tI0 iα3t( 11138571113872 1113873

lowastt sin t

(A17)



lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A18)



lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A19)

Lminus 1 e

minus bq2minus a2

radic

q2

minus a2

1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

0 0lt tlt b bgt 0

I0 a

t2

minus b2

1113969

1113888 1113889 tgt bRe(q)gtRe(a)

⎧⎪⎪⎨

⎪⎪⎩(A20)

Lminus 1


(A21)


q + a

radic)

q minus b1113890 1113891 e


Φ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A23)


q + a

radic)

q minus b1113890 1113891 e


Ψ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A25)

Data Availability




Acknowledgments


References































0

008

01

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = tet

(a)

04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = tet

(b)

0

008

01

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et sint

(c)

04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = et sint

(d)

Tem

pera

ture

0

002

0015

001

5 times 10ndash3


y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t sint

(e)

0

008

01

006

004

002

ndash002

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2

g (t) = t sint

(f )



Tem

pera

ture

15

1

05

00

y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = H (t)

(a)

Tem

pera

ture

12

1

08

06

04

02

00

y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = H (t)

(b)

0

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t

(c)

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t

(d)

0

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = sin t

(e)

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = sint

(f )

Figure 6 Continued


0y

S = 45S = 30S = 05

S = 15

05 1 15 2

05

04

03

02

01

0

Tem

pera

ture

g (t) = et

(g)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

Tem

pera

ture

1

08

06

04

02

0

g (t) = et

(h)


0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = tet

(a)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

04

03

02

01

0

ndash01

Tem

pera

ture

g (t) = tet

(b)

0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = et sint

(c)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

04

03

02

01

0

Tem

pera

ture

g (t) = et sint

(d)

Figure 7 Continued


Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

015

001

002

5 times 10ndash3

0

g (t) = t sint

(e)

0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t sint

(f )


0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = 1 g(t) = 1 h(t) = 1

(a)

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = t g(t) = 1 h(t) = 1

(b)

0 1 2 3 4 5y

Velo

city

2

15

1

05

0

M = 05M = 10

M = 15M = 25

f (t) = et g(t) = 1 h(t) = 1

(c)

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = sin t g(t) = 1 h(t) = 1

(d)

Figure 8 Continued






0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = tet g(t) = 1 h(t) = 1

(e)

0 1 2 3 4 5y

6

4

2

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = et sin t g(t) = 1 h(t) = 1

(f )

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = t sin t g(t) = 1 h(t) = 1

(g)



3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = sin t g (t) = 1 h (t) = 1

(c)

2

1

0

ndash1

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = et g (t) = 1 h (t) = 1

(d)

20 1 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

f (t) = tet g (t) = 1 h (t) = 1

(e)

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

5

4

3

2

1

0

Velo

city

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 9 Continued


2

1

0

ndash1

Velo

city

20 1 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


f (t) = 1 g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(a)

f (t) = t g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(b)

4

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

f (t) = sin t g (t) = 1 h (t) = 1

(c)

f (t) = et g (t) = 1 h (t) = 1

4

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(d)

Figure 10 Continued


f (t) = tet g (t) = 1 h (t) = 1

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

(e)

f (t) = et sin t g (t) = 1 h (t) = 1

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

5

4

3

2

1

0

Velo

city

(f )

f (t) = t sin t g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(g)


0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

4

3

2

1

0

Velo

city

f (t) = 1 g (t) = 1 h (t) = 1

(a)

GC = 80GC = 60GC = 20

GC = 40

0 1 2 3 54y

4

3

2

1

0

Velo

city

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 11 Continued


0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

4

3

2

1

0

Velo

city

f (t) = sin t g (t) = 1 h (t) = 1

(c)

GC = 80GC = 60GC = 20

GC = 40

0 1 2 3 54y

4

3

2

1

0

Velo

city

f (t) = et g (t) = 1 h (t) = 1

(d)

0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

f (t) = tet g (t) = 1 h (t) = 1

(e)

GC = 80GC = 60GC = 20

GC = 40

5

4

3

2

1

0

Velo


0 1 2 3 54y

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

f (t) = t sint g (t) = 1 h (t) = 1

(g)



3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = t g (t) = 1 h (t) = 1

(b)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = sin t g (t) = 1 h (t) = 1

(c)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = et g (t) = 1 h (t) = 1

(d)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

f (t) = tet g (t) = 1 h (t) = 1

(e)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

5

4

3

2

1

0

Velo

city

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 12 Continued


0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0f (t) = t sin t g (t) = 1 h (t) = 1

(g)


Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(a)

Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(b)

005

004

003

002

001

0

Conc

entr

atio

n

g (t) = t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(c)

Conc

entr

atio

n

02

015

01

005

0

g (t) = t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(d)

Figure 13 Continued


005

004

003

002

001

0

Conc

entr

atio

ng (t) = sin t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(e)

Conc

entr

atio

n

02

015

01

005

0

g (t) = sin t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(f )

03

04

02

01

0

Conc

entr

atio

n

g (t) = et

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(g)

Conc

entr

atio

n

08

06

04

02

0

g (t) = et

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(h)


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = tet

(a)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = tet

(b)

Figure 14 Continued


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = et sint

(c)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = et sint

(d)

001

8 times 10ndash3

6 times 10ndash3

4 times 10ndash3

2 times 10ndash3

0

Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = t sin t

(e)

0

008

006

004

002

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = t sin t

(f )


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = 1 g (t) = 1 h (t) = 1

(a)

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 15 Continued


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = sin t g (t) = 1 h (t) = 1

(c)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 105

0

ndash1 times 105

ndash2 times 105

ndash3 times 105

f (t) = et g (t) = 1 h (t) = 1

(d)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 106

0

ndash1 times 106

ndash2 times 106

ndash3 times 106

f (t) = tet g (t) = 1 h (t) = 1

(e)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

100

0

ndash100

ndash200

ndash300

ndash400

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t sin t g (t) = 1 h (t) = 1

(g)



3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = sin t g (t) = 1 h (t) = 1

(c)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et g (t) = 1 h (t) = 1

(d)

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = tet g (t) = 1 h (t) = 1

(e)

5

4

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 16 Continued


3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


3

2

1

00 1 2 3 54

y

M = 20M = 15M = 05

M = 10


Vel

ocity

(a)

1

0

ndash1

ndash2

ndash30 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

f (t) = 1 g (t) = sin t h (t) = tet

Vel

ocity

(b)

2

0

ndash2

ndash40 1 2 3 54

y

Gc = 80Gc = 60Gc = 20

Gc = 40


Vel

ocity

(c)

6

4

2

00 1 2 3 54

y

GT = 10GT = 60GT = 10

GT = 35

f (t) = 1 g (t) = sin t h (t) = t

Vel

ocity

(d)

Figure 17 Continued


2

1

0

ndash1

ndash2

ndash30 1 2 3 54

y

Sc = 078Sc = 060Sc = 022

Sc = 030


eloc

ity

(e)

3

2

1

0

ndash10 1 2 3 54

y

S = 40S = 30S = 10

S = 20

f (t) = t g (t) = et h (t) = 1

Vel

ocity

(f )


04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(a)

06

04

02

0

ndash02

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(b)



6 Conclusions






04

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(a)

06

04

02

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(b)


08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(a)

1

08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(b)







Appendix

B1(y t)

0 0lt tltyλ

eminus α1t

I0 iα3

t2

minus λy2

1113969

1113874 1113875 tgtyλ

⎧⎪⎪⎨

⎪⎪⎩(A1)

BII2 (t) e



I0 iα3t( 11138571113872 1113873lowastt (A2)

BII3 (t) (1 + λH(t))

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BII4 (t) (1 + λH(t)

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIII2 (t) e






Alowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste




Dlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIV2 (t) e


lowastH(t) + α1( 1113857

lowaste

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t (A8)

BIV3 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BIV4 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste

minus α1tI0 iα3t( 1113857 (A10)

BV2 (t) e


lowaste

t+ 1 + α1( 1113857te

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowastte

t (A11)

BV3 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A12)

BV4 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A13)

BVI2 (t) e


lowaste

t cos t + 1 + α1( 1113857et sin t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t sin t(A14)

BVI3 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A15)


BVI4 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A16)

BVII2 (t) e



minus α1tI0 iα3t( 11138571113872 1113873

lowastt sin t

(A17)



lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A18)



lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A19)

Lminus 1 e

minus bq2minus a2

radic

q2

minus a2

1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

0 0lt tlt b bgt 0

I0 a

t2

minus b2

1113969

1113888 1113889 tgt bRe(q)gtRe(a)

⎧⎪⎪⎨

⎪⎪⎩(A20)

Lminus 1


(A21)


q + a

radic)

q minus b1113890 1113891 e


Φ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A23)


q + a

radic)

q minus b1113890 1113891 e


Ψ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A25)

Data Availability




Acknowledgments


References































Tem

pera

ture

15

1

05

00

y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = H (t)

(a)

Tem

pera

ture

12

1

08

06

04

02

00

y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = H (t)

(b)

0

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t

(c)

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t

(d)

0

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = sin t

(e)

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = sint

(f )

Figure 6 Continued


0y

S = 45S = 30S = 05

S = 15

05 1 15 2

05

04

03

02

01

0

Tem

pera

ture

g (t) = et

(g)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

Tem

pera

ture

1

08

06

04

02

0

g (t) = et

(h)


0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = tet

(a)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

04

03

02

01

0

ndash01

Tem

pera

ture

g (t) = tet

(b)

0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = et sint

(c)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

04

03

02

01

0

Tem

pera

ture

g (t) = et sint

(d)

Figure 7 Continued


Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

015

001

002

5 times 10ndash3

0

g (t) = t sint

(e)

0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t sint

(f )


0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = 1 g(t) = 1 h(t) = 1

(a)

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = t g(t) = 1 h(t) = 1

(b)

0 1 2 3 4 5y

Velo

city

2

15

1

05

0

M = 05M = 10

M = 15M = 25

f (t) = et g(t) = 1 h(t) = 1

(c)

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = sin t g(t) = 1 h(t) = 1

(d)

Figure 8 Continued






0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = tet g(t) = 1 h(t) = 1

(e)

0 1 2 3 4 5y

6

4

2

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = et sin t g(t) = 1 h(t) = 1

(f )

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = t sin t g(t) = 1 h(t) = 1

(g)



3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = sin t g (t) = 1 h (t) = 1

(c)

2

1

0

ndash1

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = et g (t) = 1 h (t) = 1

(d)

20 1 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

f (t) = tet g (t) = 1 h (t) = 1

(e)

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

5

4

3

2

1

0

Velo

city

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 9 Continued


2

1

0

ndash1

Velo

city

20 1 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


f (t) = 1 g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(a)

f (t) = t g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(b)

4

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

f (t) = sin t g (t) = 1 h (t) = 1

(c)

f (t) = et g (t) = 1 h (t) = 1

4

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(d)

Figure 10 Continued


f (t) = tet g (t) = 1 h (t) = 1

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

(e)

f (t) = et sin t g (t) = 1 h (t) = 1

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

5

4

3

2

1

0

Velo

city

(f )

f (t) = t sin t g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(g)


0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

4

3

2

1

0

Velo

city

f (t) = 1 g (t) = 1 h (t) = 1

(a)

GC = 80GC = 60GC = 20

GC = 40

0 1 2 3 54y

4

3

2

1

0

Velo

city

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 11 Continued


0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

4

3

2

1

0

Velo

city

f (t) = sin t g (t) = 1 h (t) = 1

(c)

GC = 80GC = 60GC = 20

GC = 40

0 1 2 3 54y

4

3

2

1

0

Velo

city

f (t) = et g (t) = 1 h (t) = 1

(d)

0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

f (t) = tet g (t) = 1 h (t) = 1

(e)

GC = 80GC = 60GC = 20

GC = 40

5

4

3

2

1

0

Velo


0 1 2 3 54y

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

f (t) = t sint g (t) = 1 h (t) = 1

(g)



3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = t g (t) = 1 h (t) = 1

(b)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = sin t g (t) = 1 h (t) = 1

(c)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = et g (t) = 1 h (t) = 1

(d)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

f (t) = tet g (t) = 1 h (t) = 1

(e)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

5

4

3

2

1

0

Velo

city

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 12 Continued


0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0f (t) = t sin t g (t) = 1 h (t) = 1

(g)


Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(a)

Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(b)

005

004

003

002

001

0

Conc

entr

atio

n

g (t) = t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(c)

Conc

entr

atio

n

02

015

01

005

0

g (t) = t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(d)

Figure 13 Continued


005

004

003

002

001

0

Conc

entr

atio

ng (t) = sin t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(e)

Conc

entr

atio

n

02

015

01

005

0

g (t) = sin t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(f )

03

04

02

01

0

Conc

entr

atio

n

g (t) = et

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(g)

Conc

entr

atio

n

08

06

04

02

0

g (t) = et

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(h)


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = tet

(a)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = tet

(b)

Figure 14 Continued


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = et sint

(c)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = et sint

(d)

001

8 times 10ndash3

6 times 10ndash3

4 times 10ndash3

2 times 10ndash3

0

Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = t sin t

(e)

0

008

006

004

002

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = t sin t

(f )


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = 1 g (t) = 1 h (t) = 1

(a)

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 15 Continued


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = sin t g (t) = 1 h (t) = 1

(c)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 105

0

ndash1 times 105

ndash2 times 105

ndash3 times 105

f (t) = et g (t) = 1 h (t) = 1

(d)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 106

0

ndash1 times 106

ndash2 times 106

ndash3 times 106

f (t) = tet g (t) = 1 h (t) = 1

(e)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

100

0

ndash100

ndash200

ndash300

ndash400

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t sin t g (t) = 1 h (t) = 1

(g)



3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = sin t g (t) = 1 h (t) = 1

(c)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et g (t) = 1 h (t) = 1

(d)

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = tet g (t) = 1 h (t) = 1

(e)

5

4

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 16 Continued


3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


3

2

1

00 1 2 3 54

y

M = 20M = 15M = 05

M = 10


Vel

ocity

(a)

1

0

ndash1

ndash2

ndash30 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

f (t) = 1 g (t) = sin t h (t) = tet

Vel

ocity

(b)

2

0

ndash2

ndash40 1 2 3 54

y

Gc = 80Gc = 60Gc = 20

Gc = 40


Vel

ocity

(c)

6

4

2

00 1 2 3 54

y

GT = 10GT = 60GT = 10

GT = 35

f (t) = 1 g (t) = sin t h (t) = t

Vel

ocity

(d)

Figure 17 Continued


2

1

0

ndash1

ndash2

ndash30 1 2 3 54

y

Sc = 078Sc = 060Sc = 022

Sc = 030


eloc

ity

(e)

3

2

1

0

ndash10 1 2 3 54

y

S = 40S = 30S = 10

S = 20

f (t) = t g (t) = et h (t) = 1

Vel

ocity

(f )


04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(a)

06

04

02

0

ndash02

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(b)



6 Conclusions






04

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(a)

06

04

02

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(b)


08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(a)

1

08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(b)







Appendix

B1(y t)

0 0lt tltyλ

eminus α1t

I0 iα3

t2

minus λy2

1113969

1113874 1113875 tgtyλ

⎧⎪⎪⎨

⎪⎪⎩(A1)

BII2 (t) e



I0 iα3t( 11138571113872 1113873lowastt (A2)

BII3 (t) (1 + λH(t))

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BII4 (t) (1 + λH(t)

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIII2 (t) e






Alowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste




Dlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIV2 (t) e


lowastH(t) + α1( 1113857

lowaste

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t (A8)

BIV3 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BIV4 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste

minus α1tI0 iα3t( 1113857 (A10)

BV2 (t) e


lowaste

t+ 1 + α1( 1113857te

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowastte

t (A11)

BV3 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A12)

BV4 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A13)

BVI2 (t) e


lowaste

t cos t + 1 + α1( 1113857et sin t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t sin t(A14)

BVI3 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A15)


BVI4 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A16)

BVII2 (t) e



minus α1tI0 iα3t( 11138571113872 1113873

lowastt sin t

(A17)



lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A18)



lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A19)

Lminus 1 e

minus bq2minus a2

radic

q2

minus a2

1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

0 0lt tlt b bgt 0

I0 a

t2

minus b2

1113969

1113888 1113889 tgt bRe(q)gtRe(a)

⎧⎪⎪⎨

⎪⎪⎩(A20)

Lminus 1


(A21)


q + a

radic)

q minus b1113890 1113891 e


Φ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A23)


q + a

radic)

q minus b1113890 1113891 e


Ψ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A25)

Data Availability




Acknowledgments


References































0y

S = 45S = 30S = 05

S = 15

05 1 15 2

05

04

03

02

01

0

Tem

pera

ture

g (t) = et

(g)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

Tem

pera

ture

1

08

06

04

02

0

g (t) = et

(h)


0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = tet

(a)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

04

03

02

01

0

ndash01

Tem

pera

ture

g (t) = tet

(b)

0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = et sint

(c)

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

04

03

02

01

0

Tem

pera

ture

g (t) = et sint

(d)

Figure 7 Continued


Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

015

001

002

5 times 10ndash3

0

g (t) = t sint

(e)

0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t sint

(f )


0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = 1 g(t) = 1 h(t) = 1

(a)

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = t g(t) = 1 h(t) = 1

(b)

0 1 2 3 4 5y

Velo

city

2

15

1

05

0

M = 05M = 10

M = 15M = 25

f (t) = et g(t) = 1 h(t) = 1

(c)

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = sin t g(t) = 1 h(t) = 1

(d)

Figure 8 Continued






0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = tet g(t) = 1 h(t) = 1

(e)

0 1 2 3 4 5y

6

4

2

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = et sin t g(t) = 1 h(t) = 1

(f )

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = t sin t g(t) = 1 h(t) = 1

(g)



3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = sin t g (t) = 1 h (t) = 1

(c)

2

1

0

ndash1

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = et g (t) = 1 h (t) = 1

(d)

20 1 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

f (t) = tet g (t) = 1 h (t) = 1

(e)

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

5

4

3

2

1

0

Velo

city

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 9 Continued


2

1

0

ndash1

Velo

city

20 1 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


f (t) = 1 g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(a)

f (t) = t g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(b)

4

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

f (t) = sin t g (t) = 1 h (t) = 1

(c)

f (t) = et g (t) = 1 h (t) = 1

4

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(d)

Figure 10 Continued


f (t) = tet g (t) = 1 h (t) = 1

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

(e)

f (t) = et sin t g (t) = 1 h (t) = 1

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

5

4

3

2

1

0

Velo

city

(f )

f (t) = t sin t g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(g)


0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

4

3

2

1

0

Velo

city

f (t) = 1 g (t) = 1 h (t) = 1

(a)

GC = 80GC = 60GC = 20

GC = 40

0 1 2 3 54y

4

3

2

1

0

Velo

city

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 11 Continued


0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

4

3

2

1

0

Velo

city

f (t) = sin t g (t) = 1 h (t) = 1

(c)

GC = 80GC = 60GC = 20

GC = 40

0 1 2 3 54y

4

3

2

1

0

Velo

city

f (t) = et g (t) = 1 h (t) = 1

(d)

0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

f (t) = tet g (t) = 1 h (t) = 1

(e)

GC = 80GC = 60GC = 20

GC = 40

5

4

3

2

1

0

Velo


0 1 2 3 54y

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

f (t) = t sint g (t) = 1 h (t) = 1

(g)



3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = t g (t) = 1 h (t) = 1

(b)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = sin t g (t) = 1 h (t) = 1

(c)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = et g (t) = 1 h (t) = 1

(d)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

f (t) = tet g (t) = 1 h (t) = 1

(e)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

5

4

3

2

1

0

Velo

city

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 12 Continued


0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0f (t) = t sin t g (t) = 1 h (t) = 1

(g)


Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(a)

Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(b)

005

004

003

002

001

0

Conc

entr

atio

n

g (t) = t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(c)

Conc

entr

atio

n

02

015

01

005

0

g (t) = t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(d)

Figure 13 Continued


005

004

003

002

001

0

Conc

entr

atio

ng (t) = sin t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(e)

Conc

entr

atio

n

02

015

01

005

0

g (t) = sin t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(f )

03

04

02

01

0

Conc

entr

atio

n

g (t) = et

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(g)

Conc

entr

atio

n

08

06

04

02

0

g (t) = et

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(h)


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = tet

(a)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = tet

(b)

Figure 14 Continued


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = et sint

(c)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = et sint

(d)

001

8 times 10ndash3

6 times 10ndash3

4 times 10ndash3

2 times 10ndash3

0

Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = t sin t

(e)

0

008

006

004

002

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = t sin t

(f )


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = 1 g (t) = 1 h (t) = 1

(a)

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 15 Continued


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = sin t g (t) = 1 h (t) = 1

(c)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 105

0

ndash1 times 105

ndash2 times 105

ndash3 times 105

f (t) = et g (t) = 1 h (t) = 1

(d)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 106

0

ndash1 times 106

ndash2 times 106

ndash3 times 106

f (t) = tet g (t) = 1 h (t) = 1

(e)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

100

0

ndash100

ndash200

ndash300

ndash400

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t sin t g (t) = 1 h (t) = 1

(g)



3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = sin t g (t) = 1 h (t) = 1

(c)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et g (t) = 1 h (t) = 1

(d)

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = tet g (t) = 1 h (t) = 1

(e)

5

4

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 16 Continued


3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


3

2

1

00 1 2 3 54

y

M = 20M = 15M = 05

M = 10


Vel

ocity

(a)

1

0

ndash1

ndash2

ndash30 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

f (t) = 1 g (t) = sin t h (t) = tet

Vel

ocity

(b)

2

0

ndash2

ndash40 1 2 3 54

y

Gc = 80Gc = 60Gc = 20

Gc = 40


Vel

ocity

(c)

6

4

2

00 1 2 3 54

y

GT = 10GT = 60GT = 10

GT = 35

f (t) = 1 g (t) = sin t h (t) = t

Vel

ocity

(d)

Figure 17 Continued


2

1

0

ndash1

ndash2

ndash30 1 2 3 54

y

Sc = 078Sc = 060Sc = 022

Sc = 030


eloc

ity

(e)

3

2

1

0

ndash10 1 2 3 54

y

S = 40S = 30S = 10

S = 20

f (t) = t g (t) = et h (t) = 1

Vel

ocity

(f )


04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(a)

06

04

02

0

ndash02

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(b)



6 Conclusions






04

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(a)

06

04

02

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(b)


08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(a)

1

08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(b)







Appendix

B1(y t)

0 0lt tltyλ

eminus α1t

I0 iα3

t2

minus λy2

1113969

1113874 1113875 tgtyλ

⎧⎪⎪⎨

⎪⎪⎩(A1)

BII2 (t) e



I0 iα3t( 11138571113872 1113873lowastt (A2)

BII3 (t) (1 + λH(t))

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BII4 (t) (1 + λH(t)

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIII2 (t) e






Alowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste




Dlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIV2 (t) e


lowastH(t) + α1( 1113857

lowaste

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t (A8)

BIV3 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BIV4 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste

minus α1tI0 iα3t( 1113857 (A10)

BV2 (t) e


lowaste

t+ 1 + α1( 1113857te

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowastte

t (A11)

BV3 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A12)

BV4 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A13)

BVI2 (t) e


lowaste

t cos t + 1 + α1( 1113857et sin t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t sin t(A14)

BVI3 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A15)


BVI4 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A16)

BVII2 (t) e



minus α1tI0 iα3t( 11138571113872 1113873

lowastt sin t

(A17)



lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A18)



lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A19)

Lminus 1 e

minus bq2minus a2

radic

q2

minus a2

1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

0 0lt tlt b bgt 0

I0 a

t2

minus b2

1113969

1113888 1113889 tgt bRe(q)gtRe(a)

⎧⎪⎪⎨

⎪⎪⎩(A20)

Lminus 1


(A21)


q + a

radic)

q minus b1113890 1113891 e


Φ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A23)


q + a

radic)

q minus b1113890 1113891 e


Ψ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A25)

Data Availability




Acknowledgments


References































Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

015

001

002

5 times 10ndash3

0

g (t) = t sint

(e)

0

01

008

006

004

002

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2

g (t) = t sint

(f )


0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = 1 g(t) = 1 h(t) = 1

(a)

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = t g(t) = 1 h(t) = 1

(b)

0 1 2 3 4 5y

Velo

city

2

15

1

05

0

M = 05M = 10

M = 15M = 25

f (t) = et g(t) = 1 h(t) = 1

(c)

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = sin t g(t) = 1 h(t) = 1

(d)

Figure 8 Continued






0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = tet g(t) = 1 h(t) = 1

(e)

0 1 2 3 4 5y

6

4

2

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = et sin t g(t) = 1 h(t) = 1

(f )

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = t sin t g(t) = 1 h(t) = 1

(g)



3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = sin t g (t) = 1 h (t) = 1

(c)

2

1

0

ndash1

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = et g (t) = 1 h (t) = 1

(d)

20 1 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

f (t) = tet g (t) = 1 h (t) = 1

(e)

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

5

4

3

2

1

0

Velo

city

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 9 Continued


2

1

0

ndash1

Velo

city

20 1 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


f (t) = 1 g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(a)

f (t) = t g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(b)

4

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

f (t) = sin t g (t) = 1 h (t) = 1

(c)

f (t) = et g (t) = 1 h (t) = 1

4

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(d)

Figure 10 Continued


f (t) = tet g (t) = 1 h (t) = 1

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

(e)

f (t) = et sin t g (t) = 1 h (t) = 1

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

5

4

3

2

1

0

Velo

city

(f )

f (t) = t sin t g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(g)


0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

4

3

2

1

0

Velo

city

f (t) = 1 g (t) = 1 h (t) = 1

(a)

GC = 80GC = 60GC = 20

GC = 40

0 1 2 3 54y

4

3

2

1

0

Velo

city

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 11 Continued


0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

4

3

2

1

0

Velo

city

f (t) = sin t g (t) = 1 h (t) = 1

(c)

GC = 80GC = 60GC = 20

GC = 40

0 1 2 3 54y

4

3

2

1

0

Velo

city

f (t) = et g (t) = 1 h (t) = 1

(d)

0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

f (t) = tet g (t) = 1 h (t) = 1

(e)

GC = 80GC = 60GC = 20

GC = 40

5

4

3

2

1

0

Velo


0 1 2 3 54y

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

f (t) = t sint g (t) = 1 h (t) = 1

(g)



3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = t g (t) = 1 h (t) = 1

(b)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = sin t g (t) = 1 h (t) = 1

(c)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = et g (t) = 1 h (t) = 1

(d)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

f (t) = tet g (t) = 1 h (t) = 1

(e)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

5

4

3

2

1

0

Velo

city

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 12 Continued


0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0f (t) = t sin t g (t) = 1 h (t) = 1

(g)


Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(a)

Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(b)

005

004

003

002

001

0

Conc

entr

atio

n

g (t) = t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(c)

Conc

entr

atio

n

02

015

01

005

0

g (t) = t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(d)

Figure 13 Continued


005

004

003

002

001

0

Conc

entr

atio

ng (t) = sin t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(e)

Conc

entr

atio

n

02

015

01

005

0

g (t) = sin t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(f )

03

04

02

01

0

Conc

entr

atio

n

g (t) = et

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(g)

Conc

entr

atio

n

08

06

04

02

0

g (t) = et

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(h)


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = tet

(a)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = tet

(b)

Figure 14 Continued


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = et sint

(c)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = et sint

(d)

001

8 times 10ndash3

6 times 10ndash3

4 times 10ndash3

2 times 10ndash3

0

Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = t sin t

(e)

0

008

006

004

002

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = t sin t

(f )


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = 1 g (t) = 1 h (t) = 1

(a)

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 15 Continued


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = sin t g (t) = 1 h (t) = 1

(c)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 105

0

ndash1 times 105

ndash2 times 105

ndash3 times 105

f (t) = et g (t) = 1 h (t) = 1

(d)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 106

0

ndash1 times 106

ndash2 times 106

ndash3 times 106

f (t) = tet g (t) = 1 h (t) = 1

(e)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

100

0

ndash100

ndash200

ndash300

ndash400

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t sin t g (t) = 1 h (t) = 1

(g)



3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = sin t g (t) = 1 h (t) = 1

(c)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et g (t) = 1 h (t) = 1

(d)

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = tet g (t) = 1 h (t) = 1

(e)

5

4

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 16 Continued


3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


3

2

1

00 1 2 3 54

y

M = 20M = 15M = 05

M = 10


Vel

ocity

(a)

1

0

ndash1

ndash2

ndash30 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

f (t) = 1 g (t) = sin t h (t) = tet

Vel

ocity

(b)

2

0

ndash2

ndash40 1 2 3 54

y

Gc = 80Gc = 60Gc = 20

Gc = 40


Vel

ocity

(c)

6

4

2

00 1 2 3 54

y

GT = 10GT = 60GT = 10

GT = 35

f (t) = 1 g (t) = sin t h (t) = t

Vel

ocity

(d)

Figure 17 Continued


2

1

0

ndash1

ndash2

ndash30 1 2 3 54

y

Sc = 078Sc = 060Sc = 022

Sc = 030


eloc

ity

(e)

3

2

1

0

ndash10 1 2 3 54

y

S = 40S = 30S = 10

S = 20

f (t) = t g (t) = et h (t) = 1

Vel

ocity

(f )


04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(a)

06

04

02

0

ndash02

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(b)



6 Conclusions






04

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(a)

06

04

02

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(b)


08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(a)

1

08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(b)







Appendix

B1(y t)

0 0lt tltyλ

eminus α1t

I0 iα3

t2

minus λy2

1113969

1113874 1113875 tgtyλ

⎧⎪⎪⎨

⎪⎪⎩(A1)

BII2 (t) e



I0 iα3t( 11138571113872 1113873lowastt (A2)

BII3 (t) (1 + λH(t))

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BII4 (t) (1 + λH(t)

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIII2 (t) e






Alowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste




Dlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIV2 (t) e


lowastH(t) + α1( 1113857

lowaste

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t (A8)

BIV3 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BIV4 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste

minus α1tI0 iα3t( 1113857 (A10)

BV2 (t) e


lowaste

t+ 1 + α1( 1113857te

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowastte

t (A11)

BV3 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A12)

BV4 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A13)

BVI2 (t) e


lowaste

t cos t + 1 + α1( 1113857et sin t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t sin t(A14)

BVI3 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A15)


BVI4 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A16)

BVII2 (t) e



minus α1tI0 iα3t( 11138571113872 1113873

lowastt sin t

(A17)



lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A18)



lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A19)

Lminus 1 e

minus bq2minus a2

radic

q2

minus a2

1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

0 0lt tlt b bgt 0

I0 a

t2

minus b2

1113969

1113888 1113889 tgt bRe(q)gtRe(a)

⎧⎪⎪⎨

⎪⎪⎩(A20)

Lminus 1


(A21)


q + a

radic)

q minus b1113890 1113891 e


Φ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A23)


q + a

radic)

q minus b1113890 1113891 e


Ψ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A25)

Data Availability




Acknowledgments


References



































0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = tet g(t) = 1 h(t) = 1

(e)

0 1 2 3 4 5y

6

4

2

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = et sin t g(t) = 1 h(t) = 1

(f )

0 1 2 3 4 5y

3

2

1

0

Velo

city

M = 05M = 10

M = 15M = 25

f (t) = t sin t g(t) = 1 h(t) = 1

(g)



3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = sin t g (t) = 1 h (t) = 1

(c)

2

1

0

ndash1

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = et g (t) = 1 h (t) = 1

(d)

20 1 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

f (t) = tet g (t) = 1 h (t) = 1

(e)

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

5

4

3

2

1

0

Velo

city

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 9 Continued


2

1

0

ndash1

Velo

city

20 1 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


f (t) = 1 g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(a)

f (t) = t g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(b)

4

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

f (t) = sin t g (t) = 1 h (t) = 1

(c)

f (t) = et g (t) = 1 h (t) = 1

4

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(d)

Figure 10 Continued


f (t) = tet g (t) = 1 h (t) = 1

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

(e)

f (t) = et sin t g (t) = 1 h (t) = 1

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

5

4

3

2

1

0

Velo

city

(f )

f (t) = t sin t g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(g)


0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

4

3

2

1

0

Velo

city

f (t) = 1 g (t) = 1 h (t) = 1

(a)

GC = 80GC = 60GC = 20

GC = 40

0 1 2 3 54y

4

3

2

1

0

Velo

city

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 11 Continued


0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

4

3

2

1

0

Velo

city

f (t) = sin t g (t) = 1 h (t) = 1

(c)

GC = 80GC = 60GC = 20

GC = 40

0 1 2 3 54y

4

3

2

1

0

Velo

city

f (t) = et g (t) = 1 h (t) = 1

(d)

0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

f (t) = tet g (t) = 1 h (t) = 1

(e)

GC = 80GC = 60GC = 20

GC = 40

5

4

3

2

1

0

Velo


0 1 2 3 54y

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

f (t) = t sint g (t) = 1 h (t) = 1

(g)



3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = t g (t) = 1 h (t) = 1

(b)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = sin t g (t) = 1 h (t) = 1

(c)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = et g (t) = 1 h (t) = 1

(d)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

f (t) = tet g (t) = 1 h (t) = 1

(e)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

5

4

3

2

1

0

Velo

city

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 12 Continued


0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0f (t) = t sin t g (t) = 1 h (t) = 1

(g)


Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(a)

Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(b)

005

004

003

002

001

0

Conc

entr

atio

n

g (t) = t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(c)

Conc

entr

atio

n

02

015

01

005

0

g (t) = t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(d)

Figure 13 Continued


005

004

003

002

001

0

Conc

entr

atio

ng (t) = sin t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(e)

Conc

entr

atio

n

02

015

01

005

0

g (t) = sin t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(f )

03

04

02

01

0

Conc

entr

atio

n

g (t) = et

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(g)

Conc

entr

atio

n

08

06

04

02

0

g (t) = et

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(h)


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = tet

(a)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = tet

(b)

Figure 14 Continued


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = et sint

(c)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = et sint

(d)

001

8 times 10ndash3

6 times 10ndash3

4 times 10ndash3

2 times 10ndash3

0

Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = t sin t

(e)

0

008

006

004

002

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = t sin t

(f )


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = 1 g (t) = 1 h (t) = 1

(a)

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 15 Continued


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = sin t g (t) = 1 h (t) = 1

(c)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 105

0

ndash1 times 105

ndash2 times 105

ndash3 times 105

f (t) = et g (t) = 1 h (t) = 1

(d)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 106

0

ndash1 times 106

ndash2 times 106

ndash3 times 106

f (t) = tet g (t) = 1 h (t) = 1

(e)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

100

0

ndash100

ndash200

ndash300

ndash400

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t sin t g (t) = 1 h (t) = 1

(g)



3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = sin t g (t) = 1 h (t) = 1

(c)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et g (t) = 1 h (t) = 1

(d)

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = tet g (t) = 1 h (t) = 1

(e)

5

4

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 16 Continued


3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


3

2

1

00 1 2 3 54

y

M = 20M = 15M = 05

M = 10


Vel

ocity

(a)

1

0

ndash1

ndash2

ndash30 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

f (t) = 1 g (t) = sin t h (t) = tet

Vel

ocity

(b)

2

0

ndash2

ndash40 1 2 3 54

y

Gc = 80Gc = 60Gc = 20

Gc = 40


Vel

ocity

(c)

6

4

2

00 1 2 3 54

y

GT = 10GT = 60GT = 10

GT = 35

f (t) = 1 g (t) = sin t h (t) = t

Vel

ocity

(d)

Figure 17 Continued


2

1

0

ndash1

ndash2

ndash30 1 2 3 54

y

Sc = 078Sc = 060Sc = 022

Sc = 030


eloc

ity

(e)

3

2

1

0

ndash10 1 2 3 54

y

S = 40S = 30S = 10

S = 20

f (t) = t g (t) = et h (t) = 1

Vel

ocity

(f )


04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(a)

06

04

02

0

ndash02

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(b)



6 Conclusions






04

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(a)

06

04

02

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(b)


08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(a)

1

08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(b)







Appendix

B1(y t)

0 0lt tltyλ

eminus α1t

I0 iα3

t2

minus λy2

1113969

1113874 1113875 tgtyλ

⎧⎪⎪⎨

⎪⎪⎩(A1)

BII2 (t) e



I0 iα3t( 11138571113872 1113873lowastt (A2)

BII3 (t) (1 + λH(t))

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BII4 (t) (1 + λH(t)

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIII2 (t) e






Alowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste




Dlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIV2 (t) e


lowastH(t) + α1( 1113857

lowaste

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t (A8)

BIV3 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BIV4 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste

minus α1tI0 iα3t( 1113857 (A10)

BV2 (t) e


lowaste

t+ 1 + α1( 1113857te

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowastte

t (A11)

BV3 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A12)

BV4 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A13)

BVI2 (t) e


lowaste

t cos t + 1 + α1( 1113857et sin t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t sin t(A14)

BVI3 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A15)


BVI4 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A16)

BVII2 (t) e



minus α1tI0 iα3t( 11138571113872 1113873

lowastt sin t

(A17)



lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A18)



lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A19)

Lminus 1 e

minus bq2minus a2

radic

q2

minus a2

1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

0 0lt tlt b bgt 0

I0 a

t2

minus b2

1113969

1113888 1113889 tgt bRe(q)gtRe(a)

⎧⎪⎪⎨

⎪⎪⎩(A20)

Lminus 1


(A21)


q + a

radic)

q minus b1113890 1113891 e


Φ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A23)


q + a

radic)

q minus b1113890 1113891 e


Ψ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A25)

Data Availability




Acknowledgments


References































3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

ndash1

ndash2

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = sin t g (t) = 1 h (t) = 1

(c)

2

1

0

ndash1

Velo

city

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = et g (t) = 1 h (t) = 1

(d)

20 1 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

f (t) = tet g (t) = 1 h (t) = 1

(e)

0 1 2 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

5

4

3

2

1

0

Velo

city

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 9 Continued


2

1

0

ndash1

Velo

city

20 1 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


f (t) = 1 g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(a)

f (t) = t g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(b)

4

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

f (t) = sin t g (t) = 1 h (t) = 1

(c)

f (t) = et g (t) = 1 h (t) = 1

4

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(d)

Figure 10 Continued


f (t) = tet g (t) = 1 h (t) = 1

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

(e)

f (t) = et sin t g (t) = 1 h (t) = 1

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

5

4

3

2

1

0

Velo

city

(f )

f (t) = t sin t g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(g)


0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

4

3

2

1

0

Velo

city

f (t) = 1 g (t) = 1 h (t) = 1

(a)

GC = 80GC = 60GC = 20

GC = 40

0 1 2 3 54y

4

3

2

1

0

Velo

city

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 11 Continued


0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

4

3

2

1

0

Velo

city

f (t) = sin t g (t) = 1 h (t) = 1

(c)

GC = 80GC = 60GC = 20

GC = 40

0 1 2 3 54y

4

3

2

1

0

Velo

city

f (t) = et g (t) = 1 h (t) = 1

(d)

0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

f (t) = tet g (t) = 1 h (t) = 1

(e)

GC = 80GC = 60GC = 20

GC = 40

5

4

3

2

1

0

Velo


0 1 2 3 54y

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

f (t) = t sint g (t) = 1 h (t) = 1

(g)



3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = t g (t) = 1 h (t) = 1

(b)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = sin t g (t) = 1 h (t) = 1

(c)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = et g (t) = 1 h (t) = 1

(d)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

f (t) = tet g (t) = 1 h (t) = 1

(e)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

5

4

3

2

1

0

Velo

city

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 12 Continued


0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0f (t) = t sin t g (t) = 1 h (t) = 1

(g)


Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(a)

Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(b)

005

004

003

002

001

0

Conc

entr

atio

n

g (t) = t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(c)

Conc

entr

atio

n

02

015

01

005

0

g (t) = t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(d)

Figure 13 Continued


005

004

003

002

001

0

Conc

entr

atio

ng (t) = sin t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(e)

Conc

entr

atio

n

02

015

01

005

0

g (t) = sin t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(f )

03

04

02

01

0

Conc

entr

atio

n

g (t) = et

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(g)

Conc

entr

atio

n

08

06

04

02

0

g (t) = et

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(h)


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = tet

(a)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = tet

(b)

Figure 14 Continued


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = et sint

(c)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = et sint

(d)

001

8 times 10ndash3

6 times 10ndash3

4 times 10ndash3

2 times 10ndash3

0

Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = t sin t

(e)

0

008

006

004

002

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = t sin t

(f )


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = 1 g (t) = 1 h (t) = 1

(a)

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 15 Continued


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = sin t g (t) = 1 h (t) = 1

(c)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 105

0

ndash1 times 105

ndash2 times 105

ndash3 times 105

f (t) = et g (t) = 1 h (t) = 1

(d)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 106

0

ndash1 times 106

ndash2 times 106

ndash3 times 106

f (t) = tet g (t) = 1 h (t) = 1

(e)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

100

0

ndash100

ndash200

ndash300

ndash400

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t sin t g (t) = 1 h (t) = 1

(g)



3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = sin t g (t) = 1 h (t) = 1

(c)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et g (t) = 1 h (t) = 1

(d)

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = tet g (t) = 1 h (t) = 1

(e)

5

4

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 16 Continued


3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


3

2

1

00 1 2 3 54

y

M = 20M = 15M = 05

M = 10


Vel

ocity

(a)

1

0

ndash1

ndash2

ndash30 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

f (t) = 1 g (t) = sin t h (t) = tet

Vel

ocity

(b)

2

0

ndash2

ndash40 1 2 3 54

y

Gc = 80Gc = 60Gc = 20

Gc = 40


Vel

ocity

(c)

6

4

2

00 1 2 3 54

y

GT = 10GT = 60GT = 10

GT = 35

f (t) = 1 g (t) = sin t h (t) = t

Vel

ocity

(d)

Figure 17 Continued


2

1

0

ndash1

ndash2

ndash30 1 2 3 54

y

Sc = 078Sc = 060Sc = 022

Sc = 030


eloc

ity

(e)

3

2

1

0

ndash10 1 2 3 54

y

S = 40S = 30S = 10

S = 20

f (t) = t g (t) = et h (t) = 1

Vel

ocity

(f )


04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(a)

06

04

02

0

ndash02

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(b)



6 Conclusions






04

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(a)

06

04

02

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(b)


08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(a)

1

08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(b)







Appendix

B1(y t)

0 0lt tltyλ

eminus α1t

I0 iα3

t2

minus λy2

1113969

1113874 1113875 tgtyλ

⎧⎪⎪⎨

⎪⎪⎩(A1)

BII2 (t) e



I0 iα3t( 11138571113872 1113873lowastt (A2)

BII3 (t) (1 + λH(t))

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BII4 (t) (1 + λH(t)

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIII2 (t) e






Alowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste




Dlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIV2 (t) e


lowastH(t) + α1( 1113857

lowaste

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t (A8)

BIV3 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BIV4 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste

minus α1tI0 iα3t( 1113857 (A10)

BV2 (t) e


lowaste

t+ 1 + α1( 1113857te

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowastte

t (A11)

BV3 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A12)

BV4 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A13)

BVI2 (t) e


lowaste

t cos t + 1 + α1( 1113857et sin t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t sin t(A14)

BVI3 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A15)


BVI4 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A16)

BVII2 (t) e



minus α1tI0 iα3t( 11138571113872 1113873

lowastt sin t

(A17)



lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A18)



lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A19)

Lminus 1 e

minus bq2minus a2

radic

q2

minus a2

1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

0 0lt tlt b bgt 0

I0 a

t2

minus b2

1113969

1113888 1113889 tgt bRe(q)gtRe(a)

⎧⎪⎪⎨

⎪⎪⎩(A20)

Lminus 1


(A21)


q + a

radic)

q minus b1113890 1113891 e


Φ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A23)


q + a

radic)

q minus b1113890 1113891 e


Ψ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A25)

Data Availability




Acknowledgments


References































2

1

0

ndash1

Velo

city

20 1 3 4y

Pr = 70Pr = 50Pr = 071

Pr = 15

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


f (t) = 1 g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(a)

f (t) = t g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(b)

4

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

f (t) = sin t g (t) = 1 h (t) = 1

(c)

f (t) = et g (t) = 1 h (t) = 1

4

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(d)

Figure 10 Continued


f (t) = tet g (t) = 1 h (t) = 1

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

(e)

f (t) = et sin t g (t) = 1 h (t) = 1

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

5

4

3

2

1

0

Velo

city

(f )

f (t) = t sin t g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(g)


0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

4

3

2

1

0

Velo

city

f (t) = 1 g (t) = 1 h (t) = 1

(a)

GC = 80GC = 60GC = 20

GC = 40

0 1 2 3 54y

4

3

2

1

0

Velo

city

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 11 Continued


0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

4

3

2

1

0

Velo

city

f (t) = sin t g (t) = 1 h (t) = 1

(c)

GC = 80GC = 60GC = 20

GC = 40

0 1 2 3 54y

4

3

2

1

0

Velo

city

f (t) = et g (t) = 1 h (t) = 1

(d)

0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

f (t) = tet g (t) = 1 h (t) = 1

(e)

GC = 80GC = 60GC = 20

GC = 40

5

4

3

2

1

0

Velo


0 1 2 3 54y

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

f (t) = t sint g (t) = 1 h (t) = 1

(g)



3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = t g (t) = 1 h (t) = 1

(b)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = sin t g (t) = 1 h (t) = 1

(c)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = et g (t) = 1 h (t) = 1

(d)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

f (t) = tet g (t) = 1 h (t) = 1

(e)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

5

4

3

2

1

0

Velo

city

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 12 Continued


0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0f (t) = t sin t g (t) = 1 h (t) = 1

(g)


Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(a)

Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(b)

005

004

003

002

001

0

Conc

entr

atio

n

g (t) = t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(c)

Conc

entr

atio

n

02

015

01

005

0

g (t) = t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(d)

Figure 13 Continued


005

004

003

002

001

0

Conc

entr

atio

ng (t) = sin t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(e)

Conc

entr

atio

n

02

015

01

005

0

g (t) = sin t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(f )

03

04

02

01

0

Conc

entr

atio

n

g (t) = et

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(g)

Conc

entr

atio

n

08

06

04

02

0

g (t) = et

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(h)


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = tet

(a)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = tet

(b)

Figure 14 Continued


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = et sint

(c)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = et sint

(d)

001

8 times 10ndash3

6 times 10ndash3

4 times 10ndash3

2 times 10ndash3

0

Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = t sin t

(e)

0

008

006

004

002

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = t sin t

(f )


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = 1 g (t) = 1 h (t) = 1

(a)

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 15 Continued


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = sin t g (t) = 1 h (t) = 1

(c)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 105

0

ndash1 times 105

ndash2 times 105

ndash3 times 105

f (t) = et g (t) = 1 h (t) = 1

(d)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 106

0

ndash1 times 106

ndash2 times 106

ndash3 times 106

f (t) = tet g (t) = 1 h (t) = 1

(e)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

100

0

ndash100

ndash200

ndash300

ndash400

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t sin t g (t) = 1 h (t) = 1

(g)



3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = sin t g (t) = 1 h (t) = 1

(c)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et g (t) = 1 h (t) = 1

(d)

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = tet g (t) = 1 h (t) = 1

(e)

5

4

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 16 Continued


3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


3

2

1

00 1 2 3 54

y

M = 20M = 15M = 05

M = 10


Vel

ocity

(a)

1

0

ndash1

ndash2

ndash30 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

f (t) = 1 g (t) = sin t h (t) = tet

Vel

ocity

(b)

2

0

ndash2

ndash40 1 2 3 54

y

Gc = 80Gc = 60Gc = 20

Gc = 40


Vel

ocity

(c)

6

4

2

00 1 2 3 54

y

GT = 10GT = 60GT = 10

GT = 35

f (t) = 1 g (t) = sin t h (t) = t

Vel

ocity

(d)

Figure 17 Continued


2

1

0

ndash1

ndash2

ndash30 1 2 3 54

y

Sc = 078Sc = 060Sc = 022

Sc = 030


eloc

ity

(e)

3

2

1

0

ndash10 1 2 3 54

y

S = 40S = 30S = 10

S = 20

f (t) = t g (t) = et h (t) = 1

Vel

ocity

(f )


04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(a)

06

04

02

0

ndash02

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(b)



6 Conclusions






04

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(a)

06

04

02

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(b)


08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(a)

1

08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(b)







Appendix

B1(y t)

0 0lt tltyλ

eminus α1t

I0 iα3

t2

minus λy2

1113969

1113874 1113875 tgtyλ

⎧⎪⎪⎨

⎪⎪⎩(A1)

BII2 (t) e



I0 iα3t( 11138571113872 1113873lowastt (A2)

BII3 (t) (1 + λH(t))

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BII4 (t) (1 + λH(t)

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIII2 (t) e






Alowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste




Dlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIV2 (t) e


lowastH(t) + α1( 1113857

lowaste

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t (A8)

BIV3 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BIV4 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste

minus α1tI0 iα3t( 1113857 (A10)

BV2 (t) e


lowaste

t+ 1 + α1( 1113857te

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowastte

t (A11)

BV3 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A12)

BV4 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A13)

BVI2 (t) e


lowaste

t cos t + 1 + α1( 1113857et sin t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t sin t(A14)

BVI3 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A15)


BVI4 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A16)

BVII2 (t) e



minus α1tI0 iα3t( 11138571113872 1113873

lowastt sin t

(A17)



lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A18)



lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A19)

Lminus 1 e

minus bq2minus a2

radic

q2

minus a2

1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

0 0lt tlt b bgt 0

I0 a

t2

minus b2

1113969

1113888 1113889 tgt bRe(q)gtRe(a)

⎧⎪⎪⎨

⎪⎪⎩(A20)

Lminus 1


(A21)


q + a

radic)

q minus b1113890 1113891 e


Φ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A23)


q + a

radic)

q minus b1113890 1113891 e


Ψ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A25)

Data Availability




Acknowledgments


References































f (t) = tet g (t) = 1 h (t) = 1

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

(e)

f (t) = et sin t g (t) = 1 h (t) = 1

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

5

4

3

2

1

0

Velo

city

(f )

f (t) = t sin t g (t) = 1 h (t) = 14

3

2

1

0

Velo

city

0 1 2 3 54y

GT = 10GT = 60GT = 10

GT = 35

(g)


0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

4

3

2

1

0

Velo

city

f (t) = 1 g (t) = 1 h (t) = 1

(a)

GC = 80GC = 60GC = 20

GC = 40

0 1 2 3 54y

4

3

2

1

0

Velo

city

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 11 Continued


0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

4

3

2

1

0

Velo

city

f (t) = sin t g (t) = 1 h (t) = 1

(c)

GC = 80GC = 60GC = 20

GC = 40

0 1 2 3 54y

4

3

2

1

0

Velo

city

f (t) = et g (t) = 1 h (t) = 1

(d)

0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

f (t) = tet g (t) = 1 h (t) = 1

(e)

GC = 80GC = 60GC = 20

GC = 40

5

4

3

2

1

0

Velo


0 1 2 3 54y

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

f (t) = t sint g (t) = 1 h (t) = 1

(g)



3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = t g (t) = 1 h (t) = 1

(b)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = sin t g (t) = 1 h (t) = 1

(c)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = et g (t) = 1 h (t) = 1

(d)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

f (t) = tet g (t) = 1 h (t) = 1

(e)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

5

4

3

2

1

0

Velo

city

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 12 Continued


0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0f (t) = t sin t g (t) = 1 h (t) = 1

(g)


Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(a)

Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(b)

005

004

003

002

001

0

Conc

entr

atio

n

g (t) = t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(c)

Conc

entr

atio

n

02

015

01

005

0

g (t) = t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(d)

Figure 13 Continued


005

004

003

002

001

0

Conc

entr

atio

ng (t) = sin t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(e)

Conc

entr

atio

n

02

015

01

005

0

g (t) = sin t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(f )

03

04

02

01

0

Conc

entr

atio

n

g (t) = et

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(g)

Conc

entr

atio

n

08

06

04

02

0

g (t) = et

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(h)


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = tet

(a)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = tet

(b)

Figure 14 Continued


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = et sint

(c)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = et sint

(d)

001

8 times 10ndash3

6 times 10ndash3

4 times 10ndash3

2 times 10ndash3

0

Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = t sin t

(e)

0

008

006

004

002

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = t sin t

(f )


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = 1 g (t) = 1 h (t) = 1

(a)

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 15 Continued


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = sin t g (t) = 1 h (t) = 1

(c)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 105

0

ndash1 times 105

ndash2 times 105

ndash3 times 105

f (t) = et g (t) = 1 h (t) = 1

(d)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 106

0

ndash1 times 106

ndash2 times 106

ndash3 times 106

f (t) = tet g (t) = 1 h (t) = 1

(e)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

100

0

ndash100

ndash200

ndash300

ndash400

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t sin t g (t) = 1 h (t) = 1

(g)



3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = sin t g (t) = 1 h (t) = 1

(c)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et g (t) = 1 h (t) = 1

(d)

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = tet g (t) = 1 h (t) = 1

(e)

5

4

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 16 Continued


3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


3

2

1

00 1 2 3 54

y

M = 20M = 15M = 05

M = 10


Vel

ocity

(a)

1

0

ndash1

ndash2

ndash30 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

f (t) = 1 g (t) = sin t h (t) = tet

Vel

ocity

(b)

2

0

ndash2

ndash40 1 2 3 54

y

Gc = 80Gc = 60Gc = 20

Gc = 40


Vel

ocity

(c)

6

4

2

00 1 2 3 54

y

GT = 10GT = 60GT = 10

GT = 35

f (t) = 1 g (t) = sin t h (t) = t

Vel

ocity

(d)

Figure 17 Continued


2

1

0

ndash1

ndash2

ndash30 1 2 3 54

y

Sc = 078Sc = 060Sc = 022

Sc = 030


eloc

ity

(e)

3

2

1

0

ndash10 1 2 3 54

y

S = 40S = 30S = 10

S = 20

f (t) = t g (t) = et h (t) = 1

Vel

ocity

(f )


04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(a)

06

04

02

0

ndash02

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(b)



6 Conclusions






04

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(a)

06

04

02

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(b)


08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(a)

1

08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(b)







Appendix

B1(y t)

0 0lt tltyλ

eminus α1t

I0 iα3

t2

minus λy2

1113969

1113874 1113875 tgtyλ

⎧⎪⎪⎨

⎪⎪⎩(A1)

BII2 (t) e



I0 iα3t( 11138571113872 1113873lowastt (A2)

BII3 (t) (1 + λH(t))

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BII4 (t) (1 + λH(t)

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIII2 (t) e






Alowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste




Dlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIV2 (t) e


lowastH(t) + α1( 1113857

lowaste

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t (A8)

BIV3 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BIV4 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste

minus α1tI0 iα3t( 1113857 (A10)

BV2 (t) e


lowaste

t+ 1 + α1( 1113857te

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowastte

t (A11)

BV3 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A12)

BV4 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A13)

BVI2 (t) e


lowaste

t cos t + 1 + α1( 1113857et sin t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t sin t(A14)

BVI3 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A15)


BVI4 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A16)

BVII2 (t) e



minus α1tI0 iα3t( 11138571113872 1113873

lowastt sin t

(A17)



lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A18)



lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A19)

Lminus 1 e

minus bq2minus a2

radic

q2

minus a2

1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

0 0lt tlt b bgt 0

I0 a

t2

minus b2

1113969

1113888 1113889 tgt bRe(q)gtRe(a)

⎧⎪⎪⎨

⎪⎪⎩(A20)

Lminus 1


(A21)


q + a

radic)

q minus b1113890 1113891 e


Φ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A23)


q + a

radic)

q minus b1113890 1113891 e


Ψ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A25)

Data Availability




Acknowledgments


References































0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

4

3

2

1

0

Velo

city

f (t) = sin t g (t) = 1 h (t) = 1

(c)

GC = 80GC = 60GC = 20

GC = 40

0 1 2 3 54y

4

3

2

1

0

Velo

city

f (t) = et g (t) = 1 h (t) = 1

(d)

0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

f (t) = tet g (t) = 1 h (t) = 1

(e)

GC = 80GC = 60GC = 20

GC = 40

5

4

3

2

1

0

Velo


0 1 2 3 54y

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

GC = 80GC = 60GC = 20

GC = 40

f (t) = t sint g (t) = 1 h (t) = 1

(g)



3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = t g (t) = 1 h (t) = 1

(b)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = sin t g (t) = 1 h (t) = 1

(c)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = et g (t) = 1 h (t) = 1

(d)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

f (t) = tet g (t) = 1 h (t) = 1

(e)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

5

4

3

2

1

0

Velo

city

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 12 Continued


0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0f (t) = t sin t g (t) = 1 h (t) = 1

(g)


Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(a)

Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(b)

005

004

003

002

001

0

Conc

entr

atio

n

g (t) = t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(c)

Conc

entr

atio

n

02

015

01

005

0

g (t) = t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(d)

Figure 13 Continued


005

004

003

002

001

0

Conc

entr

atio

ng (t) = sin t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(e)

Conc

entr

atio

n

02

015

01

005

0

g (t) = sin t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(f )

03

04

02

01

0

Conc

entr

atio

n

g (t) = et

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(g)

Conc

entr

atio

n

08

06

04

02

0

g (t) = et

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(h)


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = tet

(a)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = tet

(b)

Figure 14 Continued


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = et sint

(c)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = et sint

(d)

001

8 times 10ndash3

6 times 10ndash3

4 times 10ndash3

2 times 10ndash3

0

Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = t sin t

(e)

0

008

006

004

002

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = t sin t

(f )


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = 1 g (t) = 1 h (t) = 1

(a)

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 15 Continued


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = sin t g (t) = 1 h (t) = 1

(c)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 105

0

ndash1 times 105

ndash2 times 105

ndash3 times 105

f (t) = et g (t) = 1 h (t) = 1

(d)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 106

0

ndash1 times 106

ndash2 times 106

ndash3 times 106

f (t) = tet g (t) = 1 h (t) = 1

(e)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

100

0

ndash100

ndash200

ndash300

ndash400

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t sin t g (t) = 1 h (t) = 1

(g)



3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = sin t g (t) = 1 h (t) = 1

(c)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et g (t) = 1 h (t) = 1

(d)

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = tet g (t) = 1 h (t) = 1

(e)

5

4

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 16 Continued


3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


3

2

1

00 1 2 3 54

y

M = 20M = 15M = 05

M = 10


Vel

ocity

(a)

1

0

ndash1

ndash2

ndash30 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

f (t) = 1 g (t) = sin t h (t) = tet

Vel

ocity

(b)

2

0

ndash2

ndash40 1 2 3 54

y

Gc = 80Gc = 60Gc = 20

Gc = 40


Vel

ocity

(c)

6

4

2

00 1 2 3 54

y

GT = 10GT = 60GT = 10

GT = 35

f (t) = 1 g (t) = sin t h (t) = t

Vel

ocity

(d)

Figure 17 Continued


2

1

0

ndash1

ndash2

ndash30 1 2 3 54

y

Sc = 078Sc = 060Sc = 022

Sc = 030


eloc

ity

(e)

3

2

1

0

ndash10 1 2 3 54

y

S = 40S = 30S = 10

S = 20

f (t) = t g (t) = et h (t) = 1

Vel

ocity

(f )


04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(a)

06

04

02

0

ndash02

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(b)



6 Conclusions






04

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(a)

06

04

02

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(b)


08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(a)

1

08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(b)







Appendix

B1(y t)

0 0lt tltyλ

eminus α1t

I0 iα3

t2

minus λy2

1113969

1113874 1113875 tgtyλ

⎧⎪⎪⎨

⎪⎪⎩(A1)

BII2 (t) e



I0 iα3t( 11138571113872 1113873lowastt (A2)

BII3 (t) (1 + λH(t))

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BII4 (t) (1 + λH(t)

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIII2 (t) e






Alowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste




Dlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIV2 (t) e


lowastH(t) + α1( 1113857

lowaste

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t (A8)

BIV3 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BIV4 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste

minus α1tI0 iα3t( 1113857 (A10)

BV2 (t) e


lowaste

t+ 1 + α1( 1113857te

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowastte

t (A11)

BV3 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A12)

BV4 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A13)

BVI2 (t) e


lowaste

t cos t + 1 + α1( 1113857et sin t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t sin t(A14)

BVI3 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A15)


BVI4 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A16)

BVII2 (t) e



minus α1tI0 iα3t( 11138571113872 1113873

lowastt sin t

(A17)



lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A18)



lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A19)

Lminus 1 e

minus bq2minus a2

radic

q2

minus a2

1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

0 0lt tlt b bgt 0

I0 a

t2

minus b2

1113969

1113888 1113889 tgt bRe(q)gtRe(a)

⎧⎪⎪⎨

⎪⎪⎩(A20)

Lminus 1


(A21)


q + a

radic)

q minus b1113890 1113891 e


Φ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A23)


q + a

radic)

q minus b1113890 1113891 e


Ψ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A25)

Data Availability




Acknowledgments


References































3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

00 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

f (t) = t g (t) = 1 h (t) = 1

(b)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = sin t g (t) = 1 h (t) = 1

(c)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0

f (t) = et g (t) = 1 h (t) = 1

(d)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

0

ndash2

ndash4

ndash6

ndash8

f (t) = tet g (t) = 1 h (t) = 1

(e)

0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

5

4

3

2

1

0

Velo

city

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 12 Continued


0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0f (t) = t sin t g (t) = 1 h (t) = 1

(g)


Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(a)

Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(b)

005

004

003

002

001

0

Conc

entr

atio

n

g (t) = t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(c)

Conc

entr

atio

n

02

015

01

005

0

g (t) = t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(d)

Figure 13 Continued


005

004

003

002

001

0

Conc

entr

atio

ng (t) = sin t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(e)

Conc

entr

atio

n

02

015

01

005

0

g (t) = sin t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(f )

03

04

02

01

0

Conc

entr

atio

n

g (t) = et

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(g)

Conc

entr

atio

n

08

06

04

02

0

g (t) = et

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(h)


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = tet

(a)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = tet

(b)

Figure 14 Continued


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = et sint

(c)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = et sint

(d)

001

8 times 10ndash3

6 times 10ndash3

4 times 10ndash3

2 times 10ndash3

0

Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = t sin t

(e)

0

008

006

004

002

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = t sin t

(f )


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = 1 g (t) = 1 h (t) = 1

(a)

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 15 Continued


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = sin t g (t) = 1 h (t) = 1

(c)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 105

0

ndash1 times 105

ndash2 times 105

ndash3 times 105

f (t) = et g (t) = 1 h (t) = 1

(d)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 106

0

ndash1 times 106

ndash2 times 106

ndash3 times 106

f (t) = tet g (t) = 1 h (t) = 1

(e)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

100

0

ndash100

ndash200

ndash300

ndash400

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t sin t g (t) = 1 h (t) = 1

(g)



3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = sin t g (t) = 1 h (t) = 1

(c)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et g (t) = 1 h (t) = 1

(d)

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = tet g (t) = 1 h (t) = 1

(e)

5

4

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 16 Continued


3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


3

2

1

00 1 2 3 54

y

M = 20M = 15M = 05

M = 10


Vel

ocity

(a)

1

0

ndash1

ndash2

ndash30 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

f (t) = 1 g (t) = sin t h (t) = tet

Vel

ocity

(b)

2

0

ndash2

ndash40 1 2 3 54

y

Gc = 80Gc = 60Gc = 20

Gc = 40


Vel

ocity

(c)

6

4

2

00 1 2 3 54

y

GT = 10GT = 60GT = 10

GT = 35

f (t) = 1 g (t) = sin t h (t) = t

Vel

ocity

(d)

Figure 17 Continued


2

1

0

ndash1

ndash2

ndash30 1 2 3 54

y

Sc = 078Sc = 060Sc = 022

Sc = 030


eloc

ity

(e)

3

2

1

0

ndash10 1 2 3 54

y

S = 40S = 30S = 10

S = 20

f (t) = t g (t) = et h (t) = 1

Vel

ocity

(f )


04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(a)

06

04

02

0

ndash02

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(b)



6 Conclusions






04

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(a)

06

04

02

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(b)


08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(a)

1

08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(b)







Appendix

B1(y t)

0 0lt tltyλ

eminus α1t

I0 iα3

t2

minus λy2

1113969

1113874 1113875 tgtyλ

⎧⎪⎪⎨

⎪⎪⎩(A1)

BII2 (t) e



I0 iα3t( 11138571113872 1113873lowastt (A2)

BII3 (t) (1 + λH(t))

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BII4 (t) (1 + λH(t)

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIII2 (t) e






Alowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste




Dlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIV2 (t) e


lowastH(t) + α1( 1113857

lowaste

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t (A8)

BIV3 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BIV4 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste

minus α1tI0 iα3t( 1113857 (A10)

BV2 (t) e


lowaste

t+ 1 + α1( 1113857te

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowastte

t (A11)

BV3 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A12)

BV4 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A13)

BVI2 (t) e


lowaste

t cos t + 1 + α1( 1113857et sin t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t sin t(A14)

BVI3 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A15)


BVI4 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A16)

BVII2 (t) e



minus α1tI0 iα3t( 11138571113872 1113873

lowastt sin t

(A17)



lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A18)



lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A19)

Lminus 1 e

minus bq2minus a2

radic

q2

minus a2

1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

0 0lt tlt b bgt 0

I0 a

t2

minus b2

1113969

1113888 1113889 tgt bRe(q)gtRe(a)

⎧⎪⎪⎨

⎪⎪⎩(A20)

Lminus 1


(A21)


q + a

radic)

q minus b1113890 1113891 e


Φ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A23)


q + a

radic)

q minus b1113890 1113891 e


Ψ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A25)

Data Availability




Acknowledgments


References































0 1 2 3 4y

Kc = 65Kc = 45Kc = 05

Kc = 25

Velo

city

2

15

1

05

0f (t) = t sin t g (t) = 1 h (t) = 1

(g)


Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(a)

Conc

entr

atio

n

12

1

08

06

04

02

0

g (t) = H (t)

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(b)

005

004

003

002

001

0

Conc

entr

atio

n

g (t) = t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(c)

Conc

entr

atio

n

02

015

01

005

0

g (t) = t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(d)

Figure 13 Continued


005

004

003

002

001

0

Conc

entr

atio

ng (t) = sin t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(e)

Conc

entr

atio

n

02

015

01

005

0

g (t) = sin t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(f )

03

04

02

01

0

Conc

entr

atio

n

g (t) = et

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(g)

Conc

entr

atio

n

08

06

04

02

0

g (t) = et

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(h)


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = tet

(a)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = tet

(b)

Figure 14 Continued


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = et sint

(c)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = et sint

(d)

001

8 times 10ndash3

6 times 10ndash3

4 times 10ndash3

2 times 10ndash3

0

Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = t sin t

(e)

0

008

006

004

002

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = t sin t

(f )


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = 1 g (t) = 1 h (t) = 1

(a)

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 15 Continued


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = sin t g (t) = 1 h (t) = 1

(c)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 105

0

ndash1 times 105

ndash2 times 105

ndash3 times 105

f (t) = et g (t) = 1 h (t) = 1

(d)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 106

0

ndash1 times 106

ndash2 times 106

ndash3 times 106

f (t) = tet g (t) = 1 h (t) = 1

(e)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

100

0

ndash100

ndash200

ndash300

ndash400

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t sin t g (t) = 1 h (t) = 1

(g)



3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = sin t g (t) = 1 h (t) = 1

(c)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et g (t) = 1 h (t) = 1

(d)

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = tet g (t) = 1 h (t) = 1

(e)

5

4

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 16 Continued


3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


3

2

1

00 1 2 3 54

y

M = 20M = 15M = 05

M = 10


Vel

ocity

(a)

1

0

ndash1

ndash2

ndash30 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

f (t) = 1 g (t) = sin t h (t) = tet

Vel

ocity

(b)

2

0

ndash2

ndash40 1 2 3 54

y

Gc = 80Gc = 60Gc = 20

Gc = 40


Vel

ocity

(c)

6

4

2

00 1 2 3 54

y

GT = 10GT = 60GT = 10

GT = 35

f (t) = 1 g (t) = sin t h (t) = t

Vel

ocity

(d)

Figure 17 Continued


2

1

0

ndash1

ndash2

ndash30 1 2 3 54

y

Sc = 078Sc = 060Sc = 022

Sc = 030


eloc

ity

(e)

3

2

1

0

ndash10 1 2 3 54

y

S = 40S = 30S = 10

S = 20

f (t) = t g (t) = et h (t) = 1

Vel

ocity

(f )


04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(a)

06

04

02

0

ndash02

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(b)



6 Conclusions






04

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(a)

06

04

02

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(b)


08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(a)

1

08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(b)







Appendix

B1(y t)

0 0lt tltyλ

eminus α1t

I0 iα3

t2

minus λy2

1113969

1113874 1113875 tgtyλ

⎧⎪⎪⎨

⎪⎪⎩(A1)

BII2 (t) e



I0 iα3t( 11138571113872 1113873lowastt (A2)

BII3 (t) (1 + λH(t))

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BII4 (t) (1 + λH(t)

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIII2 (t) e






Alowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste




Dlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIV2 (t) e


lowastH(t) + α1( 1113857

lowaste

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t (A8)

BIV3 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BIV4 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste

minus α1tI0 iα3t( 1113857 (A10)

BV2 (t) e


lowaste

t+ 1 + α1( 1113857te

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowastte

t (A11)

BV3 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A12)

BV4 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A13)

BVI2 (t) e


lowaste

t cos t + 1 + α1( 1113857et sin t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t sin t(A14)

BVI3 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A15)


BVI4 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A16)

BVII2 (t) e



minus α1tI0 iα3t( 11138571113872 1113873

lowastt sin t

(A17)



lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A18)



lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A19)

Lminus 1 e

minus bq2minus a2

radic

q2

minus a2

1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

0 0lt tlt b bgt 0

I0 a

t2

minus b2

1113969

1113888 1113889 tgt bRe(q)gtRe(a)

⎧⎪⎪⎨

⎪⎪⎩(A20)

Lminus 1


(A21)


q + a

radic)

q minus b1113890 1113891 e


Φ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A23)


q + a

radic)

q minus b1113890 1113891 e


Ψ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A25)

Data Availability




Acknowledgments


References































005

004

003

002

001

0

Conc

entr

atio

ng (t) = sin t

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(e)

Conc

entr

atio

n

02

015

01

005

0

g (t) = sin t

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(f )

03

04

02

01

0

Conc

entr

atio

n

g (t) = et

0 1 2 3y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(g)

Conc

entr

atio

n

08

06

04

02

0

g (t) = et

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

(h)


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = tet

(a)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = tet

(b)

Figure 14 Continued


0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = et sint

(c)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = et sint

(d)

001

8 times 10ndash3

6 times 10ndash3

4 times 10ndash3

2 times 10ndash3

0

Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = t sin t

(e)

0

008

006

004

002

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = t sin t

(f )


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = 1 g (t) = 1 h (t) = 1

(a)

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 15 Continued


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = sin t g (t) = 1 h (t) = 1

(c)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 105

0

ndash1 times 105

ndash2 times 105

ndash3 times 105

f (t) = et g (t) = 1 h (t) = 1

(d)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 106

0

ndash1 times 106

ndash2 times 106

ndash3 times 106

f (t) = tet g (t) = 1 h (t) = 1

(e)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

100

0

ndash100

ndash200

ndash300

ndash400

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t sin t g (t) = 1 h (t) = 1

(g)



3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = sin t g (t) = 1 h (t) = 1

(c)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et g (t) = 1 h (t) = 1

(d)

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = tet g (t) = 1 h (t) = 1

(e)

5

4

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 16 Continued


3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


3

2

1

00 1 2 3 54

y

M = 20M = 15M = 05

M = 10


Vel

ocity

(a)

1

0

ndash1

ndash2

ndash30 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

f (t) = 1 g (t) = sin t h (t) = tet

Vel

ocity

(b)

2

0

ndash2

ndash40 1 2 3 54

y

Gc = 80Gc = 60Gc = 20

Gc = 40


Vel

ocity

(c)

6

4

2

00 1 2 3 54

y

GT = 10GT = 60GT = 10

GT = 35

f (t) = 1 g (t) = sin t h (t) = t

Vel

ocity

(d)

Figure 17 Continued


2

1

0

ndash1

ndash2

ndash30 1 2 3 54

y

Sc = 078Sc = 060Sc = 022

Sc = 030


eloc

ity

(e)

3

2

1

0

ndash10 1 2 3 54

y

S = 40S = 30S = 10

S = 20

f (t) = t g (t) = et h (t) = 1

Vel

ocity

(f )


04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(a)

06

04

02

0

ndash02

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(b)



6 Conclusions






04

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(a)

06

04

02

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(b)


08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(a)

1

08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(b)







Appendix

B1(y t)

0 0lt tltyλ

eminus α1t

I0 iα3

t2

minus λy2

1113969

1113874 1113875 tgtyλ

⎧⎪⎪⎨

⎪⎪⎩(A1)

BII2 (t) e



I0 iα3t( 11138571113872 1113873lowastt (A2)

BII3 (t) (1 + λH(t))

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BII4 (t) (1 + λH(t)

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIII2 (t) e






Alowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste




Dlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIV2 (t) e


lowastH(t) + α1( 1113857

lowaste

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t (A8)

BIV3 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BIV4 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste

minus α1tI0 iα3t( 1113857 (A10)

BV2 (t) e


lowaste

t+ 1 + α1( 1113857te

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowastte

t (A11)

BV3 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A12)

BV4 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A13)

BVI2 (t) e


lowaste

t cos t + 1 + α1( 1113857et sin t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t sin t(A14)

BVI3 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A15)


BVI4 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A16)

BVII2 (t) e



minus α1tI0 iα3t( 11138571113872 1113873

lowastt sin t

(A17)



lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A18)



lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A19)

Lminus 1 e

minus bq2minus a2

radic

q2

minus a2

1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

0 0lt tlt b bgt 0

I0 a

t2

minus b2

1113969

1113888 1113889 tgt bRe(q)gtRe(a)

⎧⎪⎪⎨

⎪⎪⎩(A20)

Lminus 1


(A21)


q + a

radic)

q minus b1113890 1113891 e


Φ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A23)


q + a

radic)

q minus b1113890 1113891 e


Ψ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A25)

Data Availability




Acknowledgments


References































0

006

004

002Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = et sint

(c)

03

02

01

0

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = et sint

(d)

001

8 times 10ndash3

6 times 10ndash3

4 times 10ndash3

2 times 10ndash3

0

Con

cent

ratio

n

0 1 2 3y

Kc = 29Kc = 20Kc = 05

Kc = 14

g (t) = t sin t

(e)

0

008

006

004

002

Con

cent

ratio

n

0 1 2 3 4y

Kc = 05Kc = 14 Kc = 29

Kc = 20

g (t) = t sin t

(f )


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = 1 g (t) = 1 h (t) = 1

(a)

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t g (t) = 1 h (t) = 1

(b)

Figure 15 Continued


4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = sin t g (t) = 1 h (t) = 1

(c)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 105

0

ndash1 times 105

ndash2 times 105

ndash3 times 105

f (t) = et g (t) = 1 h (t) = 1

(d)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 106

0

ndash1 times 106

ndash2 times 106

ndash3 times 106

f (t) = tet g (t) = 1 h (t) = 1

(e)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

100

0

ndash100

ndash200

ndash300

ndash400

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t sin t g (t) = 1 h (t) = 1

(g)



3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = sin t g (t) = 1 h (t) = 1

(c)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et g (t) = 1 h (t) = 1

(d)

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = tet g (t) = 1 h (t) = 1

(e)

5

4

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 16 Continued


3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


3

2

1

00 1 2 3 54

y

M = 20M = 15M = 05

M = 10


Vel

ocity

(a)

1

0

ndash1

ndash2

ndash30 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

f (t) = 1 g (t) = sin t h (t) = tet

Vel

ocity

(b)

2

0

ndash2

ndash40 1 2 3 54

y

Gc = 80Gc = 60Gc = 20

Gc = 40


Vel

ocity

(c)

6

4

2

00 1 2 3 54

y

GT = 10GT = 60GT = 10

GT = 35

f (t) = 1 g (t) = sin t h (t) = t

Vel

ocity

(d)

Figure 17 Continued


2

1

0

ndash1

ndash2

ndash30 1 2 3 54

y

Sc = 078Sc = 060Sc = 022

Sc = 030


eloc

ity

(e)

3

2

1

0

ndash10 1 2 3 54

y

S = 40S = 30S = 10

S = 20

f (t) = t g (t) = et h (t) = 1

Vel

ocity

(f )


04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(a)

06

04

02

0

ndash02

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(b)



6 Conclusions






04

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(a)

06

04

02

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(b)


08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(a)

1

08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(b)







Appendix

B1(y t)

0 0lt tltyλ

eminus α1t

I0 iα3

t2

minus λy2

1113969

1113874 1113875 tgtyλ

⎧⎪⎪⎨

⎪⎪⎩(A1)

BII2 (t) e



I0 iα3t( 11138571113872 1113873lowastt (A2)

BII3 (t) (1 + λH(t))

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BII4 (t) (1 + λH(t)

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIII2 (t) e






Alowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste




Dlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIV2 (t) e


lowastH(t) + α1( 1113857

lowaste

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t (A8)

BIV3 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BIV4 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste

minus α1tI0 iα3t( 1113857 (A10)

BV2 (t) e


lowaste

t+ 1 + α1( 1113857te

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowastte

t (A11)

BV3 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A12)

BV4 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A13)

BVI2 (t) e


lowaste

t cos t + 1 + α1( 1113857et sin t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t sin t(A14)

BVI3 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A15)


BVI4 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A16)

BVII2 (t) e



minus α1tI0 iα3t( 11138571113872 1113873

lowastt sin t

(A17)



lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A18)



lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A19)

Lminus 1 e

minus bq2minus a2

radic

q2

minus a2

1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

0 0lt tlt b bgt 0

I0 a

t2

minus b2

1113969

1113888 1113889 tgt bRe(q)gtRe(a)

⎧⎪⎪⎨

⎪⎪⎩(A20)

Lminus 1


(A21)


q + a

radic)

q minus b1113890 1113891 e


Φ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A23)


q + a

radic)

q minus b1113890 1113891 e


Ψ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A25)

Data Availability




Acknowledgments


References































4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = sin t g (t) = 1 h (t) = 1

(c)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 105

0

ndash1 times 105

ndash2 times 105

ndash3 times 105

f (t) = et g (t) = 1 h (t) = 1

(d)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

1 times 106

0

ndash1 times 106

ndash2 times 106

ndash3 times 106

f (t) = tet g (t) = 1 h (t) = 1

(e)

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

100

0

ndash100

ndash200

ndash300

ndash400

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

4

3

2

1

0

Velo

city

0 1 2 3 54y

λ = 08

λ = 02λ = 05

f (t) = t sin t g (t) = 1 h (t) = 1

(g)



3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = sin t g (t) = 1 h (t) = 1

(c)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et g (t) = 1 h (t) = 1

(d)

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = tet g (t) = 1 h (t) = 1

(e)

5

4

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 16 Continued


3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


3

2

1

00 1 2 3 54

y

M = 20M = 15M = 05

M = 10


Vel

ocity

(a)

1

0

ndash1

ndash2

ndash30 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

f (t) = 1 g (t) = sin t h (t) = tet

Vel

ocity

(b)

2

0

ndash2

ndash40 1 2 3 54

y

Gc = 80Gc = 60Gc = 20

Gc = 40


Vel

ocity

(c)

6

4

2

00 1 2 3 54

y

GT = 10GT = 60GT = 10

GT = 35

f (t) = 1 g (t) = sin t h (t) = t

Vel

ocity

(d)

Figure 17 Continued


2

1

0

ndash1

ndash2

ndash30 1 2 3 54

y

Sc = 078Sc = 060Sc = 022

Sc = 030


eloc

ity

(e)

3

2

1

0

ndash10 1 2 3 54

y

S = 40S = 30S = 10

S = 20

f (t) = t g (t) = et h (t) = 1

Vel

ocity

(f )


04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(a)

06

04

02

0

ndash02

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(b)



6 Conclusions






04

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(a)

06

04

02

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(b)


08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(a)

1

08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(b)







Appendix

B1(y t)

0 0lt tltyλ

eminus α1t

I0 iα3

t2

minus λy2

1113969

1113874 1113875 tgtyλ

⎧⎪⎪⎨

⎪⎪⎩(A1)

BII2 (t) e



I0 iα3t( 11138571113872 1113873lowastt (A2)

BII3 (t) (1 + λH(t))

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BII4 (t) (1 + λH(t)

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIII2 (t) e






Alowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste




Dlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIV2 (t) e


lowastH(t) + α1( 1113857

lowaste

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t (A8)

BIV3 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BIV4 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste

minus α1tI0 iα3t( 1113857 (A10)

BV2 (t) e


lowaste

t+ 1 + α1( 1113857te

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowastte

t (A11)

BV3 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A12)

BV4 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A13)

BVI2 (t) e


lowaste

t cos t + 1 + α1( 1113857et sin t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t sin t(A14)

BVI3 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A15)


BVI4 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A16)

BVII2 (t) e



minus α1tI0 iα3t( 11138571113872 1113873

lowastt sin t

(A17)



lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A18)



lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A19)

Lminus 1 e

minus bq2minus a2

radic

q2

minus a2

1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

0 0lt tlt b bgt 0

I0 a

t2

minus b2

1113969

1113888 1113889 tgt bRe(q)gtRe(a)

⎧⎪⎪⎨

⎪⎪⎩(A20)

Lminus 1


(A21)


q + a

radic)

q minus b1113890 1113891 e


Φ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A23)


q + a

radic)

q minus b1113890 1113891 e


Ψ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A25)

Data Availability




Acknowledgments


References































3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = 1 g (t) = 1 h (t) = 1

(a)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t g (t) = 1 h (t) = 1

(b)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = sin t g (t) = 1 h (t) = 1

(c)

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et g (t) = 1 h (t) = 1

(d)

2

0

ndash2

ndash4

ndash6

ndash8

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = tet g (t) = 1 h (t) = 1

(e)

5

4

3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = et sin t g (t) = 1 h (t) = 1

(f )

Figure 16 Continued


3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


3

2

1

00 1 2 3 54

y

M = 20M = 15M = 05

M = 10


Vel

ocity

(a)

1

0

ndash1

ndash2

ndash30 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

f (t) = 1 g (t) = sin t h (t) = tet

Vel

ocity

(b)

2

0

ndash2

ndash40 1 2 3 54

y

Gc = 80Gc = 60Gc = 20

Gc = 40


Vel

ocity

(c)

6

4

2

00 1 2 3 54

y

GT = 10GT = 60GT = 10

GT = 35

f (t) = 1 g (t) = sin t h (t) = t

Vel

ocity

(d)

Figure 17 Continued


2

1

0

ndash1

ndash2

ndash30 1 2 3 54

y

Sc = 078Sc = 060Sc = 022

Sc = 030


eloc

ity

(e)

3

2

1

0

ndash10 1 2 3 54

y

S = 40S = 30S = 10

S = 20

f (t) = t g (t) = et h (t) = 1

Vel

ocity

(f )


04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(a)

06

04

02

0

ndash02

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(b)



6 Conclusions






04

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(a)

06

04

02

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(b)


08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(a)

1

08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(b)







Appendix

B1(y t)

0 0lt tltyλ

eminus α1t

I0 iα3

t2

minus λy2

1113969

1113874 1113875 tgtyλ

⎧⎪⎪⎨

⎪⎪⎩(A1)

BII2 (t) e



I0 iα3t( 11138571113872 1113873lowastt (A2)

BII3 (t) (1 + λH(t))

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BII4 (t) (1 + λH(t)

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIII2 (t) e






Alowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste




Dlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIV2 (t) e


lowastH(t) + α1( 1113857

lowaste

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t (A8)

BIV3 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BIV4 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste

minus α1tI0 iα3t( 1113857 (A10)

BV2 (t) e


lowaste

t+ 1 + α1( 1113857te

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowastte

t (A11)

BV3 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A12)

BV4 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A13)

BVI2 (t) e


lowaste

t cos t + 1 + α1( 1113857et sin t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t sin t(A14)

BVI3 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A15)


BVI4 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A16)

BVII2 (t) e



minus α1tI0 iα3t( 11138571113872 1113873

lowastt sin t

(A17)



lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A18)



lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A19)

Lminus 1 e

minus bq2minus a2

radic

q2

minus a2

1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

0 0lt tlt b bgt 0

I0 a

t2

minus b2

1113969

1113888 1113889 tgt bRe(q)gtRe(a)

⎧⎪⎪⎨

⎪⎪⎩(A20)

Lminus 1


(A21)


q + a

radic)

q minus b1113890 1113891 e


Φ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A23)


q + a

radic)

q minus b1113890 1113891 e


Ψ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A25)

Data Availability




Acknowledgments


References































3

2

1

0

Velo

city

0 1 2 3 54y

S = 4S = 3S = 1

S = 2

f (t) = t sin t g (t) = 1 h (t) = 1

(g)


3

2

1

00 1 2 3 54

y

M = 20M = 15M = 05

M = 10


Vel

ocity

(a)

1

0

ndash1

ndash2

ndash30 1 2 3 4

y

Kc = 65Kc = 45Kc = 05

Kc = 25

f (t) = 1 g (t) = sin t h (t) = tet

Vel

ocity

(b)

2

0

ndash2

ndash40 1 2 3 54

y

Gc = 80Gc = 60Gc = 20

Gc = 40


Vel

ocity

(c)

6

4

2

00 1 2 3 54

y

GT = 10GT = 60GT = 10

GT = 35

f (t) = 1 g (t) = sin t h (t) = t

Vel

ocity

(d)

Figure 17 Continued


2

1

0

ndash1

ndash2

ndash30 1 2 3 54

y

Sc = 078Sc = 060Sc = 022

Sc = 030


eloc

ity

(e)

3

2

1

0

ndash10 1 2 3 54

y

S = 40S = 30S = 10

S = 20

f (t) = t g (t) = et h (t) = 1

Vel

ocity

(f )


04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(a)

06

04

02

0

ndash02

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(b)



6 Conclusions






04

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(a)

06

04

02

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(b)


08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(a)

1

08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(b)







Appendix

B1(y t)

0 0lt tltyλ

eminus α1t

I0 iα3

t2

minus λy2

1113969

1113874 1113875 tgtyλ

⎧⎪⎪⎨

⎪⎪⎩(A1)

BII2 (t) e



I0 iα3t( 11138571113872 1113873lowastt (A2)

BII3 (t) (1 + λH(t))

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BII4 (t) (1 + λH(t)

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIII2 (t) e






Alowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste




Dlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIV2 (t) e


lowastH(t) + α1( 1113857

lowaste

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t (A8)

BIV3 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BIV4 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste

minus α1tI0 iα3t( 1113857 (A10)

BV2 (t) e


lowaste

t+ 1 + α1( 1113857te

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowastte

t (A11)

BV3 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A12)

BV4 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A13)

BVI2 (t) e


lowaste

t cos t + 1 + α1( 1113857et sin t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t sin t(A14)

BVI3 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A15)


BVI4 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A16)

BVII2 (t) e



minus α1tI0 iα3t( 11138571113872 1113873

lowastt sin t

(A17)



lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A18)



lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A19)

Lminus 1 e

minus bq2minus a2

radic

q2

minus a2

1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

0 0lt tlt b bgt 0

I0 a

t2

minus b2

1113969

1113888 1113889 tgt bRe(q)gtRe(a)

⎧⎪⎪⎨

⎪⎪⎩(A20)

Lminus 1


(A21)


q + a

radic)

q minus b1113890 1113891 e


Φ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A23)


q + a

radic)

q minus b1113890 1113891 e


Ψ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A25)

Data Availability




Acknowledgments


References































2

1

0

ndash1

ndash2

ndash30 1 2 3 54

y

Sc = 078Sc = 060Sc = 022

Sc = 030


eloc

ity

(e)

3

2

1

0

ndash10 1 2 3 54

y

S = 40S = 30S = 10

S = 20

f (t) = t g (t) = et h (t) = 1

Vel

ocity

(f )


04

03

02

01

0

ndash01

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(a)

06

04

02

0

ndash02

Tem

pera

ture

0y

Pr = 70Pr = 50Pr = 071

Pr = 15

05 1 15 2


(b)



6 Conclusions






04

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(a)

06

04

02

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(b)


08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(a)

1

08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(b)







Appendix

B1(y t)

0 0lt tltyλ

eminus α1t

I0 iα3

t2

minus λy2

1113969

1113874 1113875 tgtyλ

⎧⎪⎪⎨

⎪⎪⎩(A1)

BII2 (t) e



I0 iα3t( 11138571113872 1113873lowastt (A2)

BII3 (t) (1 + λH(t))

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BII4 (t) (1 + λH(t)

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIII2 (t) e






Alowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste




Dlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIV2 (t) e


lowastH(t) + α1( 1113857

lowaste

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t (A8)

BIV3 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BIV4 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste

minus α1tI0 iα3t( 1113857 (A10)

BV2 (t) e


lowaste

t+ 1 + α1( 1113857te

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowastte

t (A11)

BV3 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A12)

BV4 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A13)

BVI2 (t) e


lowaste

t cos t + 1 + α1( 1113857et sin t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t sin t(A14)

BVI3 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A15)


BVI4 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A16)

BVII2 (t) e



minus α1tI0 iα3t( 11138571113872 1113873

lowastt sin t

(A17)



lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A18)



lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A19)

Lminus 1 e

minus bq2minus a2

radic

q2

minus a2

1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

0 0lt tlt b bgt 0

I0 a

t2

minus b2

1113969

1113888 1113889 tgt bRe(q)gtRe(a)

⎧⎪⎪⎨

⎪⎪⎩(A20)

Lminus 1


(A21)


q + a

radic)

q minus b1113890 1113891 e


Φ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A23)


q + a

radic)

q minus b1113890 1113891 e


Ψ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A25)

Data Availability




Acknowledgments


References































6 Conclusions






04

03

02

01

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(a)

06

04

02

0

Tem

pera

ture

0y

S = 45S = 30S = 05

S = 15

05 1 15 2


(b)


08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(a)

1

08

06

04

02

0

ndash02

Tem

pera

ture

0y

a = 20 b = 40a = 15 b = 30a = 05 b = 10

a = 10 b = 20

05 1 15 2


(b)







Appendix

B1(y t)

0 0lt tltyλ

eminus α1t

I0 iα3

t2

minus λy2

1113969

1113874 1113875 tgtyλ

⎧⎪⎪⎨

⎪⎪⎩(A1)

BII2 (t) e



I0 iα3t( 11138571113872 1113873lowastt (A2)

BII3 (t) (1 + λH(t))

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BII4 (t) (1 + λH(t)

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIII2 (t) e






Alowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste




Dlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIV2 (t) e


lowastH(t) + α1( 1113857

lowaste

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t (A8)

BIV3 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BIV4 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste

minus α1tI0 iα3t( 1113857 (A10)

BV2 (t) e


lowaste

t+ 1 + α1( 1113857te

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowastte

t (A11)

BV3 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A12)

BV4 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A13)

BVI2 (t) e


lowaste

t cos t + 1 + α1( 1113857et sin t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t sin t(A14)

BVI3 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A15)


BVI4 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A16)

BVII2 (t) e



minus α1tI0 iα3t( 11138571113872 1113873

lowastt sin t

(A17)



lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A18)



lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A19)

Lminus 1 e

minus bq2minus a2

radic

q2

minus a2

1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

0 0lt tlt b bgt 0

I0 a

t2

minus b2

1113969

1113888 1113889 tgt bRe(q)gtRe(a)

⎧⎪⎪⎨

⎪⎪⎩(A20)

Lminus 1


(A21)


q + a

radic)

q minus b1113890 1113891 e


Φ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A23)


q + a

radic)

q minus b1113890 1113891 e


Ψ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A25)

Data Availability




Acknowledgments


References



































Appendix

B1(y t)

0 0lt tltyλ

eminus α1t

I0 iα3

t2

minus λy2

1113969

1113874 1113875 tgtyλ

⎧⎪⎪⎨

⎪⎪⎩(A1)

BII2 (t) e



I0 iα3t( 11138571113872 1113873lowastt (A2)

BII3 (t) (1 + λH(t))

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BII4 (t) (1 + λH(t)

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIII2 (t) e






Alowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste




Dlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


BIV2 (t) e


lowastH(t) + α1( 1113857

lowaste

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t (A8)

BIV3 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


BIV4 (t) e

t+ λH(t)

lowaste

t1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste

minus α1tI0 iα3t( 1113857 (A10)

BV2 (t) e


lowaste

t+ 1 + α1( 1113857te

t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowastte

t (A11)

BV3 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A12)

BV4 (t) e

t(1 + t) + λH(t)

lowaste

t(1 + t)1113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A13)

BVI2 (t) e


lowaste

t cos t + 1 + α1( 1113857et sin t1113872 1113873 + α23e

minus α1tI0 iα3t( 11138571113872 1113873

lowaste

t sin t(A14)

BVI3 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A15)


BVI4 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A16)

BVII2 (t) e



minus α1tI0 iα3t( 11138571113872 1113873

lowastt sin t

(A17)



lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A18)



lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A19)

Lminus 1 e

minus bq2minus a2

radic

q2

minus a2

1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

0 0lt tlt b bgt 0

I0 a

t2

minus b2

1113969

1113888 1113889 tgt bRe(q)gtRe(a)

⎧⎪⎪⎨

⎪⎪⎩(A20)

Lminus 1


(A21)


q + a

radic)

q minus b1113890 1113891 e


Φ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A23)


q + a

radic)

q minus b1113890 1113891 e


Ψ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A25)

Data Availability




Acknowledgments


References































BVI4 (t) e


lowaste

t cos t + et sin t1113872 11138731113872 1113873

lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A16)

BVII2 (t) e



minus α1tI0 iα3t( 11138571113872 1113873

lowastt sin t

(A17)



lowastAlowast

+ Blowaste


minus α4minus α6( )t1113874 1113875

lowaste


(A18)



lowastDlowast

+ Elowaste


minus α7minus α9( )t1113874 1113875

lowaste


(A19)

Lminus 1 e

minus bq2minus a2

radic

q2

minus a2

1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

0 0lt tlt b bgt 0

I0 a

t2

minus b2

1113969

1113888 1113889 tgt bRe(q)gtRe(a)

⎧⎪⎪⎨

⎪⎪⎩(A20)

Lminus 1


(A21)


q + a

radic)

q minus b1113890 1113891 e


Φ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A23)


q + a

radic)

q minus b1113890 1113891 e


Ψ(y t a) 12

ey

a

radic

ercfy

2t

radic +at

radic1113888 1113889 + e

minus ya

radic

ercfy

2t

radic minusat

radic1113888 11138891113896 1113897 (A25)

Data Availability




Acknowledgments


References






















































Documents

Combined Effects of Heat and Mass Transfer on MHD Free