Research ArticleUnsteady MHD Mixed Convection Flow ofChemically Reacting Micropolar Fluid between PorousParallel Plates with Soret and Dufour Effects

Odelu Ojjela and N Naresh Kumar

Department of Applied Mathematics Defence Institute of Advanced Technology (Deemed University) Pune 411025 India

Correspondence should be addressed to Odelu Ojjela odelu3yahoocoin

Received 15 November 2015 Accepted 26 January 2016

Academic Editor Oronzio Manca

Copyright copy 2016 O Ojjela and N Naresh Kumar This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

The objective of the present study is to investigate the first-order chemical reaction and Soret and Dufour effects on anincompressible MHD combined free and forced convection heat and mass transfer of a micropolar fluid through a porous mediumbetween two parallel plates Assume that there are a periodic injection and suction at the lower and upper plates The nonuniformtemperature and concentration of the plates are assumed to be varying periodically with time A suitable similarity transformationis used to reduce the governing partial differential equations into nonlinear ordinary differential equations and then solvednumerically by the quasilinearization method The fluid flow and heat and mass transfer characteristics for various parametersare analyzed in detail and shown in the form of graphs It is observed that the concentration of the fluid decreases whereas thetemperature of the fluid enhances with the increasing of chemical reaction and Soret and Dufour parameters

1 Introduction

The flow through porous boundaries has many applicationsin science and technology such as water waves over a shallowbeach mechanics of the cochlea in the human ear aerody-namic heating flow of blood in the arteries and petroleumindustry Several authors have studied theoretically the lam-inar flow in porous channels Berman [1] considered theviscous fluid and analyzed the flow characteristics when itpassed through the porous walls Later the same problem fordifferent permeability was studied by Terril and Shrestha [2]The theory of micropolar fluids was introduced by Eringen[3] which are considered as an extension of generalizedviscous fluids with microstructure Examples for micropolarfluids include lubricants colloidal suspensions porous rocksaerogels polymer blends and microemulsions The sameBerman problem with micropolar fluid was discussed bySastry and Rama Mohan Rao [4] The flow and heat transferof micropolar fluid between two porous parallel plates wasanalyzed by Ojjela and Naresh Kumar [5] Srinivasacharya

et al [6] obtained an analytical solution for the unsteadyStokes flow of micropolar fluid between two parallel platesThe effect of buoyancy parameter on flow and heat transferof micropolar fluid between two vertical parallel plates wasinvestigated by Maiti [7]

The study ofMHDheat andmass transfer through porousboundaries has attracted many researchers in the recentpast due to applications in engineering and science such asoil exploration boundary layer control and MHD powergenerators The steady incompressible free convection flowand heat transfer of an electrically conducting micropolarfluid in a vertical channel was studied by Bhargava et al[8] The laminar incompressible magnetohydrodynamic flowand heat transfer of micropolar fluid between porous diskswas analyzed numerically by Ashraf and Wehgal [9] Islamet al [10] obtained a numerical solution for an incompress-ible unsteady magnetohydrodynamic flow through verticalporous medium Nadeem et al [11] discussed the unsteadyMHD stagnation flow of a micropolar fluid through porousmediaThe effects of Hall and ion slip currents onmicropolar

Hindawi Publishing CorporationJournal of EngineeringVolume 2016 Article ID 6531948 13 pageshttpdxdoiorg10115520166531948

2 Journal of Engineering

fluid flow and heat and mass transfer in a porous mediumbetween parallel plates with chemical reaction were consid-ered by Ojjela and Naresh Kumar [12] The MHD heat andmass transfer of micropolar fluid in a porous medium withchemical reaction andHall and ion slip effects by consideringvariable viscosity and thermal diffusivity were investigated byElgazery [13] The mixed convection flow and heat transferof an electrically conducting micropolar fluid over a verticalplatewithHall and ion slip effectswas analyzed byAyano [14]

When heat and mass transfer occurs simultaneously ina moving fluid the energy flux caused by a concentrationgradient is termed as diffusion thermoeffect whereas massfluxes can also be created by temperature gradients whichis known as a thermal diffusion effect These effects arestudied as second-order phenomena andmay have significantapplications in areas like petrology hydrology and geo-sciencesThe effect of thermophoresis on an unsteady naturalconvection flow and heat and mass transfer of micropolarfluidwith Soret andDufour effects was studied byAurangzaibet al [15] Srinivasacharya and RamReddy [16] consideredthe problem of the steady MHD mixed convection heat andmass transfer of micropolar fluid through non-Darcy porousmedium over a semi-infinite vertical plate with Soret andDufour effects Influence of the Soret and Dufour numberson mixed convection flow and heat and mass transfer ofnon-Newtonian fluid in a porous medium over a verticalplate was analyzed by Mahdy [17] Hayat and Nawaz [18]investigated analytically the effects of the Hall and ion slipon the mixed convection heat and mass transfer of second-grade fluid with Soret and Dufour effects Rani and Kim [19]studied numerically the laminar flow of an incompressibleviscous fluid past an isothermal vertical cylinder with SoretandDufour effectsThe effects of chemical reaction and Soretand Dufour on the mixed convection heat and mass transferof viscous fluid over a stretching surface in the presence ofthermal radiation were analyzed by Pal and Mondal [20]Sharma et al [21] studied the mixed convective flow heatand mass transfer of viscous fluid in a porous medium past aradiative vertical plate with chemical reaction and Soret andDufour effects

In the field of fluid mechanics many fluid flow problemsare nonlinear boundary value problems To solve these prob-lems we can use a numerical technique quasilinearizationmethod which is a powerful technique having second-orderconvergence Several authors (Lee and Fan [22] Hymavathiand Shanker [23] Huang [24] Motsa et al [25] and Ojjelaand Naresh Kumar [5 12]) applied the quasilinearizationmethod to solve the nonlinear boundary layer equations

In the present study the effects of chemical reaction ontwo-dimensional mixed convection flow and heat transferof an electrically conducting micropolar fluid in a porousmedium between two parallel plates with Soret and Dufourhave been considered The reduced flow field equationsare solved using the quasilinearization method The effectsof various parameters such as Hartmann number inverseDarcyrsquos parameter Schmidt number Prandtl number chem-ical reaction rate Soret and Dufour numbers on the velocitycomponents microrotation temperature distribution and

Y

h

0

Z

X

V2ei120596t

T2ei120596t

C1ei120596t

V1ei120596t

T1ei120596t

C0ei120596t

B

Figure 1 Schematic diagram of the fluid flow between parallelporous plates

concentration are studied in detail and presented in the formof graphs

2 Formulation of the Problem

Consider a two-dimensional laminar incompressiblemicrop-olar fluid flow through an elongated rectangular channelas shown in Figure 1 Assume that the fluid is injected andaspirated periodically through the plates with injection veloc-ity 1198811119890119894120596119905 and suction velocity 119881

2119890119894120596119905 Also the nonuniform

temperature and concentration at the lower and upper platesare 1198791119890119894120596119905 1198620119890119894120596119905 and 119879

2119890119894120596119905 1198621119890119894120596119905 respectively The region

inside the parallel plates is subjected to porousmedium and aconstant external magnetic field of strength 119861

0perpendicular

to the119883119884-plane is consideredThe governing equations of the micropolar fluid flow and

heat and mass transfer in the presence of buoyancy forcesmagnetic field and in the absence of body forces body couplesare given by

nabla sdot 119902 = 0 (1)

120588 [120597119902

120597119905+ (119902 sdot nabla) 119902] = minusgrad119901 + 119896

1curl 119897

minus (120583 + 1198961) curl curl (119902)

minus120583 + 1198961

1198962

119902 + 119869 times 119861 + 119865119887

(2)

120588119895 [120597119897

120597119905+ (119902 sdot nabla) 119897] = minus2119896

1119897 + 1198961curl 119902

minus 120574 curl curl (119897) (3)

120588119888 [120597119879

120597119905+ (119902 sdot nabla) 119879] = 119896nabla

2119879 + 2120583119863 119863

+1198961

2(curl (119902) minus 2119897)

2

+ 120574nabla119897 nabla119897 +120583 + 1198961

1198962

1199022

+

1003816100381610038161003816100381611986910038161003816100381610038161003816

2

120590+1205881198631119896119879

119888119904

nabla2119862

(4)

Journal of Engineering 3

[120597119862

120597119905+ (119902 sdot nabla) 119862] = 119863

1nabla2119862 minus 1198963(119862 minus 119862

0119890119894120596119905)

+1198631119896119879

119879119898

nabla2119879

(5)

where 119865119887is the buoyancy force and it is defined as (120588119892120573

119879(119879 minus

1198791119890119894120596119905) + 120588119892120573

119862(119862 minus 119862

0119890119894120596119905))119894

Neglecting the displacement currents the Maxwell equa-tions and the generalized Ohmrsquos law are

nabla sdot 119861 = 0

nabla times 119861 = 1205831015840119869

nabla times 119864 =120597119861

120597119905

119869 = 120590 (119864 + 119902 times 119861)

(6)

where 119861 = 1198610 + 119887 119887 is induced magnetic field Assume

that the induced magnetic field is negligible compared to theapplied magnetic field so that magnetic Reynolds number issmall the electric field is zero and magnetic permeability isconstant throughout the flow field

The velocity and microrotation components are

119902 = 119906119894 + V

119897 = 119873

(7)

Following Ojjela and Naresh Kumar [5 12] the velocity andmicrorotation components are

119906 (119909 120582 119905) = (1198800

119886minus1198812119909

ℎ)1198911015840(120582) 119890119894120596119905

V (119909 120582 119905) = 1198812119891 (120582) 119890

119894120596119905

119873 (119909 120582 119905) = (1198800

119886minus1198812119909

ℎ)119860 (120582) 119890

119894120596119905

(8)

The temperature and concentration distributions can betaken as

119879 (119909 120582 119905)

= (1198791+1205831198812

120588ℎ119888[1206011(120582) + (

1198800

1198861198812

minus119909

ℎ)

2

1206012(120582)]) 119890

119894120596119905

119862 (119909 120582 119905)

= (1198620+

119899119860

ℎ120592[1198921(120582) + (

1198800

1198861198812

minus119909

ℎ)

2

1198922(120582)]) 119890

119894120596119905

(9)

where 120582 = 119910ℎ and 119891(120582)119860(120582) 1206011(120582) 1206012(120582) 1198921(120582) and 119892

2(120582)

are to be determined

The boundary conditions for the velocity microrotationtemperature and concentration are

119906 (119909 120582 119905) = 0

V (119909 120582 119905) = 1198811119890119894120596119905

119873 (119909 120582 119905) = 0

119879 (119909 120582 119905) = 1198791119890119894120596119905

119862 (119909 120582 119905) = 1198620119890119894120596119905

at 120582 = 0

119906 (119909 120582 119905) = 0

V (119909 120582 119905) = 1198812119890119894120596119905

119873 (119909 120582 119905) = 0

119879 (119909 120582 119905) = 1198792119890119894120596119905

119862 (119909 120582 119905) = 1198621119890119894120596119905

at 120582 = 1

(10)

Substituting (8) and (9) in (2) (3) (4) and (5) then we get

1198911015840119881=minus119877

1 + 11987711986010158401015840+

Re1 + 119877

(119891119891101584010158401015840minus 119891101584011989110158401015840) cos120595

+Ha2

1 + 11987711989110158401015840+ 119863minus111989110158401015840minus

EcGr(1 + 119877) 120577

(1206011015840

1+ 12057721206011015840

2)

minusShGm(1 + 119877) 120577

(1198921015840

1+ 12057721198921015840

2)

1198691(1198911198601015840minus 1198911015840119860) cos120595 = minus119904

1(2119860 + 119891

10158401015840) + 11986010158401015840

12060110158401015840

1= minus2120601

2minus Re11990421198602 cos120595 minus Re Pr ((1 + 119877)119863minus11198912

+Ha21198912 + 411989110158402 minus 11989112060110158401) cos120595 minus Du (11989210158401015840

1+ 21198922)

12060110158401015840

2= minusRe Pr(119877

2(11989110158401015840+ 2119860)

2

+1199042

Pr11986010158402

+ (1 + 119877)119863minus111989110158402+Ha211989110158402 + 119891101584010158402 + 21198911015840120601

2minus 1198911206011015840

2)

sdot cos120595 minus Du119892101584010158402

11989210158401015840

1= minus2119892

2+ Kr119892

1+ Sc Re1198911198921015840

1cos120595 minus Sc Sr (12060110158401015840

1

+ 21206012)

11989210158401015840

2= Kr119892

2+ Sc Re (1198911198921015840

2minus 211989110158401198922) cos120595 minus Sc Sr12060110158401015840

2

(11)

where the prime denotes the differentiation with respect to 120582

4 Journal of Engineering

The dimensionless forms of temperature and concentra-tion from (9) are

119879lowast=

119879 minus 1198791119890119894120596119905

(1198792minus 1198791) 119890119894120596119905

= Ec (1206011+ 12057721206012)

119862lowast=

119862 minus 1198620119890119894120596119905

(1198621minus 1198620) 119890119894120596119905

= Sh (1198921+ 12057721198922)

(12)

The boundary conditions (10) in terms of119891119860 1206011 1206012 1198921 and

1198922are

119891 (0) = 1 minus 119886

119891 (1) = 1

1198911015840(0) = 0

1198911015840(1) = 0

119860 (0) = 0

119860 (1) = 0

1206011(0) = 0

1206011(1) =

1

Ec

1206012(0) = 0

1206012(1) = 0

1198921(0) = 0

1198921(1) =

1

Sh

1198922(0) = 0

1198922(1) = 0

(13)

For micropolar fluids the shear stress 120591119896119897is given by

120591119896119897= (minus119901 + 120578V

119903119903) 120575119896119897+ 2120583119889

119896119897+ 21198961119890119896119897119903(120596119903minus 119897119903) (14)

Then the nondimensional shear stress at the lower and upperplates is

119878119891=2120591119896119897

1205881198812

2

= [2

Re(1198800

1198861198812

minus119909

ℎ) (119877 minus 1) 119891

10158401015840(120582) cos120595]

120582=01

(15)

For micropolar fluids the couple stress119872119894119895is given by

119872119894119895= 120572119897119903119903120575119894119895+ 120573119897119894119895+ 120574119897119895119894 (16)

Then the nondimensional couple stress at lower and upperplates is

119898 = [(1198800

1198861198812

minus119909

ℎ)1198921015840(120582) cos120595]

120582=01

(17)

3 Solution of the Problem

The nonlinear equations (11) are converted into the followingsystem of first-order differential equations by the substitution

(119891 1198911015840 11989110158401015840 119891101584010158401015840 119860 1198601015840 1206011 1206011015840

1 1206012 1206011015840

2 1198921 1198921015840

1 1198922 1198921015840

2)

= (1199091 1199092 1199093 1199094 1199095 1199096 1199097 1199098 1199099 11990910 11990911 11990912 11990913 11990914)

1198891199091

119889120582= 1199092

1198891199092

119889120582= 1199093

1198891199093

119889120582= 1199094

1198891199094

119889120582=minus119877

1 + 119877(1199041(1199093+ 21199095) + 1198691(11990911199096minus 11990921199095)) cos120595

+Re1 + 119877

(11990921199093minus 11990911199094) cos120595 + 119863minus1119909

3+

Ha2

1 + 1198771199093

minusEcGr(1 + 119877) 120577

(1199098+ 120577211990910) minus

ShGm(1 + 119877) 120577

(11990912

+ 120577211990914)

1198891199095

119889120582= 1199096

1198891199096

119889120582= 1199041(1199093+ 21199095) + 1198691(11990911199096minus 11990921199095) cos120595

1198891199097

119889120582= 1199098

1198891199098

119889120582= minus2119909

9minus

Re Pr1 minus Du Sc Sr

(41199092

2+ (1 + 119877)119863

minus11199092

1

+Ha211990921minus 11990911199098+1199042

Pr1199092

5) cos120595 minus KrDu

1 minus DuSc Sr11990911

+Du Sc

1 minus Du Sc SrRe119909111990912cos120595

1198891199099

119889120582= 11990910

11988911990910

119889120582= minus

Re Pr1 minus Du Sc Sr

(119877

2(1199093+ 21199095)2

+ 1199092

3+ (1 + 119877)

sdot 119863minus11199092

2+Ha21199092

2+ 211990921199099minus 119909111990910) cos120595

minusKrDu

1 minus Du Sc Sr11990913minus

Du Sc1 minus Du Sc Sr

Re (119909111990914

minus 2119909211990913) cos120595

11988911990911

119889120582= 11990912

11988911990912

119889120582= minus2119909

13+

Sc Sr Re Pr1 minus Du Sc Sr

(41199092

2+ (1 + 119877)119863

minus11199092

1

+Ha211990921minus 11990911199098+1199042

Pr1199092

5) cos120595 + Kr

1 minus DuSc Sr11990911

+Sc

1 minus Du Sc SrRe119909111990912cos120595

Journal of Engineering 5

11988911990913

119889120582= 11990914

11988911990914

119889120582=

Sc Sr Re Pr1 minus DuSc Sr

(1199092

3+ (1 + 119877)119863

minus11199092

2+Ha21199092

2

+1199042

Pr1199092

6+119877

2(1199093+ 21199095)2

+ 211990921199099minus 119909111990910) cos120595

+Kr

1 minus Du Sc Sr11990913+

Sc1 minus DuSc Sr

Re (119909111990914

minus 2119909211990913) cos120595

(18)

The boundary conditions in terms of 1199091 1199092 1199093 11990941199095 1199096 1199097

1199098 1199099 11990910 11990911 11990912 11990913 11990914are

1199091(0) = 1 minus 119886

1199092(0) = 0

1199095(0) = 0

1199097(0) = 0

1199099(0) = 0

11990911(0) = 0

11990913(0) = 0

1199091(1) = 1

1199092(1) = 0

1199095(1) = 0

1199097(1) =

1

Ec

1199099(1) = 0

11990911(1) =

1

Sh

11990913(1) = 0

(19)

The system of (18) is solved numerically subject to boundaryconditions (19) using the quasilinearization method given byBellman and Kalaba [26]

Let (119909119903119894 119894 = 1 2 14) be an approximate current

solution and let (119909119903+1119894

119894 = 1 2 14) be an improvedsolution of (18) Using Taylorrsquos series expansion about thecurrent solution by neglecting the second- and higher-orderderivative terms coupled first-order system (18) is linearizedas

119889119909119903+1

1

119889120582= 119909119903+1

2

119889119909119903+1

2

119889120582= 119909119903+1

3

119889119909119903+1

3

119889120582= 119909119903+1

4

119889119909119903+1

4

119889120582= minus

Re1 + 119877

(119909119903+1

1119909119903

4+ 119909119903+1

4119909119903

1minus 119909119903+1

2119909119903

3

minus 119909119903

2119909119903+1

3) cos120595 + 119863minus1119909119903+1

3+

Ha2

1 + 119877119909119903+1

3

minus119877

1 + 119877(1199041(119909119903+1

3+ 2119909119903+1

5) + 1198691(119909119903+1

1119909119903

6+ 119909119903

1119909119903+1

6

minus 119909119903+1

2119909119903

5minus 119909119903+1

5119909119903

2) cos120595) minus EcGr

(1 + 119877) 120577(119909119903+1

8

+ 1205772119909119903+1

10) minus

ShGm(1 + 119877) 120577

(119909119903+1

12+ 1205772119909119903+1

14)

+Re1 + 119877

(minus119909119903

2119909119903

3+ 119909119903

1119909119903

4) cos120595 + 119877119869

1

1 + 119877(119909119903

1119909119903

6

minus 119909119903

2119909119903

5) cos120595

119889119909119903+1

5

119889120582= 119909119903+1

6

119889119909119903+1

6

119889120582= 1199041(119909119903+1

3+ 2119909119903+1

5) + 1198691(119909119903+1

1119909119903

6+ 119909119903

1119909119903+1

6

minus 119909119903+1

2119909119903

5minus 119909119903+1

5119909119903

2) cos120595 minus 119869

1(119909119903

1119909119903

6minus 119909119903

2119909119903

5) cos120595

119889119909119903+1

7

119889120582= 119909119903+1

8

119889119909119903+1

8

119889120582= minus2119909

119903+1

9minus

Re Pr1 minus Du Sc Sr

(8119909119903

2119909119903+1

2+ 2 (1 + 119877)

sdot 119863minus1119909119903

1119909119903+1

1+ 2Ha2119909119903

1119909119903+1

1minus 119909119903

1119909119903+1

8minus 119909119903

8119909119903+1

1

+21199042

Pr119909119903

5119909119903+1

5) cos120595 minus KrDu

1 minus Du Sc Sr119909119903+1

11

+Du Sc

1 minus Du Sc SrRe (1199091199031119909119903+1

12+ 119909119903

12119909119903+1

1minus 119909119903

1119909119903

12)

sdot cos120595 + Re Pr1 minus DuSc Sr

(4119909119903

2119909119903

2+ (1 + 119877)119863

minus1119909119903

1119909119903

1

+Ha21199091199031119909119903

1minus 119909119903

1119909119903

8+1199042

Pr119909119903

5119909119903

5) cos120595

119889119909119903+1

9

119889120582= 119909119903+1

10

119889119909119903+1

10

119889120582= minus

Re Pr1 minus Du Sc Sr

(119877

2(2119909119903

3119909119903+1

3+ 8119909119903

5119909119903+1

5

+ 4119909119903

3119909119903+1

5+ 4119909119903

5119909119903+1

3) + 2119909

119903

3119909119903+1

3+ 2 (1 + 119877)

sdot 119863minus1119909119903

2119909119903+1

2+ 2Ha2119909119903

2119909119903+1

2+ 2119909119903

2119909119903+1

9+ 2119909119903+1

2119909119903

9

minus 119909119903

1119909119903+1

10minus 119909119903+1

1119909119903

10) cos120595 minus DuSc Re

1 minus Du Sc Sr(119909119903

1119909119903+1

14

+ 119909119903+1

1119909119903

14minus 2119909119903

2119909119903+1

13minus 2119909119903+1

2119909119903

13minus 119909119903

1119909119903

14

+ 2119909119903

2119909119903

13) cos120595 + Re Pr

1 minus Du Sc Sr(119877

2(119909119903

3119909119903

3+ 4119909119903

5119909119903

5

+ 4119909119903

3119909119903

5) + 119909119903

3119909119903

3+ (1 + 119877)119863

minus1119909119903

2119909119903

2+Ha2119909119903

2119909119903

2

+ 2119909119903

2119909119903

9minus 119909119903

1119909119903

10) cos120595 minus KrDu

1 minus Du Sc Sr119909119903+1

13

6 Journal of Engineering

119889119909119903+1

11

119889120582= 119909119903+1

12

119889119909119903+1

12

119889120582= minus2119909

119903+1

13+

Sc Sr Re Pr1 minus Du Sc Sr

(8119909119903

2119909119903+1

2+ 2 (1 + 119877)

sdot 119863minus1119909119903

1119909119903+1

1+ 2Ha2119909119903

1119909119903+1

1minus 119909119903

1119909119903+1

8minus 119909119903+1

1119909119903

8

+21199042

Pr119909119903

5119909119903+1

5) cos120595 + Kr

1 minus DuSc Sr119909119903+1

11

+Sc Re

1 minus Du Sc Sr(119909119903

1119909119903+1

12+ 119909119903+1

1119909119903

12minus 119909119903

1119909119903

12) cos120595

minusSc Sr Re Pr1 minus Du Sc Sr

(4119909119903

2119909119903

2+ (1 + 119877)119863

minus1119909119903

1119909119903

1

+Ha21199091199031119909119903

1minus 119909119903

1119909119903

8+1199042

Pr119909119903

5119909119903

5) cos120595

119889119909119903+1

13

119889120582= 119909119903+1

14

119889119909119903+1

14

119889120582=

Sc Sr Re Pr1 minus Du Sc Sr

(2119909119903+1

3119909119903

3+ 2 (1 + 119877)

sdot 119863minus1119909119903

2119909119903+1

2+ 2Ha2119909119903

2119909119903+1

2+21199042

Pr119909119903

6119909119903+1

6

+119877

2(2119909119903

3119909119903+1

3+ 8119909119903

5119909119903+1

5+ 4119909119903

3119909119903+1

5+ 4119909119903

5119909119903+1

3)

+ 2119909119903

2119909119903+1

9+ 2119909119903+1

2119909119903

9minus 119909119903

1119909119903+1

10minus 119909119903+1

1119909119903

10) cos120595

+Kr

1 minus Du Sc Sr119909119903+1

13+

Sc Re1 minus Du Sc Sr

(119909119903

1119909119903+1

14

+ 119909119903+1

1119909119903

14minus 2119909119903

2119909119903+1

13minus 2119909119903+1

2119909119903

13minus 119909119903

1119909119903

14

+ 2119909119903

2119909119903

13) cos120595 minus Sc Sr Re Pr

1 minus Du Sc Sr(119909119903

3119909119903

3+ (1 + 119877)

sdot 119863minus1119909119903

2119909119903

2+Ha2119909119903

2119909119903

2+1199042

Pr119909119903

6119909119903

6+119877

2(119909119903

3119909119903

3

+ 4119909119903

5119909119903

5+ 4119909119903

3119909119903

5) + 2119909

119903

2119909119903

9minus 119909119903

1119909119903

10) cos120595

(20)

To solve for (119909119903+1119894

119894 = 1 2 14) the solutions to sevenseparate initial value problems denoted by 119909ℎ1

119894(120582) 119909ℎ2

119894(120582)

119909ℎ3

119894(120582) 119909ℎ4

119894(120582) 119909ℎ5

119894(120582) 119909ℎ6

119894(120582) and 119909ℎ7

119894(120582) (which are the

solutions of the homogeneous system corresponding to (20))and 1199091199011

119894(120582) (which is the particular solution of (20)) with

the following initial conditions are obtained by using the 4th-order Runge-Kutta method

119909ℎ1

3(0) = 1

119909ℎ1

119894(0) = 0 for 119894 = 3

119909ℎ2

4(0) = 1

119909ℎ2

119894(0) = 0 for 119894 = 4

119909ℎ3

6(0) = 1

119909ℎ3

119894(0) = 0 for 119894 = 6

119909ℎ4

8(0) = 1

119909ℎ4

119894(0) = 0 for 119894 = 8

119909ℎ5

10(0) = 1

119909ℎ5

119894(0) = 0 for 119894 = 10

119909ℎ6

12(0) = 1

119909ℎ6

119894(0) = 0 for 119894 = 12

119909ℎ7

14(0) = 1

119909ℎ7

119894(0) = 0 for 119894 = 14

1199091199011

1(0) = 1 minus 119886

1199091199011

2(0) = 119909

1199011

3(0) = 119909

1199011

4(0) = 119909

1199011

5(0) = 0

1199091199011

6(0) = 119909

1199011

7(0) = 119909

1199011

8(0) = 119909

1199011

9(0) = 0

1199091199011

10(0) = 119909

1199011

11(0) = 119909

1199011

12(0) = 0

1199091199011

13(0) = 119909

1199011

14(0) = 0

(21)

By using the principle of superposition the general solutioncan be written as

119909119899+1

119894(120582) = 119862

1119909ℎ1

119894(120582) + 119862

2119909ℎ2

119894(120582) + 119862

3119909ℎ3

119894(120582)

+ 1198624119909ℎ4

119894(120582) + 119862

5119909ℎ5

119894(120582) + 119862

6119909ℎ6

119894(120582)

+ 1198627119909ℎ7

119894(120582) + 119909

1199011

119894(120582)

(22)

where 1198621 1198622 1198623 1198624 1198625 1198626 and 119862

7are the unknown

constants and are determined by considering the boundaryconditions at 120582 = 1 This solution (119909119903+1

119894 119894 = 1 2 14)

is then compared with solution at the previous step (119909119903119894

119894 = 1 2 14) and further iteration is performed if theconvergence has not been achieved

4 Results and Discussion

The system of nonlinear differential equations (18) subject toboundary conditions (19) is solved numerically by the quasi-linearizationmethodThe influences of various fluid and geo-metric parameters such as Soret number Sr Dufour numberDu Hartmann number Ha chemical reaction parameter KrSchmidt number Sc Eckert number Ec and Prandtl numberPr on nondimensional velocity components microrotationtemperature distribution and concentration are analyzedthrough graphs in the domain [0 1]

Figures 2 and 3 show the influence of Sr and Du ontemperature and concentration From these figures it is

Journal of Engineering 7

0

05

1

15

2

25

3

0 02 04 06 08 1

Tem

pera

ture

Sr = 01Sr = 05

Sr = 1Sr = 15

120582

(a)

0

02

04

06

08

1

12

0 02 04 06 08 1

Con

cent

ratio

n

120582

Sr = 01Sr = 05

Sr = 1Sr = 15

(b)

Figure 2 Effect of Sr on (a) temperature and (b) concentration for Kr = 2 Gr = 4 Gm = 4 Re = 2 Du = 2 Sc = 022 Pr = 1 119886 = 02 120595 = 02119877 = 02 119869

1= 02 119904

1= 2 119904

2= 10 Ha = 1119863minus1 = 2 and Ec = 1

0 02 04 06 08 1120582

0

05

1

15

2

25

3

35

Tem

pera

ture

Du = 01Du = 1

Du = 2Du = 3

(a)

Du = 01Du = 1

Du = 2Du = 3

0

02

04

06

08

1

12C

once

ntra

tion

02 04 06 08 10120582

(b)

Figure 3 Effect of Du on (a) temperature and (b) concentration for Kr = 2 Gr = 4 Gm = 4 Re = 2 Sr = 1 Sc = 022 Pr = 1 119886 = 02 120595 = 02119877 = 02 119869

1= 02 119904

1= 2 119904

2= 10 Ha = 1119863minus1 = 2 and Ec = 1

evident that the temperature of the fluid increases whereasthe concentration decreases with the increasing of Sr and DuThis is because of the difference between the temperatures ofthe fluid and surface as well as the difference between concen-trations of the fluid and surface concentrations are increasedwith Sr and Du Figure 4 describes the behavior of the tem-perature distribution and concentration for the various valuesof Kr AsKr increases the temperature distribution of the fluidalso increases whereas the concentration decreases from thelower plate to the upper plate It is clear that the increase inthe Kr produces a decrease in the species concentrationThiscauses the concentration buoyancy effects to decrease as Krincreases The effect of Ec on velocity components micro-rotation and temperature is presented in Figure 5 As Ec

increases the radial velocity microrotation and temperatureare decreasing towards the upper plate However the axialvelocity decreases towards the center of the channel and thenincreases Since the Eckert number is the relation betweenthe kinetic energy and enthalpy as enthalpy increases thetemperature distribution decreases Figure 6 elucidates thechange in velocity components microrotation temperaturedistribution and concentration for different values of PrIt is observed that the axial velocity reaches highest valuenear the hot plate and the radial velocity microrotation andconcentration decrease whereas the temperature increaseswith the increasing value of Pr Physically if Pr increases thethermal diffusivity decreases and this leads to the decreasein the heat transfer ability at the thermal boundary layer

8 Journal of Engineering

0

02

04

06

08

1

12

14

16

18Te

mpe

ratu

re

02 04 06 08 10120582

Kr = 2Kr = 4

Kr = 6Kr = 8

(a)

0

01

02

03

04

05

06

07

08

Con

cent

ratio

n

02 04 06 08 10120582

Kr = 2Kr = 4

Kr = 6Kr = 8

(b)

Figure 4 Effect of Du on (a) temperature and (b) concentration for Du = 1 Gr = 4 Gm = 4 Re = 2 Sr = 02 Sc = 022 Pr = 1 119886 = 02120595 = 08119877 = 2 119869

1= 02 119904

1= 2 119904

2= 10 Ha = 2119863minus1 = 2 and Ec = 1

Axi

al v

eloc

ity

0

02

04

06

08

1

12

02 04 06 08 10120582

Ec = 02Ec = 06

Ec = 1Ec = 14

(a)

Radi

al v

eloc

ity

Ec = 02Ec = 06

Ec = 1Ec = 14

055

057

059

061

063

065

067

069

0 04 06 08 102120582

(b)

minus025

minus02

minus015

minus01

minus005

0

005

01

Mic

roro

tatio

n 02 04 06 08 10120582

Ec = 02Ec = 06

Ec = 1Ec = 14

(c)

Ec = 02Ec = 06

Ec = 1Ec = 14

0

05

1

15

2

25

3

35

4

Tem

pera

ture

02 04 06 08 10120582

(d)

Figure 5 Effect of Ec on (a) axial velocity (b) radial velocity (c) microrotation and (d) temperature for 119863minus1 = 2 Du = 02 Gr = 4 Re = 2Sr = 02 Ha = 2 Sc = 022 119886 = 02 120595 = 08 119877 = 2 119869

1= 2 119904

1= 2 119904

2= 1 Gm = 4 Kr = 2 and Pr = 071

Journal of Engineering 9

0

02

04

06

08

1

12A

xial

vel

ocity

02 04 06 08 10120582

Pr = 02Pr = 04

Pr = 071Pr = 1

(a)

Pr = 02Pr = 04

Pr = 071Pr = 1

055

057

059

061

063

065

067

069

071

Radi

al v

eloc

ity

02 04 06 08 10120582

(b)

0

005

01

015

0 02 04 06 08 1

Mic

roro

tatio

n

120582

Pr = 02Pr = 04

Pr = 071Pr = 1

minus02

minus015

minus01

minus005

(c)

Pr = 02Pr = 04

Pr = 071Pr = 1

0

02

04

06

08

1

12

14

16Te

mpe

ratu

re

02 04 06 08 10120582

(d)

Pr = 02Pr = 04

Pr = 071Pr = 1

0

01

02

03

04

05

06

07

Con

cent

ratio

n

02 04 06 08 10120582

(e)

Figure 6 Effect of Pr on (a) axial velocity (b) radial velocity (c) microrotation (d) temperature and (e) concentration for 119863minus1 = 2 Du =02 Gr = 4 Re = 2 Sr = 02 Ha = 2 Sc = 022 119886 = 02 120595 = 08 119877 = 2 119869

1= 02 119904

1= 2 119904

2= 1 Gm = 4 Kr = 2 and Ec = 1

10 Journal of Engineering

0 02 04 06 08 1

Axi

al v

eloc

ity

0

02

04

06

08

1

12

14

16

120582

Ha = 1Ha = 2

Ha = 3Ha = 4

(a)

058

063

068

073

078

083

088

093

098

0 02 04 06 08 1

Radi

al v

eloc

ity

120582

Ha = 1Ha = 2

Ha = 3Ha = 4

(b)

0

005

01

015

0 02 04 06 08 1

Mic

roro

tatio

n

120582

Ha = 1Ha = 2

Ha = 3Ha = 4

minus025

minus02

minus015

minus01

minus005

(c)

0

02

04

06

08

1

12

14

0 02 04 06 08 1

Tem

pera

ture

120582

Ha = 1Ha = 2

Ha = 3Ha = 4

(d)

0 02 04 06 08 1120582

Ha = 1Ha = 2

Ha = 3Ha = 4

0

02

01

03

05

07

09

04

06

08

1

Con

cent

ratio

n

(e)

Figure 7 Effect of Ha on (a) axial velocity (b) radial velocity (c) microrotation (d) temperature and (e) concentration for Kr = 2 Du = 002Gr = 4 Re = 2 Sr = 02 Sc = 1 Pr = 02 119886 = 04 120595 = 02 119877 = 2 119869

1= 2 119904

1= 2 119904

2= 2 Gm = 4119863minus1 = 02 and Ec = 1

Journal of Engineering 11

Figure 7 displays the change in the velocity componentsmicrorotation temperature distribution and concentrationfor several values of Ha From this it is observed that whenHa increases the temperature distribution also increaseswhereas the concentration decreases from the lower plate toupper plate and the axial velocity attains maximum valueat the center of the plates However the radial velocity andmicrorotation increase towards the center of the plates andthen decrease This is due to the fact that the magnetic forceretards the flow in both axial and radial directions Thevariations in the velocity components microrotation tem-perature distribution and concentration for different valuesof 119863minus1 are shown in Figure 8 From these one can deducethat the temperature distribution is increasing with 119863minus1whereas the radial velocity microrotation and concentrationare decreasing towards the upper plate and the axial velocitydecreases towards the center of the plates and then increasesbecause the resistance offered by the porosity of the mediumis more than the resistance due to the magnetic lines of force

5 Conclusions

The thermal diffusion and diffusion thermoeffects on com-bined free and forced convection magnetohydrodynamicflow of micropolar fluid in a porous medium between twoparallel plates with chemical reaction are considered Thenumerical solution of the transformed governing equations isobtained by the method of quasilinearization and the resultsare analyzed for various fluid and geometric parametersthrough graphs From the results the following is concluded

(i) The influences of Sr and Du on temperature andconcentration are similar

(ii) The temperature of the fluid is enhanced whereasthe concentration of the fluid is decreased with theincreasing of Ha and119863minus1

(iii) Kr reduces the concentration and enhances the tem-perature of the fluid

(iv) Ec and Pr exhibit similar effects on the velocitycomponents and microrotation whereas it is oppositein the case of temperature

Nomenclature

119886 Injection suction ratio 1 minus 11988111198812

119905 Timeℎ Distance between two parallel plates1198811119890119894120596119905 Injection velocity

1198812119890119894120596119905 Suction velocity

119901 Fluid pressure119902 Velocity vector119888 Specific heat at constant temperature119897 Microrotation vector119873 Microrotation componentEc Eckert number 120583119881

2120588ℎ119888(119879

2minus 1198791)

119896 Thermal conductivity

1198961 Viscosity parameter

1198962 Permeability of the medium

1198963 Chemical reaction rate

119906 Velocity component in 119909-directionV Velocity component in 119910-directionPr Prandtl number 120583119888119896Re Suction Reynolds number 120588119881

2ℎ120583

119895 Gyration parameter119869 Current density1198691 Nondimensional gyration parameter

120588119895ℎ1198812120574

119861 Total magnetic field119887 Induced magnetic field1198610 Magnetic flux density

119863 Rate of deformation tensor119864 Electric fieldHa Hartmann number 119861

0ℎradic120590120583

119863minus1 Inverse Darcy parameter ℎ2119896

2

119877 Nondimensional viscosity parameter 1198961120583

1199041 Nondimensional micropolar parameter

1198961ℎ2120574

1199042 Nondimensional micropolar parameter

120574119888ℎ2119896

119879 Temperature1198791119890119894120596119905 Temperature of the lower plate

1198792119890119894120596119905 Temperature of the upper plate

119879lowast Dimensionless temperature

(119879 minus 1198791119890119894120596119905)(1198792minus 1198791)119890119894120596119905

119862 Concentration119862lowast Nondimensional concentration

(119862 minus 1198620119890119894120596119905)(1198621minus 1198620)119890119894120596119905

1198620119890119894120596119905 Concentration of the lower plate

1198621119890119894120596119905 Concentration of the upper plate

1198631 Mass diffusivity

Kr Nondimensional chemical reaction param-eter 119896

3ℎ21198631

Sc Schmidt number 1205921198631

Gr Thermal Grashof number 120588119892120573119879(1198792minus

1198791)ℎ21205831198812

Gm Solutal Grashof number 120588119892120573119862(1198621minus

1198620)ℎ21205831198812

Sh Sherwood number 119899119860ℎ120592(119862

1minus 1198620)

119899119860 Mass transfer rate

Sr Soret number11986311198961198791205921198812119888119879119898119899119860

Du Dufour number119863111989611987911989911986012058811988812059221198812119888119904119896

119879119898 Mean temperature

119896119879 Thermal diffusion ratio

119888119904 Concentration susceptibility

Greek Letters

120582 Dimensionless 119910 coordinate 119910ℎ120572 120573 120574 Gyro viscosity parameters120577 Dimensionless axial variable (119880

01198861198812minus 119909ℎ)

120592 Kinematic viscosity120588 Fluid density120583 Fluid viscosity

12 Journal of Engineering

0

02

04

06

08

1

12

14

16

0 02 04 06 08 1

Axi

al v

eloc

ity

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

(a)

078

088

098

0 02 04 06 08 1

Radi

al v

eloc

ity

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

(b)

0

005

01

015

0 02 04 06 08 1

Mic

roro

tatio

n

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

minus025

minus02

minus03

minus015

minus01

minus005

(c)

0

05

1

15

2

25

3

35

4

45

0 02 04 06 08 1

Tem

pera

ture

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

(d)

0

02

04

06

08

1

12

0 02 04 06 08 1

Con

cent

ratio

n

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

(e)

Figure 8 Effect of 119863minus1 on (a) axial velocity (b) radial velocity (c) microrotation (d) temperature and (e) concentration for Kr = 2 Du =002 Gr = 4 Re = 2 Sr = 02 Ha = 1 Sc = 02 Pr = 02 119886 = 02 120595 = 02 119877 = 2 119869

1= 2 119904

1= 2 119904

2= 2 Gm = 4 and Ec = 1

Journal of Engineering 13

1205831015840 Magnetic permeability120590 Electric conductivity120595 Nondimensional frequency parameter 120596119905

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank referees for their suggestions which haveresulted in the improvement of the paper Also one ofthe authors (N Naresh Kumar) is grateful to the DefenceResearch and Development Organization Government ofIndia for providing senior research fellowship

References

[1] A S Berman ldquoLaminar flow in channels with porous wallsrdquoJournal of Applied Physics vol 24 pp 1232ndash1235 1953

[2] R M Terril and G M Shrestha ldquoLaminar flow throughparallel and uniformly porous walls of different permeabilityrdquoZeitschrift fur Angewandte Mathematik und Physik vol 16 pp470ndash482 1965

[3] A C Eringen ldquoTheory of micropolar fluidsrdquo Journal of Mathe-matics and Mechanics vol 16 pp 1ndash18 1966

[4] V U K Sastry and V Rama Mohan Rao ldquoNumerical solutionof micropolar fluid flow in a channel with porous wallsrdquoInternational Journal of Engineering Science vol 20 no 5 pp631ndash642 1982

[5] O Ojjela and N Naresh Kumar ldquoUnsteady MHD flow andheat transfer of micropolar fluid in a porous medium betweenparallel platesrdquo Canadian Journal of Physics vol 93 no 8 pp880ndash887 2015

[6] D Srinivasacharya J V RamanaMurthy and D VenugopalamldquoUnsteady stokes flow of micropolar fluid between two parallelporous platesrdquo International Journal of Engineering Science vol39 no 14 pp 1557ndash1563 2001

[7] G Maiti ldquoConvective heat transfer in micropolar fluid flowthrough a horizontal parallel plate channelrdquo ZAMMmdashJournalof Applied Mathematics and Mechanics vol 55 no 2 pp 105ndash111 1975

[8] R Bhargava L Kumar andH S Takhar ldquoNumerical solution offree convection MHD micropolar fluid flow between two par-allel porous vertical platesrdquo International Journal of EngineeringScience vol 41 no 2 pp 123ndash136 2003

[9] M Ashraf and A R Wehgal ldquoMHD flow and heat transfer ofmicropolar fluid between two porous disksrdquoAppliedMathemat-ics and Mechanics vol 33 no 1 pp 51ndash64 2012

[10] A IslamMd H A Biswas Md R Islam and SMMohiuddinldquoMHDmicropolar fluid flow through vertical porous mediumrdquoAcademic Research International vol 1 no 3 pp 381ndash393 2011

[11] S Nadeem M Hussain and M Naz ldquoMHD stagnation flow ofa micropolar fluid through a porous mediumrdquo Meccanica vol45 no 6 pp 869ndash880 2010

[12] O Ojjela and N Naresh Kumar ldquoUnsteady heat and masstransfer of chemically reacting micropolar fluid in a porous

channel with hall and ion slip currentsrdquo International ScholarlyResearch Notices vol 2014 Article ID 646957 11 pages 2014

[13] N S Elgazery ldquoThe effects of chemical reaction Hall andion-slip currents on MHD flow with temperature dependentviscosity and thermal diffusivityrdquoCommunications in NonlinearScience and Numerical Simulation vol 14 no 4 pp 1267ndash12832009

[14] M S Ayano ldquoMixed convection flow micropolar fluid over avertical plate subject to hall and ion-slip currentsrdquo InternationalJournal of Engineering amp Applied Sciences vol 5 no 3 pp 38ndash52 2013

[15] A R M K Aurangzaib N F Mohammad and S Shafie ldquoSoretand dufour effects on unsteadyMHD flow of a micropolar fluidin the presence of thermophoresis deposition particlerdquo WorldApplied Sciences Journal vol 21 no 5 pp 766ndash773 2013

[16] D Srinivasacharya and C RamReddy ldquoSoret and Dufoureffects on mixed convection in a non-Darcy porous mediumsaturated with micropolar fluidrdquo Nonlinear Analysis Modellingand Control vol 16 no 1 pp 100ndash115 2011

[17] A Mahdy ldquoSoret and Dufour effect on double diffusion mixedconvection from a vertical surface in a porous medium satu-rated with a non-Newtonian fluidrdquo Journal of Non-NewtonianFluid Mechanics vol 165 no 11-12 pp 568ndash575 2010

[18] T Hayat andM Nawaz ldquoSoret and Dufour effects on the mixedconvection flow of a second grade fluid subject to Hall and ion-slip currentsrdquo International Journal for Numerical Methods inFluids vol 67 no 9 pp 1073ndash1099 2011

[19] H P Rani and C N Kim ldquoA numerical study of the dufourand soret effects on unsteady natural convection flow pastan isothermal vertical cylinderrdquo Korean Journal of ChemicalEngineering vol 26 no 4 pp 946ndash954 2009

[20] D Pal and H Mondal ldquoEffects of Soret Dufour chemicalreaction and thermal radiation on MHD non-Darcy unsteadymixed convective heat and mass transfer over a stretchingsheetrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 4 pp 1942ndash1958 2011

[21] B K Sharma K Yadav N K Mishra and R C ChaudharyldquoSoret and Dufour effects on unsteady MHDmixed convectionflow past a radiative vertical porous plate embedded in a porousmedium with chemical reactionrdquo Applied Mathematics vol 3no 7 pp 717ndash723 2012

[22] E S Lee and L T Fan ldquoQuasilinearization technique forsolution of boundary layer equationsrdquoThe Canadian Journal ofChemical Engineering vol 46 no 3 pp 200ndash204 1968

[23] T Hymavathi and B Shanker ldquoA quasilinearization approachto heat transfer in MHD visco-elastic fluid flowrdquo AppliedMathematics and Computation vol 215 no 6 pp 2045ndash20542009

[24] C L Huang ldquoApplication of quasilinearization technique to thevertical channel flowandheat convectionrdquo International Journalof Non-Linear Mechanics vol 13 no 2 pp 55ndash60 1978

[25] S S Motsa P Sibanda and S Shateyi ldquoOn a new quasi-linearization method for systems of nonlinear boundary valueproblemsrdquo Mathematical Methods in the Applied Sciences vol34 no 11 pp 1406ndash1413 2011

[26] R E Bellman and R E Kalaba Quasilinearization andBoundary-Value Problems Elsevier New York NY USA 1965

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DistributedSensor Networks

International Journal of

2 Journal of Engineering

fluid flow and heat and mass transfer in a porous mediumbetween parallel plates with chemical reaction were consid-ered by Ojjela and Naresh Kumar [12] The MHD heat andmass transfer of micropolar fluid in a porous medium withchemical reaction andHall and ion slip effects by consideringvariable viscosity and thermal diffusivity were investigated byElgazery [13] The mixed convection flow and heat transferof an electrically conducting micropolar fluid over a verticalplatewithHall and ion slip effectswas analyzed byAyano [14]

When heat and mass transfer occurs simultaneously ina moving fluid the energy flux caused by a concentrationgradient is termed as diffusion thermoeffect whereas massfluxes can also be created by temperature gradients whichis known as a thermal diffusion effect These effects arestudied as second-order phenomena andmay have significantapplications in areas like petrology hydrology and geo-sciencesThe effect of thermophoresis on an unsteady naturalconvection flow and heat and mass transfer of micropolarfluidwith Soret andDufour effects was studied byAurangzaibet al [15] Srinivasacharya and RamReddy [16] consideredthe problem of the steady MHD mixed convection heat andmass transfer of micropolar fluid through non-Darcy porousmedium over a semi-infinite vertical plate with Soret andDufour effects Influence of the Soret and Dufour numberson mixed convection flow and heat and mass transfer ofnon-Newtonian fluid in a porous medium over a verticalplate was analyzed by Mahdy [17] Hayat and Nawaz [18]investigated analytically the effects of the Hall and ion slipon the mixed convection heat and mass transfer of second-grade fluid with Soret and Dufour effects Rani and Kim [19]studied numerically the laminar flow of an incompressibleviscous fluid past an isothermal vertical cylinder with SoretandDufour effectsThe effects of chemical reaction and Soretand Dufour on the mixed convection heat and mass transferof viscous fluid over a stretching surface in the presence ofthermal radiation were analyzed by Pal and Mondal [20]Sharma et al [21] studied the mixed convective flow heatand mass transfer of viscous fluid in a porous medium past aradiative vertical plate with chemical reaction and Soret andDufour effects

In the field of fluid mechanics many fluid flow problemsare nonlinear boundary value problems To solve these prob-lems we can use a numerical technique quasilinearizationmethod which is a powerful technique having second-orderconvergence Several authors (Lee and Fan [22] Hymavathiand Shanker [23] Huang [24] Motsa et al [25] and Ojjelaand Naresh Kumar [5 12]) applied the quasilinearizationmethod to solve the nonlinear boundary layer equations

In the present study the effects of chemical reaction ontwo-dimensional mixed convection flow and heat transferof an electrically conducting micropolar fluid in a porousmedium between two parallel plates with Soret and Dufourhave been considered The reduced flow field equationsare solved using the quasilinearization method The effectsof various parameters such as Hartmann number inverseDarcyrsquos parameter Schmidt number Prandtl number chem-ical reaction rate Soret and Dufour numbers on the velocitycomponents microrotation temperature distribution and

Y

h

0

Z

X

V2ei120596t

T2ei120596t

C1ei120596t

V1ei120596t

T1ei120596t

C0ei120596t

B

Figure 1 Schematic diagram of the fluid flow between parallelporous plates

concentration are studied in detail and presented in the formof graphs

2 Formulation of the Problem

Consider a two-dimensional laminar incompressiblemicrop-olar fluid flow through an elongated rectangular channelas shown in Figure 1 Assume that the fluid is injected andaspirated periodically through the plates with injection veloc-ity 1198811119890119894120596119905 and suction velocity 119881

2119890119894120596119905 Also the nonuniform

temperature and concentration at the lower and upper platesare 1198791119890119894120596119905 1198620119890119894120596119905 and 119879

2119890119894120596119905 1198621119890119894120596119905 respectively The region

inside the parallel plates is subjected to porousmedium and aconstant external magnetic field of strength 119861

0perpendicular

to the119883119884-plane is consideredThe governing equations of the micropolar fluid flow and

heat and mass transfer in the presence of buoyancy forcesmagnetic field and in the absence of body forces body couplesare given by

nabla sdot 119902 = 0 (1)

120588 [120597119902

120597119905+ (119902 sdot nabla) 119902] = minusgrad119901 + 119896

1curl 119897

minus (120583 + 1198961) curl curl (119902)

minus120583 + 1198961

1198962

119902 + 119869 times 119861 + 119865119887

(2)

120588119895 [120597119897

120597119905+ (119902 sdot nabla) 119897] = minus2119896

1119897 + 1198961curl 119902

minus 120574 curl curl (119897) (3)

120588119888 [120597119879

120597119905+ (119902 sdot nabla) 119879] = 119896nabla

2119879 + 2120583119863 119863

+1198961

2(curl (119902) minus 2119897)

2

+ 120574nabla119897 nabla119897 +120583 + 1198961

1198962

1199022

+

1003816100381610038161003816100381611986910038161003816100381610038161003816

2

120590+1205881198631119896119879

119888119904

nabla2119862

(4)

Journal of Engineering 3

[120597119862

120597119905+ (119902 sdot nabla) 119862] = 119863

1nabla2119862 minus 1198963(119862 minus 119862

0119890119894120596119905)

+1198631119896119879

119879119898

nabla2119879

(5)

where 119865119887is the buoyancy force and it is defined as (120588119892120573

119879(119879 minus

1198791119890119894120596119905) + 120588119892120573

119862(119862 minus 119862

0119890119894120596119905))119894

Neglecting the displacement currents the Maxwell equa-tions and the generalized Ohmrsquos law are

nabla sdot 119861 = 0

nabla times 119861 = 1205831015840119869

nabla times 119864 =120597119861

120597119905

119869 = 120590 (119864 + 119902 times 119861)

(6)

where 119861 = 1198610 + 119887 119887 is induced magnetic field Assume

that the induced magnetic field is negligible compared to theapplied magnetic field so that magnetic Reynolds number issmall the electric field is zero and magnetic permeability isconstant throughout the flow field

The velocity and microrotation components are

119902 = 119906119894 + V

119897 = 119873

(7)

Following Ojjela and Naresh Kumar [5 12] the velocity andmicrorotation components are

119906 (119909 120582 119905) = (1198800

119886minus1198812119909

ℎ)1198911015840(120582) 119890119894120596119905

V (119909 120582 119905) = 1198812119891 (120582) 119890

119894120596119905

119873 (119909 120582 119905) = (1198800

119886minus1198812119909

ℎ)119860 (120582) 119890

119894120596119905

(8)

The temperature and concentration distributions can betaken as

119879 (119909 120582 119905)

= (1198791+1205831198812

120588ℎ119888[1206011(120582) + (

1198800

1198861198812

minus119909

ℎ)

2

1206012(120582)]) 119890

119894120596119905

119862 (119909 120582 119905)

= (1198620+

119899119860

ℎ120592[1198921(120582) + (

1198800

1198861198812

minus119909

ℎ)

2

1198922(120582)]) 119890

119894120596119905

(9)

where 120582 = 119910ℎ and 119891(120582)119860(120582) 1206011(120582) 1206012(120582) 1198921(120582) and 119892

2(120582)

are to be determined

The boundary conditions for the velocity microrotationtemperature and concentration are

119906 (119909 120582 119905) = 0

V (119909 120582 119905) = 1198811119890119894120596119905

119873 (119909 120582 119905) = 0

119879 (119909 120582 119905) = 1198791119890119894120596119905

119862 (119909 120582 119905) = 1198620119890119894120596119905

at 120582 = 0

119906 (119909 120582 119905) = 0

V (119909 120582 119905) = 1198812119890119894120596119905

119873 (119909 120582 119905) = 0

119879 (119909 120582 119905) = 1198792119890119894120596119905

119862 (119909 120582 119905) = 1198621119890119894120596119905

at 120582 = 1

(10)

Substituting (8) and (9) in (2) (3) (4) and (5) then we get

1198911015840119881=minus119877

1 + 11987711986010158401015840+

Re1 + 119877

(119891119891101584010158401015840minus 119891101584011989110158401015840) cos120595

+Ha2

1 + 11987711989110158401015840+ 119863minus111989110158401015840minus

EcGr(1 + 119877) 120577

(1206011015840

1+ 12057721206011015840

2)

minusShGm(1 + 119877) 120577

(1198921015840

1+ 12057721198921015840

2)

1198691(1198911198601015840minus 1198911015840119860) cos120595 = minus119904

1(2119860 + 119891

10158401015840) + 11986010158401015840

12060110158401015840

1= minus2120601

2minus Re11990421198602 cos120595 minus Re Pr ((1 + 119877)119863minus11198912

+Ha21198912 + 411989110158402 minus 11989112060110158401) cos120595 minus Du (11989210158401015840

1+ 21198922)

12060110158401015840

2= minusRe Pr(119877

2(11989110158401015840+ 2119860)

2

+1199042

Pr11986010158402

+ (1 + 119877)119863minus111989110158402+Ha211989110158402 + 119891101584010158402 + 21198911015840120601

2minus 1198911206011015840

2)

sdot cos120595 minus Du119892101584010158402

11989210158401015840

1= minus2119892

2+ Kr119892

1+ Sc Re1198911198921015840

1cos120595 minus Sc Sr (12060110158401015840

1

+ 21206012)

11989210158401015840

2= Kr119892

2+ Sc Re (1198911198921015840

2minus 211989110158401198922) cos120595 minus Sc Sr12060110158401015840

2

(11)

where the prime denotes the differentiation with respect to 120582

4 Journal of Engineering

The dimensionless forms of temperature and concentra-tion from (9) are

119879lowast=

119879 minus 1198791119890119894120596119905

(1198792minus 1198791) 119890119894120596119905

= Ec (1206011+ 12057721206012)

119862lowast=

119862 minus 1198620119890119894120596119905

(1198621minus 1198620) 119890119894120596119905

= Sh (1198921+ 12057721198922)

(12)

The boundary conditions (10) in terms of119891119860 1206011 1206012 1198921 and

1198922are

119891 (0) = 1 minus 119886

119891 (1) = 1

1198911015840(0) = 0

1198911015840(1) = 0

119860 (0) = 0

119860 (1) = 0

1206011(0) = 0

1206011(1) =

1

Ec

1206012(0) = 0

1206012(1) = 0

1198921(0) = 0

1198921(1) =

1

Sh

1198922(0) = 0

1198922(1) = 0

(13)

For micropolar fluids the shear stress 120591119896119897is given by

120591119896119897= (minus119901 + 120578V

119903119903) 120575119896119897+ 2120583119889

119896119897+ 21198961119890119896119897119903(120596119903minus 119897119903) (14)

Then the nondimensional shear stress at the lower and upperplates is

119878119891=2120591119896119897

1205881198812

2

= [2

Re(1198800

1198861198812

minus119909

ℎ) (119877 minus 1) 119891

10158401015840(120582) cos120595]

120582=01

(15)

For micropolar fluids the couple stress119872119894119895is given by

119872119894119895= 120572119897119903119903120575119894119895+ 120573119897119894119895+ 120574119897119895119894 (16)

Then the nondimensional couple stress at lower and upperplates is

119898 = [(1198800

1198861198812

minus119909

ℎ)1198921015840(120582) cos120595]

120582=01

(17)

3 Solution of the Problem

The nonlinear equations (11) are converted into the followingsystem of first-order differential equations by the substitution

(119891 1198911015840 11989110158401015840 119891101584010158401015840 119860 1198601015840 1206011 1206011015840

1 1206012 1206011015840

2 1198921 1198921015840

1 1198922 1198921015840

2)

= (1199091 1199092 1199093 1199094 1199095 1199096 1199097 1199098 1199099 11990910 11990911 11990912 11990913 11990914)

1198891199091

119889120582= 1199092

1198891199092

119889120582= 1199093

1198891199093

119889120582= 1199094

1198891199094

119889120582=minus119877

1 + 119877(1199041(1199093+ 21199095) + 1198691(11990911199096minus 11990921199095)) cos120595

+Re1 + 119877

(11990921199093minus 11990911199094) cos120595 + 119863minus1119909

3+

Ha2

1 + 1198771199093

minusEcGr(1 + 119877) 120577

(1199098+ 120577211990910) minus

ShGm(1 + 119877) 120577

(11990912

+ 120577211990914)

1198891199095

119889120582= 1199096

1198891199096

119889120582= 1199041(1199093+ 21199095) + 1198691(11990911199096minus 11990921199095) cos120595

1198891199097

119889120582= 1199098

1198891199098

119889120582= minus2119909

9minus

Re Pr1 minus Du Sc Sr

(41199092

2+ (1 + 119877)119863

minus11199092

1

+Ha211990921minus 11990911199098+1199042

Pr1199092

5) cos120595 minus KrDu

1 minus DuSc Sr11990911

+Du Sc

1 minus Du Sc SrRe119909111990912cos120595

1198891199099

119889120582= 11990910

11988911990910

119889120582= minus

Re Pr1 minus Du Sc Sr

(119877

2(1199093+ 21199095)2

+ 1199092

3+ (1 + 119877)

sdot 119863minus11199092

2+Ha21199092

2+ 211990921199099minus 119909111990910) cos120595

minusKrDu

1 minus Du Sc Sr11990913minus

Du Sc1 minus Du Sc Sr

Re (119909111990914

minus 2119909211990913) cos120595

11988911990911

119889120582= 11990912

11988911990912

119889120582= minus2119909

13+

Sc Sr Re Pr1 minus Du Sc Sr

(41199092

2+ (1 + 119877)119863

minus11199092

1

+Ha211990921minus 11990911199098+1199042

Pr1199092

5) cos120595 + Kr

1 minus DuSc Sr11990911

+Sc

1 minus Du Sc SrRe119909111990912cos120595

Journal of Engineering 5

11988911990913

119889120582= 11990914

11988911990914

119889120582=

Sc Sr Re Pr1 minus DuSc Sr

(1199092

3+ (1 + 119877)119863

minus11199092

2+Ha21199092

2

+1199042

Pr1199092

6+119877

2(1199093+ 21199095)2

+ 211990921199099minus 119909111990910) cos120595

+Kr

1 minus Du Sc Sr11990913+

Sc1 minus DuSc Sr

Re (119909111990914

minus 2119909211990913) cos120595

(18)

The boundary conditions in terms of 1199091 1199092 1199093 11990941199095 1199096 1199097

1199098 1199099 11990910 11990911 11990912 11990913 11990914are

1199091(0) = 1 minus 119886

1199092(0) = 0

1199095(0) = 0

1199097(0) = 0

1199099(0) = 0

11990911(0) = 0

11990913(0) = 0

1199091(1) = 1

1199092(1) = 0

1199095(1) = 0

1199097(1) =

1

Ec

1199099(1) = 0

11990911(1) =

1

Sh

11990913(1) = 0

(19)

The system of (18) is solved numerically subject to boundaryconditions (19) using the quasilinearization method given byBellman and Kalaba [26]

Let (119909119903119894 119894 = 1 2 14) be an approximate current

solution and let (119909119903+1119894

119894 = 1 2 14) be an improvedsolution of (18) Using Taylorrsquos series expansion about thecurrent solution by neglecting the second- and higher-orderderivative terms coupled first-order system (18) is linearizedas

119889119909119903+1

1

119889120582= 119909119903+1

2

119889119909119903+1

2

119889120582= 119909119903+1

3

119889119909119903+1

3

119889120582= 119909119903+1

4

119889119909119903+1

4

119889120582= minus

Re1 + 119877

(119909119903+1

1119909119903

4+ 119909119903+1

4119909119903

1minus 119909119903+1

2119909119903

3

minus 119909119903

2119909119903+1

3) cos120595 + 119863minus1119909119903+1

3+

Ha2

1 + 119877119909119903+1

3

minus119877

1 + 119877(1199041(119909119903+1

3+ 2119909119903+1

5) + 1198691(119909119903+1

1119909119903

6+ 119909119903

1119909119903+1

6

minus 119909119903+1

2119909119903

5minus 119909119903+1

5119909119903

2) cos120595) minus EcGr

(1 + 119877) 120577(119909119903+1

8

+ 1205772119909119903+1

10) minus

ShGm(1 + 119877) 120577

(119909119903+1

12+ 1205772119909119903+1

14)

+Re1 + 119877

(minus119909119903

2119909119903

3+ 119909119903

1119909119903

4) cos120595 + 119877119869

1

1 + 119877(119909119903

1119909119903

6

minus 119909119903

2119909119903

5) cos120595

119889119909119903+1

5

119889120582= 119909119903+1

6

119889119909119903+1

6

119889120582= 1199041(119909119903+1

3+ 2119909119903+1

5) + 1198691(119909119903+1

1119909119903

6+ 119909119903

1119909119903+1

6

minus 119909119903+1

2119909119903

5minus 119909119903+1

5119909119903

2) cos120595 minus 119869

1(119909119903

1119909119903

6minus 119909119903

2119909119903

5) cos120595

119889119909119903+1

7

119889120582= 119909119903+1

8

119889119909119903+1

8

119889120582= minus2119909

119903+1

9minus

Re Pr1 minus Du Sc Sr

(8119909119903

2119909119903+1

2+ 2 (1 + 119877)

sdot 119863minus1119909119903

1119909119903+1

1+ 2Ha2119909119903

1119909119903+1

1minus 119909119903

1119909119903+1

8minus 119909119903

8119909119903+1

1

+21199042

Pr119909119903

5119909119903+1

5) cos120595 minus KrDu

1 minus Du Sc Sr119909119903+1

11

+Du Sc

1 minus Du Sc SrRe (1199091199031119909119903+1

12+ 119909119903

12119909119903+1

1minus 119909119903

1119909119903

12)

sdot cos120595 + Re Pr1 minus DuSc Sr

(4119909119903

2119909119903

2+ (1 + 119877)119863

minus1119909119903

1119909119903

1

+Ha21199091199031119909119903

1minus 119909119903

1119909119903

8+1199042

Pr119909119903

5119909119903

5) cos120595

119889119909119903+1

9

119889120582= 119909119903+1

10

119889119909119903+1

10

119889120582= minus

Re Pr1 minus Du Sc Sr

(119877

2(2119909119903

3119909119903+1

3+ 8119909119903

5119909119903+1

5

+ 4119909119903

3119909119903+1

5+ 4119909119903

5119909119903+1

3) + 2119909

119903

3119909119903+1

3+ 2 (1 + 119877)

sdot 119863minus1119909119903

2119909119903+1

2+ 2Ha2119909119903

2119909119903+1

2+ 2119909119903

2119909119903+1

9+ 2119909119903+1

2119909119903

9

minus 119909119903

1119909119903+1

10minus 119909119903+1

1119909119903

10) cos120595 minus DuSc Re

1 minus Du Sc Sr(119909119903

1119909119903+1

14

+ 119909119903+1

1119909119903

14minus 2119909119903

2119909119903+1

13minus 2119909119903+1

2119909119903

13minus 119909119903

1119909119903

14

+ 2119909119903

2119909119903

13) cos120595 + Re Pr

1 minus Du Sc Sr(119877

2(119909119903

3119909119903

3+ 4119909119903

5119909119903

5

+ 4119909119903

3119909119903

5) + 119909119903

3119909119903

3+ (1 + 119877)119863

minus1119909119903

2119909119903

2+Ha2119909119903

2119909119903

2

+ 2119909119903

2119909119903

9minus 119909119903

1119909119903

10) cos120595 minus KrDu

1 minus Du Sc Sr119909119903+1

13

6 Journal of Engineering

119889119909119903+1

11

119889120582= 119909119903+1

12

119889119909119903+1

12

119889120582= minus2119909

119903+1

13+

Sc Sr Re Pr1 minus Du Sc Sr

(8119909119903

2119909119903+1

2+ 2 (1 + 119877)

sdot 119863minus1119909119903

1119909119903+1

1+ 2Ha2119909119903

1119909119903+1

1minus 119909119903

1119909119903+1

8minus 119909119903+1

1119909119903

8

+21199042

Pr119909119903

5119909119903+1

5) cos120595 + Kr

1 minus DuSc Sr119909119903+1

11

+Sc Re

1 minus Du Sc Sr(119909119903

1119909119903+1

12+ 119909119903+1

1119909119903

12minus 119909119903

1119909119903

12) cos120595

minusSc Sr Re Pr1 minus Du Sc Sr

(4119909119903

2119909119903

2+ (1 + 119877)119863

minus1119909119903

1119909119903

1

+Ha21199091199031119909119903

1minus 119909119903

1119909119903

8+1199042

Pr119909119903

5119909119903

5) cos120595

119889119909119903+1

13

119889120582= 119909119903+1

14

119889119909119903+1

14

119889120582=

Sc Sr Re Pr1 minus Du Sc Sr

(2119909119903+1

3119909119903

3+ 2 (1 + 119877)

sdot 119863minus1119909119903

2119909119903+1

2+ 2Ha2119909119903

2119909119903+1

2+21199042

Pr119909119903

6119909119903+1

6

+119877

2(2119909119903

3119909119903+1

3+ 8119909119903

5119909119903+1

5+ 4119909119903

3119909119903+1

5+ 4119909119903

5119909119903+1

3)

+ 2119909119903

2119909119903+1

9+ 2119909119903+1

2119909119903

9minus 119909119903

1119909119903+1

10minus 119909119903+1

1119909119903

10) cos120595

+Kr

1 minus Du Sc Sr119909119903+1

13+

Sc Re1 minus Du Sc Sr

(119909119903

1119909119903+1

14

+ 119909119903+1

1119909119903

14minus 2119909119903

2119909119903+1

13minus 2119909119903+1

2119909119903

13minus 119909119903

1119909119903

14

+ 2119909119903

2119909119903

13) cos120595 minus Sc Sr Re Pr

1 minus Du Sc Sr(119909119903

3119909119903

3+ (1 + 119877)

sdot 119863minus1119909119903

2119909119903

2+Ha2119909119903

2119909119903

2+1199042

Pr119909119903

6119909119903

6+119877

2(119909119903

3119909119903

3

+ 4119909119903

5119909119903

5+ 4119909119903

3119909119903

5) + 2119909

119903

2119909119903

9minus 119909119903

1119909119903

10) cos120595

(20)

To solve for (119909119903+1119894

119894 = 1 2 14) the solutions to sevenseparate initial value problems denoted by 119909ℎ1

119894(120582) 119909ℎ2

119894(120582)

119909ℎ3

119894(120582) 119909ℎ4

119894(120582) 119909ℎ5

119894(120582) 119909ℎ6

119894(120582) and 119909ℎ7

119894(120582) (which are the

solutions of the homogeneous system corresponding to (20))and 1199091199011

119894(120582) (which is the particular solution of (20)) with

the following initial conditions are obtained by using the 4th-order Runge-Kutta method

119909ℎ1

3(0) = 1

119909ℎ1

119894(0) = 0 for 119894 = 3

119909ℎ2

4(0) = 1

119909ℎ2

119894(0) = 0 for 119894 = 4

119909ℎ3

6(0) = 1

119909ℎ3

119894(0) = 0 for 119894 = 6

119909ℎ4

8(0) = 1

119909ℎ4

119894(0) = 0 for 119894 = 8

119909ℎ5

10(0) = 1

119909ℎ5

119894(0) = 0 for 119894 = 10

119909ℎ6

12(0) = 1

119909ℎ6

119894(0) = 0 for 119894 = 12

119909ℎ7

14(0) = 1

119909ℎ7

119894(0) = 0 for 119894 = 14

1199091199011

1(0) = 1 minus 119886

1199091199011

2(0) = 119909

1199011

3(0) = 119909

1199011

4(0) = 119909

1199011

5(0) = 0

1199091199011

6(0) = 119909

1199011

7(0) = 119909

1199011

8(0) = 119909

1199011

9(0) = 0

1199091199011

10(0) = 119909

1199011

11(0) = 119909

1199011

12(0) = 0

1199091199011

13(0) = 119909

1199011

14(0) = 0

(21)

By using the principle of superposition the general solutioncan be written as

119909119899+1

119894(120582) = 119862

1119909ℎ1

119894(120582) + 119862

2119909ℎ2

119894(120582) + 119862

3119909ℎ3

119894(120582)

+ 1198624119909ℎ4

119894(120582) + 119862

5119909ℎ5

119894(120582) + 119862

6119909ℎ6

119894(120582)

+ 1198627119909ℎ7

119894(120582) + 119909

1199011

119894(120582)

(22)

where 1198621 1198622 1198623 1198624 1198625 1198626 and 119862

7are the unknown

constants and are determined by considering the boundaryconditions at 120582 = 1 This solution (119909119903+1

119894 119894 = 1 2 14)

is then compared with solution at the previous step (119909119903119894

119894 = 1 2 14) and further iteration is performed if theconvergence has not been achieved

4 Results and Discussion

The system of nonlinear differential equations (18) subject toboundary conditions (19) is solved numerically by the quasi-linearizationmethodThe influences of various fluid and geo-metric parameters such as Soret number Sr Dufour numberDu Hartmann number Ha chemical reaction parameter KrSchmidt number Sc Eckert number Ec and Prandtl numberPr on nondimensional velocity components microrotationtemperature distribution and concentration are analyzedthrough graphs in the domain [0 1]

Figures 2 and 3 show the influence of Sr and Du ontemperature and concentration From these figures it is

Journal of Engineering 7

0

05

1

15

2

25

3

0 02 04 06 08 1

Tem

pera

ture

Sr = 01Sr = 05

Sr = 1Sr = 15

120582

(a)

0

02

04

06

08

1

12

0 02 04 06 08 1

Con

cent

ratio

n

120582

Sr = 01Sr = 05

Sr = 1Sr = 15

(b)

Figure 2 Effect of Sr on (a) temperature and (b) concentration for Kr = 2 Gr = 4 Gm = 4 Re = 2 Du = 2 Sc = 022 Pr = 1 119886 = 02 120595 = 02119877 = 02 119869

1= 02 119904

1= 2 119904

2= 10 Ha = 1119863minus1 = 2 and Ec = 1

0 02 04 06 08 1120582

0

05

1

15

2

25

3

35

Tem

pera

ture

Du = 01Du = 1

Du = 2Du = 3

(a)

Du = 01Du = 1

Du = 2Du = 3

0

02

04

06

08

1

12C

once

ntra

tion

02 04 06 08 10120582

(b)

Figure 3 Effect of Du on (a) temperature and (b) concentration for Kr = 2 Gr = 4 Gm = 4 Re = 2 Sr = 1 Sc = 022 Pr = 1 119886 = 02 120595 = 02119877 = 02 119869

1= 02 119904

1= 2 119904

2= 10 Ha = 1119863minus1 = 2 and Ec = 1

evident that the temperature of the fluid increases whereasthe concentration decreases with the increasing of Sr and DuThis is because of the difference between the temperatures ofthe fluid and surface as well as the difference between concen-trations of the fluid and surface concentrations are increasedwith Sr and Du Figure 4 describes the behavior of the tem-perature distribution and concentration for the various valuesof Kr AsKr increases the temperature distribution of the fluidalso increases whereas the concentration decreases from thelower plate to the upper plate It is clear that the increase inthe Kr produces a decrease in the species concentrationThiscauses the concentration buoyancy effects to decrease as Krincreases The effect of Ec on velocity components micro-rotation and temperature is presented in Figure 5 As Ec

increases the radial velocity microrotation and temperatureare decreasing towards the upper plate However the axialvelocity decreases towards the center of the channel and thenincreases Since the Eckert number is the relation betweenthe kinetic energy and enthalpy as enthalpy increases thetemperature distribution decreases Figure 6 elucidates thechange in velocity components microrotation temperaturedistribution and concentration for different values of PrIt is observed that the axial velocity reaches highest valuenear the hot plate and the radial velocity microrotation andconcentration decrease whereas the temperature increaseswith the increasing value of Pr Physically if Pr increases thethermal diffusivity decreases and this leads to the decreasein the heat transfer ability at the thermal boundary layer

8 Journal of Engineering

0

02

04

06

08

1

12

14

16

18Te

mpe

ratu

re

02 04 06 08 10120582

Kr = 2Kr = 4

Kr = 6Kr = 8

(a)

0

01

02

03

04

05

06

07

08

Con

cent

ratio

n

02 04 06 08 10120582

Kr = 2Kr = 4

Kr = 6Kr = 8

(b)

Figure 4 Effect of Du on (a) temperature and (b) concentration for Du = 1 Gr = 4 Gm = 4 Re = 2 Sr = 02 Sc = 022 Pr = 1 119886 = 02120595 = 08119877 = 2 119869

1= 02 119904

1= 2 119904

2= 10 Ha = 2119863minus1 = 2 and Ec = 1

Axi

al v

eloc

ity

0

02

04

06

08

1

12

02 04 06 08 10120582

Ec = 02Ec = 06

Ec = 1Ec = 14

(a)

Radi

al v

eloc

ity

Ec = 02Ec = 06

Ec = 1Ec = 14

055

057

059

061

063

065

067

069

0 04 06 08 102120582

(b)

minus025

minus02

minus015

minus01

minus005

0

005

01

Mic

roro

tatio

n 02 04 06 08 10120582

Ec = 02Ec = 06

Ec = 1Ec = 14

(c)

Ec = 02Ec = 06

Ec = 1Ec = 14

0

05

1

15

2

25

3

35

4

Tem

pera

ture

02 04 06 08 10120582

(d)

Figure 5 Effect of Ec on (a) axial velocity (b) radial velocity (c) microrotation and (d) temperature for 119863minus1 = 2 Du = 02 Gr = 4 Re = 2Sr = 02 Ha = 2 Sc = 022 119886 = 02 120595 = 08 119877 = 2 119869

1= 2 119904

1= 2 119904

2= 1 Gm = 4 Kr = 2 and Pr = 071

Journal of Engineering 9

0

02

04

06

08

1

12A

xial

vel

ocity

02 04 06 08 10120582

Pr = 02Pr = 04

Pr = 071Pr = 1

(a)

Pr = 02Pr = 04

Pr = 071Pr = 1

055

057

059

061

063

065

067

069

071

Radi

al v

eloc

ity

02 04 06 08 10120582

(b)

0

005

01

015

0 02 04 06 08 1

Mic

roro

tatio

n

120582

Pr = 02Pr = 04

Pr = 071Pr = 1

minus02

minus015

minus01

minus005

(c)

Pr = 02Pr = 04

Pr = 071Pr = 1

0

02

04

06

08

1

12

14

16Te

mpe

ratu

re

02 04 06 08 10120582

(d)

Pr = 02Pr = 04

Pr = 071Pr = 1

0

01

02

03

04

05

06

07

Con

cent

ratio

n

02 04 06 08 10120582

(e)

Figure 6 Effect of Pr on (a) axial velocity (b) radial velocity (c) microrotation (d) temperature and (e) concentration for 119863minus1 = 2 Du =02 Gr = 4 Re = 2 Sr = 02 Ha = 2 Sc = 022 119886 = 02 120595 = 08 119877 = 2 119869

1= 02 119904

1= 2 119904

2= 1 Gm = 4 Kr = 2 and Ec = 1

10 Journal of Engineering

0 02 04 06 08 1

Axi

al v

eloc

ity

0

02

04

06

08

1

12

14

16

120582

Ha = 1Ha = 2

Ha = 3Ha = 4

(a)

058

063

068

073

078

083

088

093

098

0 02 04 06 08 1

Radi

al v

eloc

ity

120582

Ha = 1Ha = 2

Ha = 3Ha = 4

(b)

0

005

01

015

0 02 04 06 08 1

Mic

roro

tatio

n

120582

Ha = 1Ha = 2

Ha = 3Ha = 4

minus025

minus02

minus015

minus01

minus005

(c)

0

02

04

06

08

1

12

14

0 02 04 06 08 1

Tem

pera

ture

120582

Ha = 1Ha = 2

Ha = 3Ha = 4

(d)

0 02 04 06 08 1120582

Ha = 1Ha = 2

Ha = 3Ha = 4

0

02

01

03

05

07

09

04

06

08

1

Con

cent

ratio

n

(e)

Figure 7 Effect of Ha on (a) axial velocity (b) radial velocity (c) microrotation (d) temperature and (e) concentration for Kr = 2 Du = 002Gr = 4 Re = 2 Sr = 02 Sc = 1 Pr = 02 119886 = 04 120595 = 02 119877 = 2 119869

1= 2 119904

1= 2 119904

2= 2 Gm = 4119863minus1 = 02 and Ec = 1

Journal of Engineering 11

Figure 7 displays the change in the velocity componentsmicrorotation temperature distribution and concentrationfor several values of Ha From this it is observed that whenHa increases the temperature distribution also increaseswhereas the concentration decreases from the lower plate toupper plate and the axial velocity attains maximum valueat the center of the plates However the radial velocity andmicrorotation increase towards the center of the plates andthen decrease This is due to the fact that the magnetic forceretards the flow in both axial and radial directions Thevariations in the velocity components microrotation tem-perature distribution and concentration for different valuesof 119863minus1 are shown in Figure 8 From these one can deducethat the temperature distribution is increasing with 119863minus1whereas the radial velocity microrotation and concentrationare decreasing towards the upper plate and the axial velocitydecreases towards the center of the plates and then increasesbecause the resistance offered by the porosity of the mediumis more than the resistance due to the magnetic lines of force

5 Conclusions

The thermal diffusion and diffusion thermoeffects on com-bined free and forced convection magnetohydrodynamicflow of micropolar fluid in a porous medium between twoparallel plates with chemical reaction are considered Thenumerical solution of the transformed governing equations isobtained by the method of quasilinearization and the resultsare analyzed for various fluid and geometric parametersthrough graphs From the results the following is concluded

(i) The influences of Sr and Du on temperature andconcentration are similar

(ii) The temperature of the fluid is enhanced whereasthe concentration of the fluid is decreased with theincreasing of Ha and119863minus1

(iii) Kr reduces the concentration and enhances the tem-perature of the fluid

(iv) Ec and Pr exhibit similar effects on the velocitycomponents and microrotation whereas it is oppositein the case of temperature

Nomenclature

119886 Injection suction ratio 1 minus 11988111198812

119905 Timeℎ Distance between two parallel plates1198811119890119894120596119905 Injection velocity

1198812119890119894120596119905 Suction velocity

119901 Fluid pressure119902 Velocity vector119888 Specific heat at constant temperature119897 Microrotation vector119873 Microrotation componentEc Eckert number 120583119881

2120588ℎ119888(119879

2minus 1198791)

119896 Thermal conductivity

1198961 Viscosity parameter

1198962 Permeability of the medium

1198963 Chemical reaction rate

119906 Velocity component in 119909-directionV Velocity component in 119910-directionPr Prandtl number 120583119888119896Re Suction Reynolds number 120588119881

2ℎ120583

119895 Gyration parameter119869 Current density1198691 Nondimensional gyration parameter

120588119895ℎ1198812120574

119861 Total magnetic field119887 Induced magnetic field1198610 Magnetic flux density

119863 Rate of deformation tensor119864 Electric fieldHa Hartmann number 119861

0ℎradic120590120583

119863minus1 Inverse Darcy parameter ℎ2119896

2

119877 Nondimensional viscosity parameter 1198961120583

1199041 Nondimensional micropolar parameter

1198961ℎ2120574

1199042 Nondimensional micropolar parameter

120574119888ℎ2119896

119879 Temperature1198791119890119894120596119905 Temperature of the lower plate

1198792119890119894120596119905 Temperature of the upper plate

119879lowast Dimensionless temperature

(119879 minus 1198791119890119894120596119905)(1198792minus 1198791)119890119894120596119905

119862 Concentration119862lowast Nondimensional concentration

(119862 minus 1198620119890119894120596119905)(1198621minus 1198620)119890119894120596119905

1198620119890119894120596119905 Concentration of the lower plate

1198621119890119894120596119905 Concentration of the upper plate

1198631 Mass diffusivity

Kr Nondimensional chemical reaction param-eter 119896

3ℎ21198631

Sc Schmidt number 1205921198631

Gr Thermal Grashof number 120588119892120573119879(1198792minus

1198791)ℎ21205831198812

Gm Solutal Grashof number 120588119892120573119862(1198621minus

1198620)ℎ21205831198812

Sh Sherwood number 119899119860ℎ120592(119862

1minus 1198620)

119899119860 Mass transfer rate

Sr Soret number11986311198961198791205921198812119888119879119898119899119860

Du Dufour number119863111989611987911989911986012058811988812059221198812119888119904119896

119879119898 Mean temperature

119896119879 Thermal diffusion ratio

119888119904 Concentration susceptibility

Greek Letters

120582 Dimensionless 119910 coordinate 119910ℎ120572 120573 120574 Gyro viscosity parameters120577 Dimensionless axial variable (119880

01198861198812minus 119909ℎ)

120592 Kinematic viscosity120588 Fluid density120583 Fluid viscosity

12 Journal of Engineering

0

02

04

06

08

1

12

14

16

0 02 04 06 08 1

Axi

al v

eloc

ity

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

(a)

078

088

098

0 02 04 06 08 1

Radi

al v

eloc

ity

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

(b)

0

005

01

015

0 02 04 06 08 1

Mic

roro

tatio

n

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

minus025

minus02

minus03

minus015

minus01

minus005

(c)

0

05

1

15

2

25

3

35

4

45

0 02 04 06 08 1

Tem

pera

ture

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

(d)

0

02

04

06

08

1

12

0 02 04 06 08 1

Con

cent

ratio

n

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

(e)

Figure 8 Effect of 119863minus1 on (a) axial velocity (b) radial velocity (c) microrotation (d) temperature and (e) concentration for Kr = 2 Du =002 Gr = 4 Re = 2 Sr = 02 Ha = 1 Sc = 02 Pr = 02 119886 = 02 120595 = 02 119877 = 2 119869

1= 2 119904

1= 2 119904

2= 2 Gm = 4 and Ec = 1

Journal of Engineering 13

1205831015840 Magnetic permeability120590 Electric conductivity120595 Nondimensional frequency parameter 120596119905

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank referees for their suggestions which haveresulted in the improvement of the paper Also one ofthe authors (N Naresh Kumar) is grateful to the DefenceResearch and Development Organization Government ofIndia for providing senior research fellowship

References

[1] A S Berman ldquoLaminar flow in channels with porous wallsrdquoJournal of Applied Physics vol 24 pp 1232ndash1235 1953

[2] R M Terril and G M Shrestha ldquoLaminar flow throughparallel and uniformly porous walls of different permeabilityrdquoZeitschrift fur Angewandte Mathematik und Physik vol 16 pp470ndash482 1965

[3] A C Eringen ldquoTheory of micropolar fluidsrdquo Journal of Mathe-matics and Mechanics vol 16 pp 1ndash18 1966

[4] V U K Sastry and V Rama Mohan Rao ldquoNumerical solutionof micropolar fluid flow in a channel with porous wallsrdquoInternational Journal of Engineering Science vol 20 no 5 pp631ndash642 1982

[5] O Ojjela and N Naresh Kumar ldquoUnsteady MHD flow andheat transfer of micropolar fluid in a porous medium betweenparallel platesrdquo Canadian Journal of Physics vol 93 no 8 pp880ndash887 2015

[6] D Srinivasacharya J V RamanaMurthy and D VenugopalamldquoUnsteady stokes flow of micropolar fluid between two parallelporous platesrdquo International Journal of Engineering Science vol39 no 14 pp 1557ndash1563 2001

[7] G Maiti ldquoConvective heat transfer in micropolar fluid flowthrough a horizontal parallel plate channelrdquo ZAMMmdashJournalof Applied Mathematics and Mechanics vol 55 no 2 pp 105ndash111 1975

[8] R Bhargava L Kumar andH S Takhar ldquoNumerical solution offree convection MHD micropolar fluid flow between two par-allel porous vertical platesrdquo International Journal of EngineeringScience vol 41 no 2 pp 123ndash136 2003

[9] M Ashraf and A R Wehgal ldquoMHD flow and heat transfer ofmicropolar fluid between two porous disksrdquoAppliedMathemat-ics and Mechanics vol 33 no 1 pp 51ndash64 2012

[10] A IslamMd H A Biswas Md R Islam and SMMohiuddinldquoMHDmicropolar fluid flow through vertical porous mediumrdquoAcademic Research International vol 1 no 3 pp 381ndash393 2011

[11] S Nadeem M Hussain and M Naz ldquoMHD stagnation flow ofa micropolar fluid through a porous mediumrdquo Meccanica vol45 no 6 pp 869ndash880 2010

[12] O Ojjela and N Naresh Kumar ldquoUnsteady heat and masstransfer of chemically reacting micropolar fluid in a porous

channel with hall and ion slip currentsrdquo International ScholarlyResearch Notices vol 2014 Article ID 646957 11 pages 2014

[13] N S Elgazery ldquoThe effects of chemical reaction Hall andion-slip currents on MHD flow with temperature dependentviscosity and thermal diffusivityrdquoCommunications in NonlinearScience and Numerical Simulation vol 14 no 4 pp 1267ndash12832009

[14] M S Ayano ldquoMixed convection flow micropolar fluid over avertical plate subject to hall and ion-slip currentsrdquo InternationalJournal of Engineering amp Applied Sciences vol 5 no 3 pp 38ndash52 2013

[15] A R M K Aurangzaib N F Mohammad and S Shafie ldquoSoretand dufour effects on unsteadyMHD flow of a micropolar fluidin the presence of thermophoresis deposition particlerdquo WorldApplied Sciences Journal vol 21 no 5 pp 766ndash773 2013

[16] D Srinivasacharya and C RamReddy ldquoSoret and Dufoureffects on mixed convection in a non-Darcy porous mediumsaturated with micropolar fluidrdquo Nonlinear Analysis Modellingand Control vol 16 no 1 pp 100ndash115 2011

[17] A Mahdy ldquoSoret and Dufour effect on double diffusion mixedconvection from a vertical surface in a porous medium satu-rated with a non-Newtonian fluidrdquo Journal of Non-NewtonianFluid Mechanics vol 165 no 11-12 pp 568ndash575 2010

[18] T Hayat andM Nawaz ldquoSoret and Dufour effects on the mixedconvection flow of a second grade fluid subject to Hall and ion-slip currentsrdquo International Journal for Numerical Methods inFluids vol 67 no 9 pp 1073ndash1099 2011

[19] H P Rani and C N Kim ldquoA numerical study of the dufourand soret effects on unsteady natural convection flow pastan isothermal vertical cylinderrdquo Korean Journal of ChemicalEngineering vol 26 no 4 pp 946ndash954 2009

[20] D Pal and H Mondal ldquoEffects of Soret Dufour chemicalreaction and thermal radiation on MHD non-Darcy unsteadymixed convective heat and mass transfer over a stretchingsheetrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 4 pp 1942ndash1958 2011

[21] B K Sharma K Yadav N K Mishra and R C ChaudharyldquoSoret and Dufour effects on unsteady MHDmixed convectionflow past a radiative vertical porous plate embedded in a porousmedium with chemical reactionrdquo Applied Mathematics vol 3no 7 pp 717ndash723 2012

[22] E S Lee and L T Fan ldquoQuasilinearization technique forsolution of boundary layer equationsrdquoThe Canadian Journal ofChemical Engineering vol 46 no 3 pp 200ndash204 1968

[23] T Hymavathi and B Shanker ldquoA quasilinearization approachto heat transfer in MHD visco-elastic fluid flowrdquo AppliedMathematics and Computation vol 215 no 6 pp 2045ndash20542009

[24] C L Huang ldquoApplication of quasilinearization technique to thevertical channel flowandheat convectionrdquo International Journalof Non-Linear Mechanics vol 13 no 2 pp 55ndash60 1978

[25] S S Motsa P Sibanda and S Shateyi ldquoOn a new quasi-linearization method for systems of nonlinear boundary valueproblemsrdquo Mathematical Methods in the Applied Sciences vol34 no 11 pp 1406ndash1413 2011

[26] R E Bellman and R E Kalaba Quasilinearization andBoundary-Value Problems Elsevier New York NY USA 1965

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DistributedSensor Networks

International Journal of

Journal of Engineering 3

[120597119862

120597119905+ (119902 sdot nabla) 119862] = 119863

1nabla2119862 minus 1198963(119862 minus 119862

0119890119894120596119905)

+1198631119896119879

119879119898

nabla2119879

(5)

where 119865119887is the buoyancy force and it is defined as (120588119892120573

119879(119879 minus

1198791119890119894120596119905) + 120588119892120573

119862(119862 minus 119862

0119890119894120596119905))119894

Neglecting the displacement currents the Maxwell equa-tions and the generalized Ohmrsquos law are

nabla sdot 119861 = 0

nabla times 119861 = 1205831015840119869

nabla times 119864 =120597119861

120597119905

119869 = 120590 (119864 + 119902 times 119861)

(6)

where 119861 = 1198610 + 119887 119887 is induced magnetic field Assume

that the induced magnetic field is negligible compared to theapplied magnetic field so that magnetic Reynolds number issmall the electric field is zero and magnetic permeability isconstant throughout the flow field

The velocity and microrotation components are

119902 = 119906119894 + V

119897 = 119873

(7)

Following Ojjela and Naresh Kumar [5 12] the velocity andmicrorotation components are

119906 (119909 120582 119905) = (1198800

119886minus1198812119909

ℎ)1198911015840(120582) 119890119894120596119905

V (119909 120582 119905) = 1198812119891 (120582) 119890

119894120596119905

119873 (119909 120582 119905) = (1198800

119886minus1198812119909

ℎ)119860 (120582) 119890

119894120596119905

(8)

The temperature and concentration distributions can betaken as

119879 (119909 120582 119905)

= (1198791+1205831198812

120588ℎ119888[1206011(120582) + (

1198800

1198861198812

minus119909

ℎ)

2

1206012(120582)]) 119890

119894120596119905

119862 (119909 120582 119905)

= (1198620+

119899119860

ℎ120592[1198921(120582) + (

1198800

1198861198812

minus119909

ℎ)

2

1198922(120582)]) 119890

119894120596119905

(9)

where 120582 = 119910ℎ and 119891(120582)119860(120582) 1206011(120582) 1206012(120582) 1198921(120582) and 119892

2(120582)

are to be determined

The boundary conditions for the velocity microrotationtemperature and concentration are

119906 (119909 120582 119905) = 0

V (119909 120582 119905) = 1198811119890119894120596119905

119873 (119909 120582 119905) = 0

119879 (119909 120582 119905) = 1198791119890119894120596119905

119862 (119909 120582 119905) = 1198620119890119894120596119905

at 120582 = 0

119906 (119909 120582 119905) = 0

V (119909 120582 119905) = 1198812119890119894120596119905

119873 (119909 120582 119905) = 0

119879 (119909 120582 119905) = 1198792119890119894120596119905

119862 (119909 120582 119905) = 1198621119890119894120596119905

at 120582 = 1

(10)

Substituting (8) and (9) in (2) (3) (4) and (5) then we get

1198911015840119881=minus119877

1 + 11987711986010158401015840+

Re1 + 119877

(119891119891101584010158401015840minus 119891101584011989110158401015840) cos120595

+Ha2

1 + 11987711989110158401015840+ 119863minus111989110158401015840minus

EcGr(1 + 119877) 120577

(1206011015840

1+ 12057721206011015840

2)

minusShGm(1 + 119877) 120577

(1198921015840

1+ 12057721198921015840

2)

1198691(1198911198601015840minus 1198911015840119860) cos120595 = minus119904

1(2119860 + 119891

10158401015840) + 11986010158401015840

12060110158401015840

1= minus2120601

2minus Re11990421198602 cos120595 minus Re Pr ((1 + 119877)119863minus11198912

+Ha21198912 + 411989110158402 minus 11989112060110158401) cos120595 minus Du (11989210158401015840

1+ 21198922)

12060110158401015840

2= minusRe Pr(119877

2(11989110158401015840+ 2119860)

2

+1199042

Pr11986010158402

+ (1 + 119877)119863minus111989110158402+Ha211989110158402 + 119891101584010158402 + 21198911015840120601

2minus 1198911206011015840

2)

sdot cos120595 minus Du119892101584010158402

11989210158401015840

1= minus2119892

2+ Kr119892

1+ Sc Re1198911198921015840

1cos120595 minus Sc Sr (12060110158401015840

1

+ 21206012)

11989210158401015840

2= Kr119892

2+ Sc Re (1198911198921015840

2minus 211989110158401198922) cos120595 minus Sc Sr12060110158401015840

2

(11)

where the prime denotes the differentiation with respect to 120582

4 Journal of Engineering

The dimensionless forms of temperature and concentra-tion from (9) are

119879lowast=

119879 minus 1198791119890119894120596119905

(1198792minus 1198791) 119890119894120596119905

= Ec (1206011+ 12057721206012)

119862lowast=

119862 minus 1198620119890119894120596119905

(1198621minus 1198620) 119890119894120596119905

= Sh (1198921+ 12057721198922)

(12)

The boundary conditions (10) in terms of119891119860 1206011 1206012 1198921 and

1198922are

119891 (0) = 1 minus 119886

119891 (1) = 1

1198911015840(0) = 0

1198911015840(1) = 0

119860 (0) = 0

119860 (1) = 0

1206011(0) = 0

1206011(1) =

1

Ec

1206012(0) = 0

1206012(1) = 0

1198921(0) = 0

1198921(1) =

1

Sh

1198922(0) = 0

1198922(1) = 0

(13)

For micropolar fluids the shear stress 120591119896119897is given by

120591119896119897= (minus119901 + 120578V

119903119903) 120575119896119897+ 2120583119889

119896119897+ 21198961119890119896119897119903(120596119903minus 119897119903) (14)

Then the nondimensional shear stress at the lower and upperplates is

119878119891=2120591119896119897

1205881198812

2

= [2

Re(1198800

1198861198812

minus119909

ℎ) (119877 minus 1) 119891

10158401015840(120582) cos120595]

120582=01

(15)

For micropolar fluids the couple stress119872119894119895is given by

119872119894119895= 120572119897119903119903120575119894119895+ 120573119897119894119895+ 120574119897119895119894 (16)

Then the nondimensional couple stress at lower and upperplates is

119898 = [(1198800

1198861198812

minus119909

ℎ)1198921015840(120582) cos120595]

120582=01

(17)

3 Solution of the Problem

The nonlinear equations (11) are converted into the followingsystem of first-order differential equations by the substitution

(119891 1198911015840 11989110158401015840 119891101584010158401015840 119860 1198601015840 1206011 1206011015840

1 1206012 1206011015840

2 1198921 1198921015840

1 1198922 1198921015840

2)

= (1199091 1199092 1199093 1199094 1199095 1199096 1199097 1199098 1199099 11990910 11990911 11990912 11990913 11990914)

1198891199091

119889120582= 1199092

1198891199092

119889120582= 1199093

1198891199093

119889120582= 1199094

1198891199094

119889120582=minus119877

1 + 119877(1199041(1199093+ 21199095) + 1198691(11990911199096minus 11990921199095)) cos120595

+Re1 + 119877

(11990921199093minus 11990911199094) cos120595 + 119863minus1119909

3+

Ha2

1 + 1198771199093

minusEcGr(1 + 119877) 120577

(1199098+ 120577211990910) minus

ShGm(1 + 119877) 120577

(11990912

+ 120577211990914)

1198891199095

119889120582= 1199096

1198891199096

119889120582= 1199041(1199093+ 21199095) + 1198691(11990911199096minus 11990921199095) cos120595

1198891199097

119889120582= 1199098

1198891199098

119889120582= minus2119909

9minus

Re Pr1 minus Du Sc Sr

(41199092

2+ (1 + 119877)119863

minus11199092

1

+Ha211990921minus 11990911199098+1199042

Pr1199092

5) cos120595 minus KrDu

1 minus DuSc Sr11990911

+Du Sc

1 minus Du Sc SrRe119909111990912cos120595

1198891199099

119889120582= 11990910

11988911990910

119889120582= minus

Re Pr1 minus Du Sc Sr

(119877

2(1199093+ 21199095)2

+ 1199092

3+ (1 + 119877)

sdot 119863minus11199092

2+Ha21199092

2+ 211990921199099minus 119909111990910) cos120595

minusKrDu

1 minus Du Sc Sr11990913minus

Du Sc1 minus Du Sc Sr

Re (119909111990914

minus 2119909211990913) cos120595

11988911990911

119889120582= 11990912

11988911990912

119889120582= minus2119909

13+

Sc Sr Re Pr1 minus Du Sc Sr

(41199092

2+ (1 + 119877)119863

minus11199092

1

+Ha211990921minus 11990911199098+1199042

Pr1199092

5) cos120595 + Kr

1 minus DuSc Sr11990911

+Sc

1 minus Du Sc SrRe119909111990912cos120595

Journal of Engineering 5

11988911990913

119889120582= 11990914

11988911990914

119889120582=

Sc Sr Re Pr1 minus DuSc Sr

(1199092

3+ (1 + 119877)119863

minus11199092

2+Ha21199092

2

+1199042

Pr1199092

6+119877

2(1199093+ 21199095)2

+ 211990921199099minus 119909111990910) cos120595

+Kr

1 minus Du Sc Sr11990913+

Sc1 minus DuSc Sr

Re (119909111990914

minus 2119909211990913) cos120595

(18)

The boundary conditions in terms of 1199091 1199092 1199093 11990941199095 1199096 1199097

1199098 1199099 11990910 11990911 11990912 11990913 11990914are

1199091(0) = 1 minus 119886

1199092(0) = 0

1199095(0) = 0

1199097(0) = 0

1199099(0) = 0

11990911(0) = 0

11990913(0) = 0

1199091(1) = 1

1199092(1) = 0

1199095(1) = 0

1199097(1) =

1

Ec

1199099(1) = 0

11990911(1) =

1

Sh

11990913(1) = 0

(19)

The system of (18) is solved numerically subject to boundaryconditions (19) using the quasilinearization method given byBellman and Kalaba [26]

Let (119909119903119894 119894 = 1 2 14) be an approximate current

solution and let (119909119903+1119894

119894 = 1 2 14) be an improvedsolution of (18) Using Taylorrsquos series expansion about thecurrent solution by neglecting the second- and higher-orderderivative terms coupled first-order system (18) is linearizedas

119889119909119903+1

1

119889120582= 119909119903+1

2

119889119909119903+1

2

119889120582= 119909119903+1

3

119889119909119903+1

3

119889120582= 119909119903+1

4

119889119909119903+1

4

119889120582= minus

Re1 + 119877

(119909119903+1

1119909119903

4+ 119909119903+1

4119909119903

1minus 119909119903+1

2119909119903

3

minus 119909119903

2119909119903+1

3) cos120595 + 119863minus1119909119903+1

3+

Ha2

1 + 119877119909119903+1

3

minus119877

1 + 119877(1199041(119909119903+1

3+ 2119909119903+1

5) + 1198691(119909119903+1

1119909119903

6+ 119909119903

1119909119903+1

6

minus 119909119903+1

2119909119903

5minus 119909119903+1

5119909119903

2) cos120595) minus EcGr

(1 + 119877) 120577(119909119903+1

8

+ 1205772119909119903+1

10) minus

ShGm(1 + 119877) 120577

(119909119903+1

12+ 1205772119909119903+1

14)

+Re1 + 119877

(minus119909119903

2119909119903

3+ 119909119903

1119909119903

4) cos120595 + 119877119869

1

1 + 119877(119909119903

1119909119903

6

minus 119909119903

2119909119903

5) cos120595

119889119909119903+1

5

119889120582= 119909119903+1

6

119889119909119903+1

6

119889120582= 1199041(119909119903+1

3+ 2119909119903+1

5) + 1198691(119909119903+1

1119909119903

6+ 119909119903

1119909119903+1

6

minus 119909119903+1

2119909119903

5minus 119909119903+1

5119909119903

2) cos120595 minus 119869

1(119909119903

1119909119903

6minus 119909119903

2119909119903

5) cos120595

119889119909119903+1

7

119889120582= 119909119903+1

8

119889119909119903+1

8

119889120582= minus2119909

119903+1

9minus

Re Pr1 minus Du Sc Sr

(8119909119903

2119909119903+1

2+ 2 (1 + 119877)

sdot 119863minus1119909119903

1119909119903+1

1+ 2Ha2119909119903

1119909119903+1

1minus 119909119903

1119909119903+1

8minus 119909119903

8119909119903+1

1

+21199042

Pr119909119903

5119909119903+1

5) cos120595 minus KrDu

1 minus Du Sc Sr119909119903+1

11

+Du Sc

1 minus Du Sc SrRe (1199091199031119909119903+1

12+ 119909119903

12119909119903+1

1minus 119909119903

1119909119903

12)

sdot cos120595 + Re Pr1 minus DuSc Sr

(4119909119903

2119909119903

2+ (1 + 119877)119863

minus1119909119903

1119909119903

1

+Ha21199091199031119909119903

1minus 119909119903

1119909119903

8+1199042

Pr119909119903

5119909119903

5) cos120595

119889119909119903+1

9

119889120582= 119909119903+1

10

119889119909119903+1

10

119889120582= minus

Re Pr1 minus Du Sc Sr

(119877

2(2119909119903

3119909119903+1

3+ 8119909119903

5119909119903+1

5

+ 4119909119903

3119909119903+1

5+ 4119909119903

5119909119903+1

3) + 2119909

119903

3119909119903+1

3+ 2 (1 + 119877)

sdot 119863minus1119909119903

2119909119903+1

2+ 2Ha2119909119903

2119909119903+1

2+ 2119909119903

2119909119903+1

9+ 2119909119903+1

2119909119903

9

minus 119909119903

1119909119903+1

10minus 119909119903+1

1119909119903

10) cos120595 minus DuSc Re

1 minus Du Sc Sr(119909119903

1119909119903+1

14

+ 119909119903+1

1119909119903

14minus 2119909119903

2119909119903+1

13minus 2119909119903+1

2119909119903

13minus 119909119903

1119909119903

14

+ 2119909119903

2119909119903

13) cos120595 + Re Pr

1 minus Du Sc Sr(119877

2(119909119903

3119909119903

3+ 4119909119903

5119909119903

5

+ 4119909119903

3119909119903

5) + 119909119903

3119909119903

3+ (1 + 119877)119863

minus1119909119903

2119909119903

2+Ha2119909119903

2119909119903

2

+ 2119909119903

2119909119903

9minus 119909119903

1119909119903

10) cos120595 minus KrDu

1 minus Du Sc Sr119909119903+1

13

6 Journal of Engineering

119889119909119903+1

11

119889120582= 119909119903+1

12

119889119909119903+1

12

119889120582= minus2119909

119903+1

13+

Sc Sr Re Pr1 minus Du Sc Sr

(8119909119903

2119909119903+1

2+ 2 (1 + 119877)

sdot 119863minus1119909119903

1119909119903+1

1+ 2Ha2119909119903

1119909119903+1

1minus 119909119903

1119909119903+1

8minus 119909119903+1

1119909119903

8

+21199042

Pr119909119903

5119909119903+1

5) cos120595 + Kr

1 minus DuSc Sr119909119903+1

11

+Sc Re

1 minus Du Sc Sr(119909119903

1119909119903+1

12+ 119909119903+1

1119909119903

12minus 119909119903

1119909119903

12) cos120595

minusSc Sr Re Pr1 minus Du Sc Sr

(4119909119903

2119909119903

2+ (1 + 119877)119863

minus1119909119903

1119909119903

1

+Ha21199091199031119909119903

1minus 119909119903

1119909119903

8+1199042

Pr119909119903

5119909119903

5) cos120595

119889119909119903+1

13

119889120582= 119909119903+1

14

119889119909119903+1

14

119889120582=

Sc Sr Re Pr1 minus Du Sc Sr

(2119909119903+1

3119909119903

3+ 2 (1 + 119877)

sdot 119863minus1119909119903

2119909119903+1

2+ 2Ha2119909119903

2119909119903+1

2+21199042

Pr119909119903

6119909119903+1

6

+119877

2(2119909119903

3119909119903+1

3+ 8119909119903

5119909119903+1

5+ 4119909119903

3119909119903+1

5+ 4119909119903

5119909119903+1

3)

+ 2119909119903

2119909119903+1

9+ 2119909119903+1

2119909119903

9minus 119909119903

1119909119903+1

10minus 119909119903+1

1119909119903

10) cos120595

+Kr

1 minus Du Sc Sr119909119903+1

13+

Sc Re1 minus Du Sc Sr

(119909119903

1119909119903+1

14

+ 119909119903+1

1119909119903

14minus 2119909119903

2119909119903+1

13minus 2119909119903+1

2119909119903

13minus 119909119903

1119909119903

14

+ 2119909119903

2119909119903

13) cos120595 minus Sc Sr Re Pr

1 minus Du Sc Sr(119909119903

3119909119903

3+ (1 + 119877)

sdot 119863minus1119909119903

2119909119903

2+Ha2119909119903

2119909119903

2+1199042

Pr119909119903

6119909119903

6+119877

2(119909119903

3119909119903

3

+ 4119909119903

5119909119903

5+ 4119909119903

3119909119903

5) + 2119909

119903

2119909119903

9minus 119909119903

1119909119903

10) cos120595

(20)

To solve for (119909119903+1119894

119894 = 1 2 14) the solutions to sevenseparate initial value problems denoted by 119909ℎ1

119894(120582) 119909ℎ2

119894(120582)

119909ℎ3

119894(120582) 119909ℎ4

119894(120582) 119909ℎ5

119894(120582) 119909ℎ6

119894(120582) and 119909ℎ7

119894(120582) (which are the

solutions of the homogeneous system corresponding to (20))and 1199091199011

119894(120582) (which is the particular solution of (20)) with

the following initial conditions are obtained by using the 4th-order Runge-Kutta method

119909ℎ1

3(0) = 1

119909ℎ1

119894(0) = 0 for 119894 = 3

119909ℎ2

4(0) = 1

119909ℎ2

119894(0) = 0 for 119894 = 4

119909ℎ3

6(0) = 1

119909ℎ3

119894(0) = 0 for 119894 = 6

119909ℎ4

8(0) = 1

119909ℎ4

119894(0) = 0 for 119894 = 8

119909ℎ5

10(0) = 1

119909ℎ5

119894(0) = 0 for 119894 = 10

119909ℎ6

12(0) = 1

119909ℎ6

119894(0) = 0 for 119894 = 12

119909ℎ7

14(0) = 1

119909ℎ7

119894(0) = 0 for 119894 = 14

1199091199011

1(0) = 1 minus 119886

1199091199011

2(0) = 119909

1199011

3(0) = 119909

1199011

4(0) = 119909

1199011

5(0) = 0

1199091199011

6(0) = 119909

1199011

7(0) = 119909

1199011

8(0) = 119909

1199011

9(0) = 0

1199091199011

10(0) = 119909

1199011

11(0) = 119909

1199011

12(0) = 0

1199091199011

13(0) = 119909

1199011

14(0) = 0

(21)

By using the principle of superposition the general solutioncan be written as

119909119899+1

119894(120582) = 119862

1119909ℎ1

119894(120582) + 119862

2119909ℎ2

119894(120582) + 119862

3119909ℎ3

119894(120582)

+ 1198624119909ℎ4

119894(120582) + 119862

5119909ℎ5

119894(120582) + 119862

6119909ℎ6

119894(120582)

+ 1198627119909ℎ7

119894(120582) + 119909

1199011

119894(120582)

(22)

where 1198621 1198622 1198623 1198624 1198625 1198626 and 119862

7are the unknown

constants and are determined by considering the boundaryconditions at 120582 = 1 This solution (119909119903+1

119894 119894 = 1 2 14)

is then compared with solution at the previous step (119909119903119894

119894 = 1 2 14) and further iteration is performed if theconvergence has not been achieved

4 Results and Discussion

The system of nonlinear differential equations (18) subject toboundary conditions (19) is solved numerically by the quasi-linearizationmethodThe influences of various fluid and geo-metric parameters such as Soret number Sr Dufour numberDu Hartmann number Ha chemical reaction parameter KrSchmidt number Sc Eckert number Ec and Prandtl numberPr on nondimensional velocity components microrotationtemperature distribution and concentration are analyzedthrough graphs in the domain [0 1]

Figures 2 and 3 show the influence of Sr and Du ontemperature and concentration From these figures it is

Journal of Engineering 7

0

05

1

15

2

25

3

0 02 04 06 08 1

Tem

pera

ture

Sr = 01Sr = 05

Sr = 1Sr = 15

120582

(a)

0

02

04

06

08

1

12

0 02 04 06 08 1

Con

cent

ratio

n

120582

Sr = 01Sr = 05

Sr = 1Sr = 15

(b)

Figure 2 Effect of Sr on (a) temperature and (b) concentration for Kr = 2 Gr = 4 Gm = 4 Re = 2 Du = 2 Sc = 022 Pr = 1 119886 = 02 120595 = 02119877 = 02 119869

1= 02 119904

1= 2 119904

2= 10 Ha = 1119863minus1 = 2 and Ec = 1

0 02 04 06 08 1120582

0

05

1

15

2

25

3

35

Tem

pera

ture

Du = 01Du = 1

Du = 2Du = 3

(a)

Du = 01Du = 1

Du = 2Du = 3

0

02

04

06

08

1

12C

once

ntra

tion

02 04 06 08 10120582

(b)

Figure 3 Effect of Du on (a) temperature and (b) concentration for Kr = 2 Gr = 4 Gm = 4 Re = 2 Sr = 1 Sc = 022 Pr = 1 119886 = 02 120595 = 02119877 = 02 119869

1= 02 119904

1= 2 119904

2= 10 Ha = 1119863minus1 = 2 and Ec = 1

evident that the temperature of the fluid increases whereasthe concentration decreases with the increasing of Sr and DuThis is because of the difference between the temperatures ofthe fluid and surface as well as the difference between concen-trations of the fluid and surface concentrations are increasedwith Sr and Du Figure 4 describes the behavior of the tem-perature distribution and concentration for the various valuesof Kr AsKr increases the temperature distribution of the fluidalso increases whereas the concentration decreases from thelower plate to the upper plate It is clear that the increase inthe Kr produces a decrease in the species concentrationThiscauses the concentration buoyancy effects to decrease as Krincreases The effect of Ec on velocity components micro-rotation and temperature is presented in Figure 5 As Ec

increases the radial velocity microrotation and temperatureare decreasing towards the upper plate However the axialvelocity decreases towards the center of the channel and thenincreases Since the Eckert number is the relation betweenthe kinetic energy and enthalpy as enthalpy increases thetemperature distribution decreases Figure 6 elucidates thechange in velocity components microrotation temperaturedistribution and concentration for different values of PrIt is observed that the axial velocity reaches highest valuenear the hot plate and the radial velocity microrotation andconcentration decrease whereas the temperature increaseswith the increasing value of Pr Physically if Pr increases thethermal diffusivity decreases and this leads to the decreasein the heat transfer ability at the thermal boundary layer

8 Journal of Engineering

0

02

04

06

08

1

12

14

16

18Te

mpe

ratu

re

02 04 06 08 10120582

Kr = 2Kr = 4

Kr = 6Kr = 8

(a)

0

01

02

03

04

05

06

07

08

Con

cent

ratio

n

02 04 06 08 10120582

Kr = 2Kr = 4

Kr = 6Kr = 8

(b)

Figure 4 Effect of Du on (a) temperature and (b) concentration for Du = 1 Gr = 4 Gm = 4 Re = 2 Sr = 02 Sc = 022 Pr = 1 119886 = 02120595 = 08119877 = 2 119869

1= 02 119904

1= 2 119904

2= 10 Ha = 2119863minus1 = 2 and Ec = 1

Axi

al v

eloc

ity

0

02

04

06

08

1

12

02 04 06 08 10120582

Ec = 02Ec = 06

Ec = 1Ec = 14

(a)

Radi

al v

eloc

ity

Ec = 02Ec = 06

Ec = 1Ec = 14

055

057

059

061

063

065

067

069

0 04 06 08 102120582

(b)

minus025

minus02

minus015

minus01

minus005

0

005

01

Mic

roro

tatio

n 02 04 06 08 10120582

Ec = 02Ec = 06

Ec = 1Ec = 14

(c)

Ec = 02Ec = 06

Ec = 1Ec = 14

0

05

1

15

2

25

3

35

4

Tem

pera

ture

02 04 06 08 10120582

(d)

Figure 5 Effect of Ec on (a) axial velocity (b) radial velocity (c) microrotation and (d) temperature for 119863minus1 = 2 Du = 02 Gr = 4 Re = 2Sr = 02 Ha = 2 Sc = 022 119886 = 02 120595 = 08 119877 = 2 119869

1= 2 119904

1= 2 119904

2= 1 Gm = 4 Kr = 2 and Pr = 071

Journal of Engineering 9

0

02

04

06

08

1

12A

xial

vel

ocity

02 04 06 08 10120582

Pr = 02Pr = 04

Pr = 071Pr = 1

(a)

Pr = 02Pr = 04

Pr = 071Pr = 1

055

057

059

061

063

065

067

069

071

Radi

al v

eloc

ity

02 04 06 08 10120582

(b)

0

005

01

015

0 02 04 06 08 1

Mic

roro

tatio

n

120582

Pr = 02Pr = 04

Pr = 071Pr = 1

minus02

minus015

minus01

minus005

(c)

Pr = 02Pr = 04

Pr = 071Pr = 1

0

02

04

06

08

1

12

14

16Te

mpe

ratu

re

02 04 06 08 10120582

(d)

Pr = 02Pr = 04

Pr = 071Pr = 1

0

01

02

03

04

05

06

07

Con

cent

ratio

n

02 04 06 08 10120582

(e)

Figure 6 Effect of Pr on (a) axial velocity (b) radial velocity (c) microrotation (d) temperature and (e) concentration for 119863minus1 = 2 Du =02 Gr = 4 Re = 2 Sr = 02 Ha = 2 Sc = 022 119886 = 02 120595 = 08 119877 = 2 119869

1= 02 119904

1= 2 119904

2= 1 Gm = 4 Kr = 2 and Ec = 1

10 Journal of Engineering

0 02 04 06 08 1

Axi

al v

eloc

ity

0

02

04

06

08

1

12

14

16

120582

Ha = 1Ha = 2

Ha = 3Ha = 4

(a)

058

063

068

073

078

083

088

093

098

0 02 04 06 08 1

Radi

al v

eloc

ity

120582

Ha = 1Ha = 2

Ha = 3Ha = 4

(b)

0

005

01

015

0 02 04 06 08 1

Mic

roro

tatio

n

120582

Ha = 1Ha = 2

Ha = 3Ha = 4

minus025

minus02

minus015

minus01

minus005

(c)

0

02

04

06

08

1

12

14

0 02 04 06 08 1

Tem

pera

ture

120582

Ha = 1Ha = 2

Ha = 3Ha = 4

(d)

0 02 04 06 08 1120582

Ha = 1Ha = 2

Ha = 3Ha = 4

0

02

01

03

05

07

09

04

06

08

1

Con

cent

ratio

n

(e)

Figure 7 Effect of Ha on (a) axial velocity (b) radial velocity (c) microrotation (d) temperature and (e) concentration for Kr = 2 Du = 002Gr = 4 Re = 2 Sr = 02 Sc = 1 Pr = 02 119886 = 04 120595 = 02 119877 = 2 119869

1= 2 119904

1= 2 119904

2= 2 Gm = 4119863minus1 = 02 and Ec = 1

Journal of Engineering 11

Figure 7 displays the change in the velocity componentsmicrorotation temperature distribution and concentrationfor several values of Ha From this it is observed that whenHa increases the temperature distribution also increaseswhereas the concentration decreases from the lower plate toupper plate and the axial velocity attains maximum valueat the center of the plates However the radial velocity andmicrorotation increase towards the center of the plates andthen decrease This is due to the fact that the magnetic forceretards the flow in both axial and radial directions Thevariations in the velocity components microrotation tem-perature distribution and concentration for different valuesof 119863minus1 are shown in Figure 8 From these one can deducethat the temperature distribution is increasing with 119863minus1whereas the radial velocity microrotation and concentrationare decreasing towards the upper plate and the axial velocitydecreases towards the center of the plates and then increasesbecause the resistance offered by the porosity of the mediumis more than the resistance due to the magnetic lines of force

5 Conclusions

The thermal diffusion and diffusion thermoeffects on com-bined free and forced convection magnetohydrodynamicflow of micropolar fluid in a porous medium between twoparallel plates with chemical reaction are considered Thenumerical solution of the transformed governing equations isobtained by the method of quasilinearization and the resultsare analyzed for various fluid and geometric parametersthrough graphs From the results the following is concluded

(i) The influences of Sr and Du on temperature andconcentration are similar

(ii) The temperature of the fluid is enhanced whereasthe concentration of the fluid is decreased with theincreasing of Ha and119863minus1

(iii) Kr reduces the concentration and enhances the tem-perature of the fluid

(iv) Ec and Pr exhibit similar effects on the velocitycomponents and microrotation whereas it is oppositein the case of temperature

Nomenclature

119886 Injection suction ratio 1 minus 11988111198812

119905 Timeℎ Distance between two parallel plates1198811119890119894120596119905 Injection velocity

1198812119890119894120596119905 Suction velocity

119901 Fluid pressure119902 Velocity vector119888 Specific heat at constant temperature119897 Microrotation vector119873 Microrotation componentEc Eckert number 120583119881

2120588ℎ119888(119879

2minus 1198791)

119896 Thermal conductivity

1198961 Viscosity parameter

1198962 Permeability of the medium

1198963 Chemical reaction rate

119906 Velocity component in 119909-directionV Velocity component in 119910-directionPr Prandtl number 120583119888119896Re Suction Reynolds number 120588119881

2ℎ120583

119895 Gyration parameter119869 Current density1198691 Nondimensional gyration parameter

120588119895ℎ1198812120574

119861 Total magnetic field119887 Induced magnetic field1198610 Magnetic flux density

119863 Rate of deformation tensor119864 Electric fieldHa Hartmann number 119861

0ℎradic120590120583

119863minus1 Inverse Darcy parameter ℎ2119896

2

119877 Nondimensional viscosity parameter 1198961120583

1199041 Nondimensional micropolar parameter

1198961ℎ2120574

1199042 Nondimensional micropolar parameter

120574119888ℎ2119896

119879 Temperature1198791119890119894120596119905 Temperature of the lower plate

1198792119890119894120596119905 Temperature of the upper plate

119879lowast Dimensionless temperature

(119879 minus 1198791119890119894120596119905)(1198792minus 1198791)119890119894120596119905

119862 Concentration119862lowast Nondimensional concentration

(119862 minus 1198620119890119894120596119905)(1198621minus 1198620)119890119894120596119905

1198620119890119894120596119905 Concentration of the lower plate

1198621119890119894120596119905 Concentration of the upper plate

1198631 Mass diffusivity

Kr Nondimensional chemical reaction param-eter 119896

3ℎ21198631

Sc Schmidt number 1205921198631

Gr Thermal Grashof number 120588119892120573119879(1198792minus

1198791)ℎ21205831198812

Gm Solutal Grashof number 120588119892120573119862(1198621minus

1198620)ℎ21205831198812

Sh Sherwood number 119899119860ℎ120592(119862

1minus 1198620)

119899119860 Mass transfer rate

Sr Soret number11986311198961198791205921198812119888119879119898119899119860

Du Dufour number119863111989611987911989911986012058811988812059221198812119888119904119896

119879119898 Mean temperature

119896119879 Thermal diffusion ratio

119888119904 Concentration susceptibility

Greek Letters

120582 Dimensionless 119910 coordinate 119910ℎ120572 120573 120574 Gyro viscosity parameters120577 Dimensionless axial variable (119880

01198861198812minus 119909ℎ)

120592 Kinematic viscosity120588 Fluid density120583 Fluid viscosity

12 Journal of Engineering

0

02

04

06

08

1

12

14

16

0 02 04 06 08 1

Axi

al v

eloc

ity

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

(a)

078

088

098

0 02 04 06 08 1

Radi

al v

eloc

ity

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

(b)

0

005

01

015

0 02 04 06 08 1

Mic

roro

tatio

n

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

minus025

minus02

minus03

minus015

minus01

minus005

(c)

0

05

1

15

2

25

3

35

4

45

0 02 04 06 08 1

Tem

pera

ture

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

(d)

0

02

04

06

08

1

12

0 02 04 06 08 1

Con

cent

ratio

n

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

(e)

Figure 8 Effect of 119863minus1 on (a) axial velocity (b) radial velocity (c) microrotation (d) temperature and (e) concentration for Kr = 2 Du =002 Gr = 4 Re = 2 Sr = 02 Ha = 1 Sc = 02 Pr = 02 119886 = 02 120595 = 02 119877 = 2 119869

1= 2 119904

1= 2 119904

2= 2 Gm = 4 and Ec = 1

Journal of Engineering 13

1205831015840 Magnetic permeability120590 Electric conductivity120595 Nondimensional frequency parameter 120596119905

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank referees for their suggestions which haveresulted in the improvement of the paper Also one ofthe authors (N Naresh Kumar) is grateful to the DefenceResearch and Development Organization Government ofIndia for providing senior research fellowship

References

[1] A S Berman ldquoLaminar flow in channels with porous wallsrdquoJournal of Applied Physics vol 24 pp 1232ndash1235 1953

[2] R M Terril and G M Shrestha ldquoLaminar flow throughparallel and uniformly porous walls of different permeabilityrdquoZeitschrift fur Angewandte Mathematik und Physik vol 16 pp470ndash482 1965

[3] A C Eringen ldquoTheory of micropolar fluidsrdquo Journal of Mathe-matics and Mechanics vol 16 pp 1ndash18 1966

[4] V U K Sastry and V Rama Mohan Rao ldquoNumerical solutionof micropolar fluid flow in a channel with porous wallsrdquoInternational Journal of Engineering Science vol 20 no 5 pp631ndash642 1982

[5] O Ojjela and N Naresh Kumar ldquoUnsteady MHD flow andheat transfer of micropolar fluid in a porous medium betweenparallel platesrdquo Canadian Journal of Physics vol 93 no 8 pp880ndash887 2015

[6] D Srinivasacharya J V RamanaMurthy and D VenugopalamldquoUnsteady stokes flow of micropolar fluid between two parallelporous platesrdquo International Journal of Engineering Science vol39 no 14 pp 1557ndash1563 2001

[7] G Maiti ldquoConvective heat transfer in micropolar fluid flowthrough a horizontal parallel plate channelrdquo ZAMMmdashJournalof Applied Mathematics and Mechanics vol 55 no 2 pp 105ndash111 1975

[8] R Bhargava L Kumar andH S Takhar ldquoNumerical solution offree convection MHD micropolar fluid flow between two par-allel porous vertical platesrdquo International Journal of EngineeringScience vol 41 no 2 pp 123ndash136 2003

[9] M Ashraf and A R Wehgal ldquoMHD flow and heat transfer ofmicropolar fluid between two porous disksrdquoAppliedMathemat-ics and Mechanics vol 33 no 1 pp 51ndash64 2012

[10] A IslamMd H A Biswas Md R Islam and SMMohiuddinldquoMHDmicropolar fluid flow through vertical porous mediumrdquoAcademic Research International vol 1 no 3 pp 381ndash393 2011

[11] S Nadeem M Hussain and M Naz ldquoMHD stagnation flow ofa micropolar fluid through a porous mediumrdquo Meccanica vol45 no 6 pp 869ndash880 2010

[12] O Ojjela and N Naresh Kumar ldquoUnsteady heat and masstransfer of chemically reacting micropolar fluid in a porous

channel with hall and ion slip currentsrdquo International ScholarlyResearch Notices vol 2014 Article ID 646957 11 pages 2014

[13] N S Elgazery ldquoThe effects of chemical reaction Hall andion-slip currents on MHD flow with temperature dependentviscosity and thermal diffusivityrdquoCommunications in NonlinearScience and Numerical Simulation vol 14 no 4 pp 1267ndash12832009

[14] M S Ayano ldquoMixed convection flow micropolar fluid over avertical plate subject to hall and ion-slip currentsrdquo InternationalJournal of Engineering amp Applied Sciences vol 5 no 3 pp 38ndash52 2013

[15] A R M K Aurangzaib N F Mohammad and S Shafie ldquoSoretand dufour effects on unsteadyMHD flow of a micropolar fluidin the presence of thermophoresis deposition particlerdquo WorldApplied Sciences Journal vol 21 no 5 pp 766ndash773 2013

[16] D Srinivasacharya and C RamReddy ldquoSoret and Dufoureffects on mixed convection in a non-Darcy porous mediumsaturated with micropolar fluidrdquo Nonlinear Analysis Modellingand Control vol 16 no 1 pp 100ndash115 2011

[17] A Mahdy ldquoSoret and Dufour effect on double diffusion mixedconvection from a vertical surface in a porous medium satu-rated with a non-Newtonian fluidrdquo Journal of Non-NewtonianFluid Mechanics vol 165 no 11-12 pp 568ndash575 2010

[18] T Hayat andM Nawaz ldquoSoret and Dufour effects on the mixedconvection flow of a second grade fluid subject to Hall and ion-slip currentsrdquo International Journal for Numerical Methods inFluids vol 67 no 9 pp 1073ndash1099 2011

[19] H P Rani and C N Kim ldquoA numerical study of the dufourand soret effects on unsteady natural convection flow pastan isothermal vertical cylinderrdquo Korean Journal of ChemicalEngineering vol 26 no 4 pp 946ndash954 2009

[20] D Pal and H Mondal ldquoEffects of Soret Dufour chemicalreaction and thermal radiation on MHD non-Darcy unsteadymixed convective heat and mass transfer over a stretchingsheetrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 4 pp 1942ndash1958 2011

[21] B K Sharma K Yadav N K Mishra and R C ChaudharyldquoSoret and Dufour effects on unsteady MHDmixed convectionflow past a radiative vertical porous plate embedded in a porousmedium with chemical reactionrdquo Applied Mathematics vol 3no 7 pp 717ndash723 2012

[22] E S Lee and L T Fan ldquoQuasilinearization technique forsolution of boundary layer equationsrdquoThe Canadian Journal ofChemical Engineering vol 46 no 3 pp 200ndash204 1968

[23] T Hymavathi and B Shanker ldquoA quasilinearization approachto heat transfer in MHD visco-elastic fluid flowrdquo AppliedMathematics and Computation vol 215 no 6 pp 2045ndash20542009

[24] C L Huang ldquoApplication of quasilinearization technique to thevertical channel flowandheat convectionrdquo International Journalof Non-Linear Mechanics vol 13 no 2 pp 55ndash60 1978

[25] S S Motsa P Sibanda and S Shateyi ldquoOn a new quasi-linearization method for systems of nonlinear boundary valueproblemsrdquo Mathematical Methods in the Applied Sciences vol34 no 11 pp 1406ndash1413 2011

[26] R E Bellman and R E Kalaba Quasilinearization andBoundary-Value Problems Elsevier New York NY USA 1965

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International Journal of

4 Journal of Engineering

The dimensionless forms of temperature and concentra-tion from (9) are

119879lowast=

119879 minus 1198791119890119894120596119905

(1198792minus 1198791) 119890119894120596119905

= Ec (1206011+ 12057721206012)

119862lowast=

119862 minus 1198620119890119894120596119905

(1198621minus 1198620) 119890119894120596119905

= Sh (1198921+ 12057721198922)

(12)

The boundary conditions (10) in terms of119891119860 1206011 1206012 1198921 and

1198922are

119891 (0) = 1 minus 119886

119891 (1) = 1

1198911015840(0) = 0

1198911015840(1) = 0

119860 (0) = 0

119860 (1) = 0

1206011(0) = 0

1206011(1) =

1

Ec

1206012(0) = 0

1206012(1) = 0

1198921(0) = 0

1198921(1) =

1

Sh

1198922(0) = 0

1198922(1) = 0

(13)

For micropolar fluids the shear stress 120591119896119897is given by

120591119896119897= (minus119901 + 120578V

119903119903) 120575119896119897+ 2120583119889

119896119897+ 21198961119890119896119897119903(120596119903minus 119897119903) (14)

Then the nondimensional shear stress at the lower and upperplates is

119878119891=2120591119896119897

1205881198812

2

= [2

Re(1198800

1198861198812

minus119909

ℎ) (119877 minus 1) 119891

10158401015840(120582) cos120595]

120582=01

(15)

For micropolar fluids the couple stress119872119894119895is given by

119872119894119895= 120572119897119903119903120575119894119895+ 120573119897119894119895+ 120574119897119895119894 (16)

Then the nondimensional couple stress at lower and upperplates is

119898 = [(1198800

1198861198812

minus119909

ℎ)1198921015840(120582) cos120595]

120582=01

(17)

3 Solution of the Problem

The nonlinear equations (11) are converted into the followingsystem of first-order differential equations by the substitution

(119891 1198911015840 11989110158401015840 119891101584010158401015840 119860 1198601015840 1206011 1206011015840

1 1206012 1206011015840

2 1198921 1198921015840

1 1198922 1198921015840

2)

= (1199091 1199092 1199093 1199094 1199095 1199096 1199097 1199098 1199099 11990910 11990911 11990912 11990913 11990914)

1198891199091

119889120582= 1199092

1198891199092

119889120582= 1199093

1198891199093

119889120582= 1199094

1198891199094

119889120582=minus119877

1 + 119877(1199041(1199093+ 21199095) + 1198691(11990911199096minus 11990921199095)) cos120595

+Re1 + 119877

(11990921199093minus 11990911199094) cos120595 + 119863minus1119909

3+

Ha2

1 + 1198771199093

minusEcGr(1 + 119877) 120577

(1199098+ 120577211990910) minus

ShGm(1 + 119877) 120577

(11990912

+ 120577211990914)

1198891199095

119889120582= 1199096

1198891199096

119889120582= 1199041(1199093+ 21199095) + 1198691(11990911199096minus 11990921199095) cos120595

1198891199097

119889120582= 1199098

1198891199098

119889120582= minus2119909

9minus

Re Pr1 minus Du Sc Sr

(41199092

2+ (1 + 119877)119863

minus11199092

1

+Ha211990921minus 11990911199098+1199042

Pr1199092

5) cos120595 minus KrDu

1 minus DuSc Sr11990911

+Du Sc

1 minus Du Sc SrRe119909111990912cos120595

1198891199099

119889120582= 11990910

11988911990910

119889120582= minus

Re Pr1 minus Du Sc Sr

(119877

2(1199093+ 21199095)2

+ 1199092

3+ (1 + 119877)

sdot 119863minus11199092

2+Ha21199092

2+ 211990921199099minus 119909111990910) cos120595

minusKrDu

1 minus Du Sc Sr11990913minus

Du Sc1 minus Du Sc Sr

Re (119909111990914

minus 2119909211990913) cos120595

11988911990911

119889120582= 11990912

11988911990912

119889120582= minus2119909

13+

Sc Sr Re Pr1 minus Du Sc Sr

(41199092

2+ (1 + 119877)119863

minus11199092

1

+Ha211990921minus 11990911199098+1199042

Pr1199092

5) cos120595 + Kr

1 minus DuSc Sr11990911

+Sc

1 minus Du Sc SrRe119909111990912cos120595

Journal of Engineering 5

11988911990913

119889120582= 11990914

11988911990914

119889120582=

Sc Sr Re Pr1 minus DuSc Sr

(1199092

3+ (1 + 119877)119863

minus11199092

2+Ha21199092

2

+1199042

Pr1199092

6+119877

2(1199093+ 21199095)2

+ 211990921199099minus 119909111990910) cos120595

+Kr

1 minus Du Sc Sr11990913+

Sc1 minus DuSc Sr

Re (119909111990914

minus 2119909211990913) cos120595

(18)

The boundary conditions in terms of 1199091 1199092 1199093 11990941199095 1199096 1199097

1199098 1199099 11990910 11990911 11990912 11990913 11990914are

1199091(0) = 1 minus 119886

1199092(0) = 0

1199095(0) = 0

1199097(0) = 0

1199099(0) = 0

11990911(0) = 0

11990913(0) = 0

1199091(1) = 1

1199092(1) = 0

1199095(1) = 0

1199097(1) =

1

Ec

1199099(1) = 0

11990911(1) =

1

Sh

11990913(1) = 0

(19)

The system of (18) is solved numerically subject to boundaryconditions (19) using the quasilinearization method given byBellman and Kalaba [26]

Let (119909119903119894 119894 = 1 2 14) be an approximate current

solution and let (119909119903+1119894

119894 = 1 2 14) be an improvedsolution of (18) Using Taylorrsquos series expansion about thecurrent solution by neglecting the second- and higher-orderderivative terms coupled first-order system (18) is linearizedas

119889119909119903+1

1

119889120582= 119909119903+1

2

119889119909119903+1

2

119889120582= 119909119903+1

3

119889119909119903+1

3

119889120582= 119909119903+1

4

119889119909119903+1

4

119889120582= minus

Re1 + 119877

(119909119903+1

1119909119903

4+ 119909119903+1

4119909119903

1minus 119909119903+1

2119909119903

3

minus 119909119903

2119909119903+1

3) cos120595 + 119863minus1119909119903+1

3+

Ha2

1 + 119877119909119903+1

3

minus119877

1 + 119877(1199041(119909119903+1

3+ 2119909119903+1

5) + 1198691(119909119903+1

1119909119903

6+ 119909119903

1119909119903+1

6

minus 119909119903+1

2119909119903

5minus 119909119903+1

5119909119903

2) cos120595) minus EcGr

(1 + 119877) 120577(119909119903+1

8

+ 1205772119909119903+1

10) minus

ShGm(1 + 119877) 120577

(119909119903+1

12+ 1205772119909119903+1

14)

+Re1 + 119877

(minus119909119903

2119909119903

3+ 119909119903

1119909119903

4) cos120595 + 119877119869

1

1 + 119877(119909119903

1119909119903

6

minus 119909119903

2119909119903

5) cos120595

119889119909119903+1

5

119889120582= 119909119903+1

6

119889119909119903+1

6

119889120582= 1199041(119909119903+1

3+ 2119909119903+1

5) + 1198691(119909119903+1

1119909119903

6+ 119909119903

1119909119903+1

6

minus 119909119903+1

2119909119903

5minus 119909119903+1

5119909119903

2) cos120595 minus 119869

1(119909119903

1119909119903

6minus 119909119903

2119909119903

5) cos120595

119889119909119903+1

7

119889120582= 119909119903+1

8

119889119909119903+1

8

119889120582= minus2119909

119903+1

9minus

Re Pr1 minus Du Sc Sr

(8119909119903

2119909119903+1

2+ 2 (1 + 119877)

sdot 119863minus1119909119903

1119909119903+1

1+ 2Ha2119909119903

1119909119903+1

1minus 119909119903

1119909119903+1

8minus 119909119903

8119909119903+1

1

+21199042

Pr119909119903

5119909119903+1

5) cos120595 minus KrDu

1 minus Du Sc Sr119909119903+1

11

+Du Sc

1 minus Du Sc SrRe (1199091199031119909119903+1

12+ 119909119903

12119909119903+1

1minus 119909119903

1119909119903

12)

sdot cos120595 + Re Pr1 minus DuSc Sr

(4119909119903

2119909119903

2+ (1 + 119877)119863

minus1119909119903

1119909119903

1

+Ha21199091199031119909119903

1minus 119909119903

1119909119903

8+1199042

Pr119909119903

5119909119903

5) cos120595

119889119909119903+1

9

119889120582= 119909119903+1

10

119889119909119903+1

10

119889120582= minus

Re Pr1 minus Du Sc Sr

(119877

2(2119909119903

3119909119903+1

3+ 8119909119903

5119909119903+1

5

+ 4119909119903

3119909119903+1

5+ 4119909119903

5119909119903+1

3) + 2119909

119903

3119909119903+1

3+ 2 (1 + 119877)

sdot 119863minus1119909119903

2119909119903+1

2+ 2Ha2119909119903

2119909119903+1

2+ 2119909119903

2119909119903+1

9+ 2119909119903+1

2119909119903

9

minus 119909119903

1119909119903+1

10minus 119909119903+1

1119909119903

10) cos120595 minus DuSc Re

1 minus Du Sc Sr(119909119903

1119909119903+1

14

+ 119909119903+1

1119909119903

14minus 2119909119903

2119909119903+1

13minus 2119909119903+1

2119909119903

13minus 119909119903

1119909119903

14

+ 2119909119903

2119909119903

13) cos120595 + Re Pr

1 minus Du Sc Sr(119877

2(119909119903

3119909119903

3+ 4119909119903

5119909119903

5

+ 4119909119903

3119909119903

5) + 119909119903

3119909119903

3+ (1 + 119877)119863

minus1119909119903

2119909119903

2+Ha2119909119903

2119909119903

2

+ 2119909119903

2119909119903

9minus 119909119903

1119909119903

10) cos120595 minus KrDu

1 minus Du Sc Sr119909119903+1

13

6 Journal of Engineering

119889119909119903+1

11

119889120582= 119909119903+1

12

119889119909119903+1

12

119889120582= minus2119909

119903+1

13+

Sc Sr Re Pr1 minus Du Sc Sr

(8119909119903

2119909119903+1

2+ 2 (1 + 119877)

sdot 119863minus1119909119903

1119909119903+1

1+ 2Ha2119909119903

1119909119903+1

1minus 119909119903

1119909119903+1

8minus 119909119903+1

1119909119903

8

+21199042

Pr119909119903

5119909119903+1

5) cos120595 + Kr

1 minus DuSc Sr119909119903+1

11

+Sc Re

1 minus Du Sc Sr(119909119903

1119909119903+1

12+ 119909119903+1

1119909119903

12minus 119909119903

1119909119903

12) cos120595

minusSc Sr Re Pr1 minus Du Sc Sr

(4119909119903

2119909119903

2+ (1 + 119877)119863

minus1119909119903

1119909119903

1

+Ha21199091199031119909119903

1minus 119909119903

1119909119903

8+1199042

Pr119909119903

5119909119903

5) cos120595

119889119909119903+1

13

119889120582= 119909119903+1

14

119889119909119903+1

14

119889120582=

Sc Sr Re Pr1 minus Du Sc Sr

(2119909119903+1

3119909119903

3+ 2 (1 + 119877)

sdot 119863minus1119909119903

2119909119903+1

2+ 2Ha2119909119903

2119909119903+1

2+21199042

Pr119909119903

6119909119903+1

6

+119877

2(2119909119903

3119909119903+1

3+ 8119909119903

5119909119903+1

5+ 4119909119903

3119909119903+1

5+ 4119909119903

5119909119903+1

3)

+ 2119909119903

2119909119903+1

9+ 2119909119903+1

2119909119903

9minus 119909119903

1119909119903+1

10minus 119909119903+1

1119909119903

10) cos120595

+Kr

1 minus Du Sc Sr119909119903+1

13+

Sc Re1 minus Du Sc Sr

(119909119903

1119909119903+1

14

+ 119909119903+1

1119909119903

14minus 2119909119903

2119909119903+1

13minus 2119909119903+1

2119909119903

13minus 119909119903

1119909119903

14

+ 2119909119903

2119909119903

13) cos120595 minus Sc Sr Re Pr

1 minus Du Sc Sr(119909119903

3119909119903

3+ (1 + 119877)

sdot 119863minus1119909119903

2119909119903

2+Ha2119909119903

2119909119903

2+1199042

Pr119909119903

6119909119903

6+119877

2(119909119903

3119909119903

3

+ 4119909119903

5119909119903

5+ 4119909119903

3119909119903

5) + 2119909

119903

2119909119903

9minus 119909119903

1119909119903

10) cos120595

(20)

To solve for (119909119903+1119894

119894 = 1 2 14) the solutions to sevenseparate initial value problems denoted by 119909ℎ1

119894(120582) 119909ℎ2

119894(120582)

119909ℎ3

119894(120582) 119909ℎ4

119894(120582) 119909ℎ5

119894(120582) 119909ℎ6

119894(120582) and 119909ℎ7

119894(120582) (which are the

solutions of the homogeneous system corresponding to (20))and 1199091199011

119894(120582) (which is the particular solution of (20)) with

the following initial conditions are obtained by using the 4th-order Runge-Kutta method

119909ℎ1

3(0) = 1

119909ℎ1

119894(0) = 0 for 119894 = 3

119909ℎ2

4(0) = 1

119909ℎ2

119894(0) = 0 for 119894 = 4

119909ℎ3

6(0) = 1

119909ℎ3

119894(0) = 0 for 119894 = 6

119909ℎ4

8(0) = 1

119909ℎ4

119894(0) = 0 for 119894 = 8

119909ℎ5

10(0) = 1

119909ℎ5

119894(0) = 0 for 119894 = 10

119909ℎ6

12(0) = 1

119909ℎ6

119894(0) = 0 for 119894 = 12

119909ℎ7

14(0) = 1

119909ℎ7

119894(0) = 0 for 119894 = 14

1199091199011

1(0) = 1 minus 119886

1199091199011

2(0) = 119909

1199011

3(0) = 119909

1199011

4(0) = 119909

1199011

5(0) = 0

1199091199011

6(0) = 119909

1199011

7(0) = 119909

1199011

8(0) = 119909

1199011

9(0) = 0

1199091199011

10(0) = 119909

1199011

11(0) = 119909

1199011

12(0) = 0

1199091199011

13(0) = 119909

1199011

14(0) = 0

(21)

By using the principle of superposition the general solutioncan be written as

119909119899+1

119894(120582) = 119862

1119909ℎ1

119894(120582) + 119862

2119909ℎ2

119894(120582) + 119862

3119909ℎ3

119894(120582)

+ 1198624119909ℎ4

119894(120582) + 119862

5119909ℎ5

119894(120582) + 119862

6119909ℎ6

119894(120582)

+ 1198627119909ℎ7

119894(120582) + 119909

1199011

119894(120582)

(22)

where 1198621 1198622 1198623 1198624 1198625 1198626 and 119862

7are the unknown

constants and are determined by considering the boundaryconditions at 120582 = 1 This solution (119909119903+1

119894 119894 = 1 2 14)

is then compared with solution at the previous step (119909119903119894

119894 = 1 2 14) and further iteration is performed if theconvergence has not been achieved

4 Results and Discussion

The system of nonlinear differential equations (18) subject toboundary conditions (19) is solved numerically by the quasi-linearizationmethodThe influences of various fluid and geo-metric parameters such as Soret number Sr Dufour numberDu Hartmann number Ha chemical reaction parameter KrSchmidt number Sc Eckert number Ec and Prandtl numberPr on nondimensional velocity components microrotationtemperature distribution and concentration are analyzedthrough graphs in the domain [0 1]

Figures 2 and 3 show the influence of Sr and Du ontemperature and concentration From these figures it is

Journal of Engineering 7

0

05

1

15

2

25

3

0 02 04 06 08 1

Tem

pera

ture

Sr = 01Sr = 05

Sr = 1Sr = 15

120582

(a)

0

02

04

06

08

1

12

0 02 04 06 08 1

Con

cent

ratio

n

120582

Sr = 01Sr = 05

Sr = 1Sr = 15

(b)

Figure 2 Effect of Sr on (a) temperature and (b) concentration for Kr = 2 Gr = 4 Gm = 4 Re = 2 Du = 2 Sc = 022 Pr = 1 119886 = 02 120595 = 02119877 = 02 119869

1= 02 119904

1= 2 119904

2= 10 Ha = 1119863minus1 = 2 and Ec = 1

0 02 04 06 08 1120582

0

05

1

15

2

25

3

35

Tem

pera

ture

Du = 01Du = 1

Du = 2Du = 3

(a)

Du = 01Du = 1

Du = 2Du = 3

0

02

04

06

08

1

12C

once

ntra

tion

02 04 06 08 10120582

(b)

Figure 3 Effect of Du on (a) temperature and (b) concentration for Kr = 2 Gr = 4 Gm = 4 Re = 2 Sr = 1 Sc = 022 Pr = 1 119886 = 02 120595 = 02119877 = 02 119869

1= 02 119904

1= 2 119904

2= 10 Ha = 1119863minus1 = 2 and Ec = 1

evident that the temperature of the fluid increases whereasthe concentration decreases with the increasing of Sr and DuThis is because of the difference between the temperatures ofthe fluid and surface as well as the difference between concen-trations of the fluid and surface concentrations are increasedwith Sr and Du Figure 4 describes the behavior of the tem-perature distribution and concentration for the various valuesof Kr AsKr increases the temperature distribution of the fluidalso increases whereas the concentration decreases from thelower plate to the upper plate It is clear that the increase inthe Kr produces a decrease in the species concentrationThiscauses the concentration buoyancy effects to decrease as Krincreases The effect of Ec on velocity components micro-rotation and temperature is presented in Figure 5 As Ec

increases the radial velocity microrotation and temperatureare decreasing towards the upper plate However the axialvelocity decreases towards the center of the channel and thenincreases Since the Eckert number is the relation betweenthe kinetic energy and enthalpy as enthalpy increases thetemperature distribution decreases Figure 6 elucidates thechange in velocity components microrotation temperaturedistribution and concentration for different values of PrIt is observed that the axial velocity reaches highest valuenear the hot plate and the radial velocity microrotation andconcentration decrease whereas the temperature increaseswith the increasing value of Pr Physically if Pr increases thethermal diffusivity decreases and this leads to the decreasein the heat transfer ability at the thermal boundary layer

8 Journal of Engineering

0

02

04

06

08

1

12

14

16

18Te

mpe

ratu

re

02 04 06 08 10120582

Kr = 2Kr = 4

Kr = 6Kr = 8

(a)

0

01

02

03

04

05

06

07

08

Con

cent

ratio

n

02 04 06 08 10120582

Kr = 2Kr = 4

Kr = 6Kr = 8

(b)

Figure 4 Effect of Du on (a) temperature and (b) concentration for Du = 1 Gr = 4 Gm = 4 Re = 2 Sr = 02 Sc = 022 Pr = 1 119886 = 02120595 = 08119877 = 2 119869

1= 02 119904

1= 2 119904

2= 10 Ha = 2119863minus1 = 2 and Ec = 1

Axi

al v

eloc

ity

0

02

04

06

08

1

12

02 04 06 08 10120582

Ec = 02Ec = 06

Ec = 1Ec = 14

(a)

Radi

al v

eloc

ity

Ec = 02Ec = 06

Ec = 1Ec = 14

055

057

059

061

063

065

067

069

0 04 06 08 102120582

(b)

minus025

minus02

minus015

minus01

minus005

0

005

01

Mic

roro

tatio

n 02 04 06 08 10120582

Ec = 02Ec = 06

Ec = 1Ec = 14

(c)

Ec = 02Ec = 06

Ec = 1Ec = 14

0

05

1

15

2

25

3

35

4

Tem

pera

ture

02 04 06 08 10120582

(d)

Figure 5 Effect of Ec on (a) axial velocity (b) radial velocity (c) microrotation and (d) temperature for 119863minus1 = 2 Du = 02 Gr = 4 Re = 2Sr = 02 Ha = 2 Sc = 022 119886 = 02 120595 = 08 119877 = 2 119869

1= 2 119904

1= 2 119904

2= 1 Gm = 4 Kr = 2 and Pr = 071

Journal of Engineering 9

0

02

04

06

08

1

12A

xial

vel

ocity

02 04 06 08 10120582

Pr = 02Pr = 04

Pr = 071Pr = 1

(a)

Pr = 02Pr = 04

Pr = 071Pr = 1

055

057

059

061

063

065

067

069

071

Radi

al v

eloc

ity

02 04 06 08 10120582

(b)

0

005

01

015

0 02 04 06 08 1

Mic

roro

tatio

n

120582

Pr = 02Pr = 04

Pr = 071Pr = 1

minus02

minus015

minus01

minus005

(c)

Pr = 02Pr = 04

Pr = 071Pr = 1

0

02

04

06

08

1

12

14

16Te

mpe

ratu

re

02 04 06 08 10120582

(d)

Pr = 02Pr = 04

Pr = 071Pr = 1

0

01

02

03

04

05

06

07

Con

cent

ratio

n

02 04 06 08 10120582

(e)

Figure 6 Effect of Pr on (a) axial velocity (b) radial velocity (c) microrotation (d) temperature and (e) concentration for 119863minus1 = 2 Du =02 Gr = 4 Re = 2 Sr = 02 Ha = 2 Sc = 022 119886 = 02 120595 = 08 119877 = 2 119869

1= 02 119904

1= 2 119904

2= 1 Gm = 4 Kr = 2 and Ec = 1

10 Journal of Engineering

0 02 04 06 08 1

Axi

al v

eloc

ity

0

02

04

06

08

1

12

14

16

120582

Ha = 1Ha = 2

Ha = 3Ha = 4

(a)

058

063

068

073

078

083

088

093

098

0 02 04 06 08 1

Radi

al v

eloc

ity

120582

Ha = 1Ha = 2

Ha = 3Ha = 4

(b)

0

005

01

015

0 02 04 06 08 1

Mic

roro

tatio

n

120582

Ha = 1Ha = 2

Ha = 3Ha = 4

minus025

minus02

minus015

minus01

minus005

(c)

0

02

04

06

08

1

12

14

0 02 04 06 08 1

Tem

pera

ture

120582

Ha = 1Ha = 2

Ha = 3Ha = 4

(d)

0 02 04 06 08 1120582

Ha = 1Ha = 2

Ha = 3Ha = 4

0

02

01

03

05

07

09

04

06

08

1

Con

cent

ratio

n

(e)

Figure 7 Effect of Ha on (a) axial velocity (b) radial velocity (c) microrotation (d) temperature and (e) concentration for Kr = 2 Du = 002Gr = 4 Re = 2 Sr = 02 Sc = 1 Pr = 02 119886 = 04 120595 = 02 119877 = 2 119869

1= 2 119904

1= 2 119904

2= 2 Gm = 4119863minus1 = 02 and Ec = 1

Journal of Engineering 11

Figure 7 displays the change in the velocity componentsmicrorotation temperature distribution and concentrationfor several values of Ha From this it is observed that whenHa increases the temperature distribution also increaseswhereas the concentration decreases from the lower plate toupper plate and the axial velocity attains maximum valueat the center of the plates However the radial velocity andmicrorotation increase towards the center of the plates andthen decrease This is due to the fact that the magnetic forceretards the flow in both axial and radial directions Thevariations in the velocity components microrotation tem-perature distribution and concentration for different valuesof 119863minus1 are shown in Figure 8 From these one can deducethat the temperature distribution is increasing with 119863minus1whereas the radial velocity microrotation and concentrationare decreasing towards the upper plate and the axial velocitydecreases towards the center of the plates and then increasesbecause the resistance offered by the porosity of the mediumis more than the resistance due to the magnetic lines of force

5 Conclusions

The thermal diffusion and diffusion thermoeffects on com-bined free and forced convection magnetohydrodynamicflow of micropolar fluid in a porous medium between twoparallel plates with chemical reaction are considered Thenumerical solution of the transformed governing equations isobtained by the method of quasilinearization and the resultsare analyzed for various fluid and geometric parametersthrough graphs From the results the following is concluded

(i) The influences of Sr and Du on temperature andconcentration are similar

(ii) The temperature of the fluid is enhanced whereasthe concentration of the fluid is decreased with theincreasing of Ha and119863minus1

(iii) Kr reduces the concentration and enhances the tem-perature of the fluid

(iv) Ec and Pr exhibit similar effects on the velocitycomponents and microrotation whereas it is oppositein the case of temperature

Nomenclature

119886 Injection suction ratio 1 minus 11988111198812

119905 Timeℎ Distance between two parallel plates1198811119890119894120596119905 Injection velocity

1198812119890119894120596119905 Suction velocity

119901 Fluid pressure119902 Velocity vector119888 Specific heat at constant temperature119897 Microrotation vector119873 Microrotation componentEc Eckert number 120583119881

2120588ℎ119888(119879

2minus 1198791)

119896 Thermal conductivity

1198961 Viscosity parameter

1198962 Permeability of the medium

1198963 Chemical reaction rate

119906 Velocity component in 119909-directionV Velocity component in 119910-directionPr Prandtl number 120583119888119896Re Suction Reynolds number 120588119881

2ℎ120583

119895 Gyration parameter119869 Current density1198691 Nondimensional gyration parameter

120588119895ℎ1198812120574

119861 Total magnetic field119887 Induced magnetic field1198610 Magnetic flux density

119863 Rate of deformation tensor119864 Electric fieldHa Hartmann number 119861

0ℎradic120590120583

119863minus1 Inverse Darcy parameter ℎ2119896

2

119877 Nondimensional viscosity parameter 1198961120583

1199041 Nondimensional micropolar parameter

1198961ℎ2120574

1199042 Nondimensional micropolar parameter

120574119888ℎ2119896

119879 Temperature1198791119890119894120596119905 Temperature of the lower plate

1198792119890119894120596119905 Temperature of the upper plate

119879lowast Dimensionless temperature

(119879 minus 1198791119890119894120596119905)(1198792minus 1198791)119890119894120596119905

119862 Concentration119862lowast Nondimensional concentration

(119862 minus 1198620119890119894120596119905)(1198621minus 1198620)119890119894120596119905

1198620119890119894120596119905 Concentration of the lower plate

1198621119890119894120596119905 Concentration of the upper plate

1198631 Mass diffusivity

Kr Nondimensional chemical reaction param-eter 119896

3ℎ21198631

Sc Schmidt number 1205921198631

Gr Thermal Grashof number 120588119892120573119879(1198792minus

1198791)ℎ21205831198812

Gm Solutal Grashof number 120588119892120573119862(1198621minus

1198620)ℎ21205831198812

Sh Sherwood number 119899119860ℎ120592(119862

1minus 1198620)

119899119860 Mass transfer rate

Sr Soret number11986311198961198791205921198812119888119879119898119899119860

Du Dufour number119863111989611987911989911986012058811988812059221198812119888119904119896

119879119898 Mean temperature

119896119879 Thermal diffusion ratio

119888119904 Concentration susceptibility

Greek Letters

120582 Dimensionless 119910 coordinate 119910ℎ120572 120573 120574 Gyro viscosity parameters120577 Dimensionless axial variable (119880

01198861198812minus 119909ℎ)

120592 Kinematic viscosity120588 Fluid density120583 Fluid viscosity

12 Journal of Engineering

0

02

04

06

08

1

12

14

16

0 02 04 06 08 1

Axi

al v

eloc

ity

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

(a)

078

088

098

0 02 04 06 08 1

Radi

al v

eloc

ity

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

(b)

0

005

01

015

0 02 04 06 08 1

Mic

roro

tatio

n

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

minus025

minus02

minus03

minus015

minus01

minus005

(c)

0

05

1

15

2

25

3

35

4

45

0 02 04 06 08 1

Tem

pera

ture

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

(d)

0

02

04

06

08

1

12

0 02 04 06 08 1

Con

cent

ratio

n

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

(e)

Figure 8 Effect of 119863minus1 on (a) axial velocity (b) radial velocity (c) microrotation (d) temperature and (e) concentration for Kr = 2 Du =002 Gr = 4 Re = 2 Sr = 02 Ha = 1 Sc = 02 Pr = 02 119886 = 02 120595 = 02 119877 = 2 119869

1= 2 119904

1= 2 119904

2= 2 Gm = 4 and Ec = 1

Journal of Engineering 13

1205831015840 Magnetic permeability120590 Electric conductivity120595 Nondimensional frequency parameter 120596119905

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank referees for their suggestions which haveresulted in the improvement of the paper Also one ofthe authors (N Naresh Kumar) is grateful to the DefenceResearch and Development Organization Government ofIndia for providing senior research fellowship

References

[1] A S Berman ldquoLaminar flow in channels with porous wallsrdquoJournal of Applied Physics vol 24 pp 1232ndash1235 1953

[2] R M Terril and G M Shrestha ldquoLaminar flow throughparallel and uniformly porous walls of different permeabilityrdquoZeitschrift fur Angewandte Mathematik und Physik vol 16 pp470ndash482 1965

[3] A C Eringen ldquoTheory of micropolar fluidsrdquo Journal of Mathe-matics and Mechanics vol 16 pp 1ndash18 1966

[4] V U K Sastry and V Rama Mohan Rao ldquoNumerical solutionof micropolar fluid flow in a channel with porous wallsrdquoInternational Journal of Engineering Science vol 20 no 5 pp631ndash642 1982

[5] O Ojjela and N Naresh Kumar ldquoUnsteady MHD flow andheat transfer of micropolar fluid in a porous medium betweenparallel platesrdquo Canadian Journal of Physics vol 93 no 8 pp880ndash887 2015

[6] D Srinivasacharya J V RamanaMurthy and D VenugopalamldquoUnsteady stokes flow of micropolar fluid between two parallelporous platesrdquo International Journal of Engineering Science vol39 no 14 pp 1557ndash1563 2001

[7] G Maiti ldquoConvective heat transfer in micropolar fluid flowthrough a horizontal parallel plate channelrdquo ZAMMmdashJournalof Applied Mathematics and Mechanics vol 55 no 2 pp 105ndash111 1975

[8] R Bhargava L Kumar andH S Takhar ldquoNumerical solution offree convection MHD micropolar fluid flow between two par-allel porous vertical platesrdquo International Journal of EngineeringScience vol 41 no 2 pp 123ndash136 2003

[9] M Ashraf and A R Wehgal ldquoMHD flow and heat transfer ofmicropolar fluid between two porous disksrdquoAppliedMathemat-ics and Mechanics vol 33 no 1 pp 51ndash64 2012

[10] A IslamMd H A Biswas Md R Islam and SMMohiuddinldquoMHDmicropolar fluid flow through vertical porous mediumrdquoAcademic Research International vol 1 no 3 pp 381ndash393 2011

[11] S Nadeem M Hussain and M Naz ldquoMHD stagnation flow ofa micropolar fluid through a porous mediumrdquo Meccanica vol45 no 6 pp 869ndash880 2010

[12] O Ojjela and N Naresh Kumar ldquoUnsteady heat and masstransfer of chemically reacting micropolar fluid in a porous

channel with hall and ion slip currentsrdquo International ScholarlyResearch Notices vol 2014 Article ID 646957 11 pages 2014

[13] N S Elgazery ldquoThe effects of chemical reaction Hall andion-slip currents on MHD flow with temperature dependentviscosity and thermal diffusivityrdquoCommunications in NonlinearScience and Numerical Simulation vol 14 no 4 pp 1267ndash12832009

[14] M S Ayano ldquoMixed convection flow micropolar fluid over avertical plate subject to hall and ion-slip currentsrdquo InternationalJournal of Engineering amp Applied Sciences vol 5 no 3 pp 38ndash52 2013

[15] A R M K Aurangzaib N F Mohammad and S Shafie ldquoSoretand dufour effects on unsteadyMHD flow of a micropolar fluidin the presence of thermophoresis deposition particlerdquo WorldApplied Sciences Journal vol 21 no 5 pp 766ndash773 2013

[16] D Srinivasacharya and C RamReddy ldquoSoret and Dufoureffects on mixed convection in a non-Darcy porous mediumsaturated with micropolar fluidrdquo Nonlinear Analysis Modellingand Control vol 16 no 1 pp 100ndash115 2011

[17] A Mahdy ldquoSoret and Dufour effect on double diffusion mixedconvection from a vertical surface in a porous medium satu-rated with a non-Newtonian fluidrdquo Journal of Non-NewtonianFluid Mechanics vol 165 no 11-12 pp 568ndash575 2010

[18] T Hayat andM Nawaz ldquoSoret and Dufour effects on the mixedconvection flow of a second grade fluid subject to Hall and ion-slip currentsrdquo International Journal for Numerical Methods inFluids vol 67 no 9 pp 1073ndash1099 2011

[19] H P Rani and C N Kim ldquoA numerical study of the dufourand soret effects on unsteady natural convection flow pastan isothermal vertical cylinderrdquo Korean Journal of ChemicalEngineering vol 26 no 4 pp 946ndash954 2009

[20] D Pal and H Mondal ldquoEffects of Soret Dufour chemicalreaction and thermal radiation on MHD non-Darcy unsteadymixed convective heat and mass transfer over a stretchingsheetrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 4 pp 1942ndash1958 2011

[21] B K Sharma K Yadav N K Mishra and R C ChaudharyldquoSoret and Dufour effects on unsteady MHDmixed convectionflow past a radiative vertical porous plate embedded in a porousmedium with chemical reactionrdquo Applied Mathematics vol 3no 7 pp 717ndash723 2012

[22] E S Lee and L T Fan ldquoQuasilinearization technique forsolution of boundary layer equationsrdquoThe Canadian Journal ofChemical Engineering vol 46 no 3 pp 200ndash204 1968

[23] T Hymavathi and B Shanker ldquoA quasilinearization approachto heat transfer in MHD visco-elastic fluid flowrdquo AppliedMathematics and Computation vol 215 no 6 pp 2045ndash20542009

[24] C L Huang ldquoApplication of quasilinearization technique to thevertical channel flowandheat convectionrdquo International Journalof Non-Linear Mechanics vol 13 no 2 pp 55ndash60 1978

[25] S S Motsa P Sibanda and S Shateyi ldquoOn a new quasi-linearization method for systems of nonlinear boundary valueproblemsrdquo Mathematical Methods in the Applied Sciences vol34 no 11 pp 1406ndash1413 2011

[26] R E Bellman and R E Kalaba Quasilinearization andBoundary-Value Problems Elsevier New York NY USA 1965

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Journal of Engineering 5

11988911990913

119889120582= 11990914

11988911990914

119889120582=

Sc Sr Re Pr1 minus DuSc Sr

(1199092

3+ (1 + 119877)119863

minus11199092

2+Ha21199092

2

+1199042

Pr1199092

6+119877

2(1199093+ 21199095)2

+ 211990921199099minus 119909111990910) cos120595

+Kr

1 minus Du Sc Sr11990913+

Sc1 minus DuSc Sr

Re (119909111990914

minus 2119909211990913) cos120595

(18)

The boundary conditions in terms of 1199091 1199092 1199093 11990941199095 1199096 1199097

1199098 1199099 11990910 11990911 11990912 11990913 11990914are

1199091(0) = 1 minus 119886

1199092(0) = 0

1199095(0) = 0

1199097(0) = 0

1199099(0) = 0

11990911(0) = 0

11990913(0) = 0

1199091(1) = 1

1199092(1) = 0

1199095(1) = 0

1199097(1) =

1

Ec

1199099(1) = 0

11990911(1) =

1

Sh

11990913(1) = 0

(19)

The system of (18) is solved numerically subject to boundaryconditions (19) using the quasilinearization method given byBellman and Kalaba [26]

Let (119909119903119894 119894 = 1 2 14) be an approximate current

solution and let (119909119903+1119894

119894 = 1 2 14) be an improvedsolution of (18) Using Taylorrsquos series expansion about thecurrent solution by neglecting the second- and higher-orderderivative terms coupled first-order system (18) is linearizedas

119889119909119903+1

1

119889120582= 119909119903+1

2

119889119909119903+1

2

119889120582= 119909119903+1

3

119889119909119903+1

3

119889120582= 119909119903+1

4

119889119909119903+1

4

119889120582= minus

Re1 + 119877

(119909119903+1

1119909119903

4+ 119909119903+1

4119909119903

1minus 119909119903+1

2119909119903

3

minus 119909119903

2119909119903+1

3) cos120595 + 119863minus1119909119903+1

3+

Ha2

1 + 119877119909119903+1

3

minus119877

1 + 119877(1199041(119909119903+1

3+ 2119909119903+1

5) + 1198691(119909119903+1

1119909119903

6+ 119909119903

1119909119903+1

6

minus 119909119903+1

2119909119903

5minus 119909119903+1

5119909119903

2) cos120595) minus EcGr

(1 + 119877) 120577(119909119903+1

8

+ 1205772119909119903+1

10) minus

ShGm(1 + 119877) 120577

(119909119903+1

12+ 1205772119909119903+1

14)

+Re1 + 119877

(minus119909119903

2119909119903

3+ 119909119903

1119909119903

4) cos120595 + 119877119869

1

1 + 119877(119909119903

1119909119903

6

minus 119909119903

2119909119903

5) cos120595

119889119909119903+1

5

119889120582= 119909119903+1

6

119889119909119903+1

6

119889120582= 1199041(119909119903+1

3+ 2119909119903+1

5) + 1198691(119909119903+1

1119909119903

6+ 119909119903

1119909119903+1

6

minus 119909119903+1

2119909119903

5minus 119909119903+1

5119909119903

2) cos120595 minus 119869

1(119909119903

1119909119903

6minus 119909119903

2119909119903

5) cos120595

119889119909119903+1

7

119889120582= 119909119903+1

8

119889119909119903+1

8

119889120582= minus2119909

119903+1

9minus

Re Pr1 minus Du Sc Sr

(8119909119903

2119909119903+1

2+ 2 (1 + 119877)

sdot 119863minus1119909119903

1119909119903+1

1+ 2Ha2119909119903

1119909119903+1

1minus 119909119903

1119909119903+1

8minus 119909119903

8119909119903+1

1

+21199042

Pr119909119903

5119909119903+1

5) cos120595 minus KrDu

1 minus Du Sc Sr119909119903+1

11

+Du Sc

1 minus Du Sc SrRe (1199091199031119909119903+1

12+ 119909119903

12119909119903+1

1minus 119909119903

1119909119903

12)

sdot cos120595 + Re Pr1 minus DuSc Sr

(4119909119903

2119909119903

2+ (1 + 119877)119863

minus1119909119903

1119909119903

1

+Ha21199091199031119909119903

1minus 119909119903

1119909119903

8+1199042

Pr119909119903

5119909119903

5) cos120595

119889119909119903+1

9

119889120582= 119909119903+1

10

119889119909119903+1

10

119889120582= minus

Re Pr1 minus Du Sc Sr

(119877

2(2119909119903

3119909119903+1

3+ 8119909119903

5119909119903+1

5

+ 4119909119903

3119909119903+1

5+ 4119909119903

5119909119903+1

3) + 2119909

119903

3119909119903+1

3+ 2 (1 + 119877)

sdot 119863minus1119909119903

2119909119903+1

2+ 2Ha2119909119903

2119909119903+1

2+ 2119909119903

2119909119903+1

9+ 2119909119903+1

2119909119903

9

minus 119909119903

1119909119903+1

10minus 119909119903+1

1119909119903

10) cos120595 minus DuSc Re

1 minus Du Sc Sr(119909119903

1119909119903+1

14

+ 119909119903+1

1119909119903

14minus 2119909119903

2119909119903+1

13minus 2119909119903+1

2119909119903

13minus 119909119903

1119909119903

14

+ 2119909119903

2119909119903

13) cos120595 + Re Pr

1 minus Du Sc Sr(119877

2(119909119903

3119909119903

3+ 4119909119903

5119909119903

5

+ 4119909119903

3119909119903

5) + 119909119903

3119909119903

3+ (1 + 119877)119863

minus1119909119903

2119909119903

2+Ha2119909119903

2119909119903

2

+ 2119909119903

2119909119903

9minus 119909119903

1119909119903

10) cos120595 minus KrDu

1 minus Du Sc Sr119909119903+1

13

6 Journal of Engineering

119889119909119903+1

11

119889120582= 119909119903+1

12

119889119909119903+1

12

119889120582= minus2119909

119903+1

13+

Sc Sr Re Pr1 minus Du Sc Sr

(8119909119903

2119909119903+1

2+ 2 (1 + 119877)

sdot 119863minus1119909119903

1119909119903+1

1+ 2Ha2119909119903

1119909119903+1

1minus 119909119903

1119909119903+1

8minus 119909119903+1

1119909119903

8

+21199042

Pr119909119903

5119909119903+1

5) cos120595 + Kr

1 minus DuSc Sr119909119903+1

11

+Sc Re

1 minus Du Sc Sr(119909119903

1119909119903+1

12+ 119909119903+1

1119909119903

12minus 119909119903

1119909119903

12) cos120595

minusSc Sr Re Pr1 minus Du Sc Sr

(4119909119903

2119909119903

2+ (1 + 119877)119863

minus1119909119903

1119909119903

1

+Ha21199091199031119909119903

1minus 119909119903

1119909119903

8+1199042

Pr119909119903

5119909119903

5) cos120595

119889119909119903+1

13

119889120582= 119909119903+1

14

119889119909119903+1

14

119889120582=

Sc Sr Re Pr1 minus Du Sc Sr

(2119909119903+1

3119909119903

3+ 2 (1 + 119877)

sdot 119863minus1119909119903

2119909119903+1

2+ 2Ha2119909119903

2119909119903+1

2+21199042

Pr119909119903

6119909119903+1

6

+119877

2(2119909119903

3119909119903+1

3+ 8119909119903

5119909119903+1

5+ 4119909119903

3119909119903+1

5+ 4119909119903

5119909119903+1

3)

+ 2119909119903

2119909119903+1

9+ 2119909119903+1

2119909119903

9minus 119909119903

1119909119903+1

10minus 119909119903+1

1119909119903

10) cos120595

+Kr

1 minus Du Sc Sr119909119903+1

13+

Sc Re1 minus Du Sc Sr

(119909119903

1119909119903+1

14

+ 119909119903+1

1119909119903

14minus 2119909119903

2119909119903+1

13minus 2119909119903+1

2119909119903

13minus 119909119903

1119909119903

14

+ 2119909119903

2119909119903

13) cos120595 minus Sc Sr Re Pr

1 minus Du Sc Sr(119909119903

3119909119903

3+ (1 + 119877)

sdot 119863minus1119909119903

2119909119903

2+Ha2119909119903

2119909119903

2+1199042

Pr119909119903

6119909119903

6+119877

2(119909119903

3119909119903

3

+ 4119909119903

5119909119903

5+ 4119909119903

3119909119903

5) + 2119909

119903

2119909119903

9minus 119909119903

1119909119903

10) cos120595

(20)

To solve for (119909119903+1119894

119894 = 1 2 14) the solutions to sevenseparate initial value problems denoted by 119909ℎ1

119894(120582) 119909ℎ2

119894(120582)

119909ℎ3

119894(120582) 119909ℎ4

119894(120582) 119909ℎ5

119894(120582) 119909ℎ6

119894(120582) and 119909ℎ7

119894(120582) (which are the

solutions of the homogeneous system corresponding to (20))and 1199091199011

119894(120582) (which is the particular solution of (20)) with

the following initial conditions are obtained by using the 4th-order Runge-Kutta method

119909ℎ1

3(0) = 1

119909ℎ1

119894(0) = 0 for 119894 = 3

119909ℎ2

4(0) = 1

119909ℎ2

119894(0) = 0 for 119894 = 4

119909ℎ3

6(0) = 1

119909ℎ3

119894(0) = 0 for 119894 = 6

119909ℎ4

8(0) = 1

119909ℎ4

119894(0) = 0 for 119894 = 8

119909ℎ5

10(0) = 1

119909ℎ5

119894(0) = 0 for 119894 = 10

119909ℎ6

12(0) = 1

119909ℎ6

119894(0) = 0 for 119894 = 12

119909ℎ7

14(0) = 1

119909ℎ7

119894(0) = 0 for 119894 = 14

1199091199011

1(0) = 1 minus 119886

1199091199011

2(0) = 119909

1199011

3(0) = 119909

1199011

4(0) = 119909

1199011

5(0) = 0

1199091199011

6(0) = 119909

1199011

7(0) = 119909

1199011

8(0) = 119909

1199011

9(0) = 0

1199091199011

10(0) = 119909

1199011

11(0) = 119909

1199011

12(0) = 0

1199091199011

13(0) = 119909

1199011

14(0) = 0

(21)

By using the principle of superposition the general solutioncan be written as

119909119899+1

119894(120582) = 119862

1119909ℎ1

119894(120582) + 119862

2119909ℎ2

119894(120582) + 119862

3119909ℎ3

119894(120582)

+ 1198624119909ℎ4

119894(120582) + 119862

5119909ℎ5

119894(120582) + 119862

6119909ℎ6

119894(120582)

+ 1198627119909ℎ7

119894(120582) + 119909

1199011

119894(120582)

(22)

where 1198621 1198622 1198623 1198624 1198625 1198626 and 119862

7are the unknown

constants and are determined by considering the boundaryconditions at 120582 = 1 This solution (119909119903+1

119894 119894 = 1 2 14)

is then compared with solution at the previous step (119909119903119894

119894 = 1 2 14) and further iteration is performed if theconvergence has not been achieved

4 Results and Discussion

The system of nonlinear differential equations (18) subject toboundary conditions (19) is solved numerically by the quasi-linearizationmethodThe influences of various fluid and geo-metric parameters such as Soret number Sr Dufour numberDu Hartmann number Ha chemical reaction parameter KrSchmidt number Sc Eckert number Ec and Prandtl numberPr on nondimensional velocity components microrotationtemperature distribution and concentration are analyzedthrough graphs in the domain [0 1]

Figures 2 and 3 show the influence of Sr and Du ontemperature and concentration From these figures it is

Journal of Engineering 7

0

05

1

15

2

25

3

0 02 04 06 08 1

Tem

pera

ture

Sr = 01Sr = 05

Sr = 1Sr = 15

120582

(a)

0

02

04

06

08

1

12

0 02 04 06 08 1

Con

cent

ratio

n

120582

Sr = 01Sr = 05

Sr = 1Sr = 15

(b)

Figure 2 Effect of Sr on (a) temperature and (b) concentration for Kr = 2 Gr = 4 Gm = 4 Re = 2 Du = 2 Sc = 022 Pr = 1 119886 = 02 120595 = 02119877 = 02 119869

1= 02 119904

1= 2 119904

2= 10 Ha = 1119863minus1 = 2 and Ec = 1

0 02 04 06 08 1120582

0

05

1

15

2

25

3

35

Tem

pera

ture

Du = 01Du = 1

Du = 2Du = 3

(a)

Du = 01Du = 1

Du = 2Du = 3

0

02

04

06

08

1

12C

once

ntra

tion

02 04 06 08 10120582

(b)

Figure 3 Effect of Du on (a) temperature and (b) concentration for Kr = 2 Gr = 4 Gm = 4 Re = 2 Sr = 1 Sc = 022 Pr = 1 119886 = 02 120595 = 02119877 = 02 119869

1= 02 119904

1= 2 119904

2= 10 Ha = 1119863minus1 = 2 and Ec = 1

evident that the temperature of the fluid increases whereasthe concentration decreases with the increasing of Sr and DuThis is because of the difference between the temperatures ofthe fluid and surface as well as the difference between concen-trations of the fluid and surface concentrations are increasedwith Sr and Du Figure 4 describes the behavior of the tem-perature distribution and concentration for the various valuesof Kr AsKr increases the temperature distribution of the fluidalso increases whereas the concentration decreases from thelower plate to the upper plate It is clear that the increase inthe Kr produces a decrease in the species concentrationThiscauses the concentration buoyancy effects to decrease as Krincreases The effect of Ec on velocity components micro-rotation and temperature is presented in Figure 5 As Ec

increases the radial velocity microrotation and temperatureare decreasing towards the upper plate However the axialvelocity decreases towards the center of the channel and thenincreases Since the Eckert number is the relation betweenthe kinetic energy and enthalpy as enthalpy increases thetemperature distribution decreases Figure 6 elucidates thechange in velocity components microrotation temperaturedistribution and concentration for different values of PrIt is observed that the axial velocity reaches highest valuenear the hot plate and the radial velocity microrotation andconcentration decrease whereas the temperature increaseswith the increasing value of Pr Physically if Pr increases thethermal diffusivity decreases and this leads to the decreasein the heat transfer ability at the thermal boundary layer

8 Journal of Engineering

0

02

04

06

08

1

12

14

16

18Te

mpe

ratu

re

02 04 06 08 10120582

Kr = 2Kr = 4

Kr = 6Kr = 8

(a)

0

01

02

03

04

05

06

07

08

Con

cent

ratio

n

02 04 06 08 10120582

Kr = 2Kr = 4

Kr = 6Kr = 8

(b)

Figure 4 Effect of Du on (a) temperature and (b) concentration for Du = 1 Gr = 4 Gm = 4 Re = 2 Sr = 02 Sc = 022 Pr = 1 119886 = 02120595 = 08119877 = 2 119869

1= 02 119904

1= 2 119904

2= 10 Ha = 2119863minus1 = 2 and Ec = 1

Axi

al v

eloc

ity

0

02

04

06

08

1

12

02 04 06 08 10120582

Ec = 02Ec = 06

Ec = 1Ec = 14

(a)

Radi

al v

eloc

ity

Ec = 02Ec = 06

Ec = 1Ec = 14

055

057

059

061

063

065

067

069

0 04 06 08 102120582

(b)

minus025

minus02

minus015

minus01

minus005

0

005

01

Mic

roro

tatio

n 02 04 06 08 10120582

Ec = 02Ec = 06

Ec = 1Ec = 14

(c)

Ec = 02Ec = 06

Ec = 1Ec = 14

0

05

1

15

2

25

3

35

4

Tem

pera

ture

02 04 06 08 10120582

(d)

Figure 5 Effect of Ec on (a) axial velocity (b) radial velocity (c) microrotation and (d) temperature for 119863minus1 = 2 Du = 02 Gr = 4 Re = 2Sr = 02 Ha = 2 Sc = 022 119886 = 02 120595 = 08 119877 = 2 119869

1= 2 119904

1= 2 119904

2= 1 Gm = 4 Kr = 2 and Pr = 071

Journal of Engineering 9

0

02

04

06

08

1

12A

xial

vel

ocity

02 04 06 08 10120582

Pr = 02Pr = 04

Pr = 071Pr = 1

(a)

Pr = 02Pr = 04

Pr = 071Pr = 1

055

057

059

061

063

065

067

069

071

Radi

al v

eloc

ity

02 04 06 08 10120582

(b)

0

005

01

015

0 02 04 06 08 1

Mic

roro

tatio

n

120582

Pr = 02Pr = 04

Pr = 071Pr = 1

minus02

minus015

minus01

minus005

(c)

Pr = 02Pr = 04

Pr = 071Pr = 1

0

02

04

06

08

1

12

14

16Te

mpe

ratu

re

02 04 06 08 10120582

(d)

Pr = 02Pr = 04

Pr = 071Pr = 1

0

01

02

03

04

05

06

07

Con

cent

ratio

n

02 04 06 08 10120582

(e)

Figure 6 Effect of Pr on (a) axial velocity (b) radial velocity (c) microrotation (d) temperature and (e) concentration for 119863minus1 = 2 Du =02 Gr = 4 Re = 2 Sr = 02 Ha = 2 Sc = 022 119886 = 02 120595 = 08 119877 = 2 119869

1= 02 119904

1= 2 119904

2= 1 Gm = 4 Kr = 2 and Ec = 1

10 Journal of Engineering

0 02 04 06 08 1

Axi

al v

eloc

ity

0

02

04

06

08

1

12

14

16

120582

Ha = 1Ha = 2

Ha = 3Ha = 4

(a)

058

063

068

073

078

083

088

093

098

0 02 04 06 08 1

Radi

al v

eloc

ity

120582

Ha = 1Ha = 2

Ha = 3Ha = 4

(b)

0

005

01

015

0 02 04 06 08 1

Mic

roro

tatio

n

120582

Ha = 1Ha = 2

Ha = 3Ha = 4

minus025

minus02

minus015

minus01

minus005

(c)

0

02

04

06

08

1

12

14

0 02 04 06 08 1

Tem

pera

ture

120582

Ha = 1Ha = 2

Ha = 3Ha = 4

(d)

0 02 04 06 08 1120582

Ha = 1Ha = 2

Ha = 3Ha = 4

0

02

01

03

05

07

09

04

06

08

1

Con

cent

ratio

n

(e)

Figure 7 Effect of Ha on (a) axial velocity (b) radial velocity (c) microrotation (d) temperature and (e) concentration for Kr = 2 Du = 002Gr = 4 Re = 2 Sr = 02 Sc = 1 Pr = 02 119886 = 04 120595 = 02 119877 = 2 119869

1= 2 119904

1= 2 119904

2= 2 Gm = 4119863minus1 = 02 and Ec = 1

Journal of Engineering 11

Figure 7 displays the change in the velocity componentsmicrorotation temperature distribution and concentrationfor several values of Ha From this it is observed that whenHa increases the temperature distribution also increaseswhereas the concentration decreases from the lower plate toupper plate and the axial velocity attains maximum valueat the center of the plates However the radial velocity andmicrorotation increase towards the center of the plates andthen decrease This is due to the fact that the magnetic forceretards the flow in both axial and radial directions Thevariations in the velocity components microrotation tem-perature distribution and concentration for different valuesof 119863minus1 are shown in Figure 8 From these one can deducethat the temperature distribution is increasing with 119863minus1whereas the radial velocity microrotation and concentrationare decreasing towards the upper plate and the axial velocitydecreases towards the center of the plates and then increasesbecause the resistance offered by the porosity of the mediumis more than the resistance due to the magnetic lines of force

5 Conclusions

The thermal diffusion and diffusion thermoeffects on com-bined free and forced convection magnetohydrodynamicflow of micropolar fluid in a porous medium between twoparallel plates with chemical reaction are considered Thenumerical solution of the transformed governing equations isobtained by the method of quasilinearization and the resultsare analyzed for various fluid and geometric parametersthrough graphs From the results the following is concluded

(i) The influences of Sr and Du on temperature andconcentration are similar

(ii) The temperature of the fluid is enhanced whereasthe concentration of the fluid is decreased with theincreasing of Ha and119863minus1

(iii) Kr reduces the concentration and enhances the tem-perature of the fluid

(iv) Ec and Pr exhibit similar effects on the velocitycomponents and microrotation whereas it is oppositein the case of temperature

Nomenclature

119886 Injection suction ratio 1 minus 11988111198812

119905 Timeℎ Distance between two parallel plates1198811119890119894120596119905 Injection velocity

1198812119890119894120596119905 Suction velocity

119901 Fluid pressure119902 Velocity vector119888 Specific heat at constant temperature119897 Microrotation vector119873 Microrotation componentEc Eckert number 120583119881

2120588ℎ119888(119879

2minus 1198791)

119896 Thermal conductivity

1198961 Viscosity parameter

1198962 Permeability of the medium

1198963 Chemical reaction rate

119906 Velocity component in 119909-directionV Velocity component in 119910-directionPr Prandtl number 120583119888119896Re Suction Reynolds number 120588119881

2ℎ120583

119895 Gyration parameter119869 Current density1198691 Nondimensional gyration parameter

120588119895ℎ1198812120574

119861 Total magnetic field119887 Induced magnetic field1198610 Magnetic flux density

119863 Rate of deformation tensor119864 Electric fieldHa Hartmann number 119861

0ℎradic120590120583

119863minus1 Inverse Darcy parameter ℎ2119896

2

119877 Nondimensional viscosity parameter 1198961120583

1199041 Nondimensional micropolar parameter

1198961ℎ2120574

1199042 Nondimensional micropolar parameter

120574119888ℎ2119896

119879 Temperature1198791119890119894120596119905 Temperature of the lower plate

1198792119890119894120596119905 Temperature of the upper plate

119879lowast Dimensionless temperature

(119879 minus 1198791119890119894120596119905)(1198792minus 1198791)119890119894120596119905

119862 Concentration119862lowast Nondimensional concentration

(119862 minus 1198620119890119894120596119905)(1198621minus 1198620)119890119894120596119905

1198620119890119894120596119905 Concentration of the lower plate

1198621119890119894120596119905 Concentration of the upper plate

1198631 Mass diffusivity

Kr Nondimensional chemical reaction param-eter 119896

3ℎ21198631

Sc Schmidt number 1205921198631

Gr Thermal Grashof number 120588119892120573119879(1198792minus

1198791)ℎ21205831198812

Gm Solutal Grashof number 120588119892120573119862(1198621minus

1198620)ℎ21205831198812

Sh Sherwood number 119899119860ℎ120592(119862

1minus 1198620)

119899119860 Mass transfer rate

Sr Soret number11986311198961198791205921198812119888119879119898119899119860

Du Dufour number119863111989611987911989911986012058811988812059221198812119888119904119896

119879119898 Mean temperature

119896119879 Thermal diffusion ratio

119888119904 Concentration susceptibility

Greek Letters

120582 Dimensionless 119910 coordinate 119910ℎ120572 120573 120574 Gyro viscosity parameters120577 Dimensionless axial variable (119880

01198861198812minus 119909ℎ)

120592 Kinematic viscosity120588 Fluid density120583 Fluid viscosity

12 Journal of Engineering

0

02

04

06

08

1

12

14

16

0 02 04 06 08 1

Axi

al v

eloc

ity

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

(a)

078

088

098

0 02 04 06 08 1

Radi

al v

eloc

ity

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

(b)

0

005

01

015

0 02 04 06 08 1

Mic

roro

tatio

n

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

minus025

minus02

minus03

minus015

minus01

minus005

(c)

0

05

1

15

2

25

3

35

4

45

0 02 04 06 08 1

Tem

pera

ture

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

(d)

0

02

04

06

08

1

12

0 02 04 06 08 1

Con

cent

ratio

n

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

(e)

Figure 8 Effect of 119863minus1 on (a) axial velocity (b) radial velocity (c) microrotation (d) temperature and (e) concentration for Kr = 2 Du =002 Gr = 4 Re = 2 Sr = 02 Ha = 1 Sc = 02 Pr = 02 119886 = 02 120595 = 02 119877 = 2 119869

1= 2 119904

1= 2 119904

2= 2 Gm = 4 and Ec = 1

Journal of Engineering 13

1205831015840 Magnetic permeability120590 Electric conductivity120595 Nondimensional frequency parameter 120596119905

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank referees for their suggestions which haveresulted in the improvement of the paper Also one ofthe authors (N Naresh Kumar) is grateful to the DefenceResearch and Development Organization Government ofIndia for providing senior research fellowship

References

[1] A S Berman ldquoLaminar flow in channels with porous wallsrdquoJournal of Applied Physics vol 24 pp 1232ndash1235 1953

[2] R M Terril and G M Shrestha ldquoLaminar flow throughparallel and uniformly porous walls of different permeabilityrdquoZeitschrift fur Angewandte Mathematik und Physik vol 16 pp470ndash482 1965

[3] A C Eringen ldquoTheory of micropolar fluidsrdquo Journal of Mathe-matics and Mechanics vol 16 pp 1ndash18 1966

[4] V U K Sastry and V Rama Mohan Rao ldquoNumerical solutionof micropolar fluid flow in a channel with porous wallsrdquoInternational Journal of Engineering Science vol 20 no 5 pp631ndash642 1982

[5] O Ojjela and N Naresh Kumar ldquoUnsteady MHD flow andheat transfer of micropolar fluid in a porous medium betweenparallel platesrdquo Canadian Journal of Physics vol 93 no 8 pp880ndash887 2015

[6] D Srinivasacharya J V RamanaMurthy and D VenugopalamldquoUnsteady stokes flow of micropolar fluid between two parallelporous platesrdquo International Journal of Engineering Science vol39 no 14 pp 1557ndash1563 2001

[7] G Maiti ldquoConvective heat transfer in micropolar fluid flowthrough a horizontal parallel plate channelrdquo ZAMMmdashJournalof Applied Mathematics and Mechanics vol 55 no 2 pp 105ndash111 1975

[8] R Bhargava L Kumar andH S Takhar ldquoNumerical solution offree convection MHD micropolar fluid flow between two par-allel porous vertical platesrdquo International Journal of EngineeringScience vol 41 no 2 pp 123ndash136 2003

[9] M Ashraf and A R Wehgal ldquoMHD flow and heat transfer ofmicropolar fluid between two porous disksrdquoAppliedMathemat-ics and Mechanics vol 33 no 1 pp 51ndash64 2012

[10] A IslamMd H A Biswas Md R Islam and SMMohiuddinldquoMHDmicropolar fluid flow through vertical porous mediumrdquoAcademic Research International vol 1 no 3 pp 381ndash393 2011

[11] S Nadeem M Hussain and M Naz ldquoMHD stagnation flow ofa micropolar fluid through a porous mediumrdquo Meccanica vol45 no 6 pp 869ndash880 2010

[12] O Ojjela and N Naresh Kumar ldquoUnsteady heat and masstransfer of chemically reacting micropolar fluid in a porous

channel with hall and ion slip currentsrdquo International ScholarlyResearch Notices vol 2014 Article ID 646957 11 pages 2014

[13] N S Elgazery ldquoThe effects of chemical reaction Hall andion-slip currents on MHD flow with temperature dependentviscosity and thermal diffusivityrdquoCommunications in NonlinearScience and Numerical Simulation vol 14 no 4 pp 1267ndash12832009

[14] M S Ayano ldquoMixed convection flow micropolar fluid over avertical plate subject to hall and ion-slip currentsrdquo InternationalJournal of Engineering amp Applied Sciences vol 5 no 3 pp 38ndash52 2013

[15] A R M K Aurangzaib N F Mohammad and S Shafie ldquoSoretand dufour effects on unsteadyMHD flow of a micropolar fluidin the presence of thermophoresis deposition particlerdquo WorldApplied Sciences Journal vol 21 no 5 pp 766ndash773 2013

[16] D Srinivasacharya and C RamReddy ldquoSoret and Dufoureffects on mixed convection in a non-Darcy porous mediumsaturated with micropolar fluidrdquo Nonlinear Analysis Modellingand Control vol 16 no 1 pp 100ndash115 2011

[17] A Mahdy ldquoSoret and Dufour effect on double diffusion mixedconvection from a vertical surface in a porous medium satu-rated with a non-Newtonian fluidrdquo Journal of Non-NewtonianFluid Mechanics vol 165 no 11-12 pp 568ndash575 2010

[18] T Hayat andM Nawaz ldquoSoret and Dufour effects on the mixedconvection flow of a second grade fluid subject to Hall and ion-slip currentsrdquo International Journal for Numerical Methods inFluids vol 67 no 9 pp 1073ndash1099 2011

[19] H P Rani and C N Kim ldquoA numerical study of the dufourand soret effects on unsteady natural convection flow pastan isothermal vertical cylinderrdquo Korean Journal of ChemicalEngineering vol 26 no 4 pp 946ndash954 2009

[20] D Pal and H Mondal ldquoEffects of Soret Dufour chemicalreaction and thermal radiation on MHD non-Darcy unsteadymixed convective heat and mass transfer over a stretchingsheetrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 4 pp 1942ndash1958 2011

[21] B K Sharma K Yadav N K Mishra and R C ChaudharyldquoSoret and Dufour effects on unsteady MHDmixed convectionflow past a radiative vertical porous plate embedded in a porousmedium with chemical reactionrdquo Applied Mathematics vol 3no 7 pp 717ndash723 2012

[22] E S Lee and L T Fan ldquoQuasilinearization technique forsolution of boundary layer equationsrdquoThe Canadian Journal ofChemical Engineering vol 46 no 3 pp 200ndash204 1968

[23] T Hymavathi and B Shanker ldquoA quasilinearization approachto heat transfer in MHD visco-elastic fluid flowrdquo AppliedMathematics and Computation vol 215 no 6 pp 2045ndash20542009

[24] C L Huang ldquoApplication of quasilinearization technique to thevertical channel flowandheat convectionrdquo International Journalof Non-Linear Mechanics vol 13 no 2 pp 55ndash60 1978

[25] S S Motsa P Sibanda and S Shateyi ldquoOn a new quasi-linearization method for systems of nonlinear boundary valueproblemsrdquo Mathematical Methods in the Applied Sciences vol34 no 11 pp 1406ndash1413 2011

[26] R E Bellman and R E Kalaba Quasilinearization andBoundary-Value Problems Elsevier New York NY USA 1965

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Shock and Vibration

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Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

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Navigation and Observation

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DistributedSensor Networks

International Journal of

6 Journal of Engineering

119889119909119903+1

11

119889120582= 119909119903+1

12

119889119909119903+1

12

119889120582= minus2119909

119903+1

13+

Sc Sr Re Pr1 minus Du Sc Sr

(8119909119903

2119909119903+1

2+ 2 (1 + 119877)

sdot 119863minus1119909119903

1119909119903+1

1+ 2Ha2119909119903

1119909119903+1

1minus 119909119903

1119909119903+1

8minus 119909119903+1

1119909119903

8

+21199042

Pr119909119903

5119909119903+1

5) cos120595 + Kr

1 minus DuSc Sr119909119903+1

11

+Sc Re

1 minus Du Sc Sr(119909119903

1119909119903+1

12+ 119909119903+1

1119909119903

12minus 119909119903

1119909119903

12) cos120595

minusSc Sr Re Pr1 minus Du Sc Sr

(4119909119903

2119909119903

2+ (1 + 119877)119863

minus1119909119903

1119909119903

1

+Ha21199091199031119909119903

1minus 119909119903

1119909119903

8+1199042

Pr119909119903

5119909119903

5) cos120595

119889119909119903+1

13

119889120582= 119909119903+1

14

119889119909119903+1

14

119889120582=

Sc Sr Re Pr1 minus Du Sc Sr

(2119909119903+1

3119909119903

3+ 2 (1 + 119877)

sdot 119863minus1119909119903

2119909119903+1

2+ 2Ha2119909119903

2119909119903+1

2+21199042

Pr119909119903

6119909119903+1

6

+119877

2(2119909119903

3119909119903+1

3+ 8119909119903

5119909119903+1

5+ 4119909119903

3119909119903+1

5+ 4119909119903

5119909119903+1

3)

+ 2119909119903

2119909119903+1

9+ 2119909119903+1

2119909119903

9minus 119909119903

1119909119903+1

10minus 119909119903+1

1119909119903

10) cos120595

+Kr

1 minus Du Sc Sr119909119903+1

13+

Sc Re1 minus Du Sc Sr

(119909119903

1119909119903+1

14

+ 119909119903+1

1119909119903

14minus 2119909119903

2119909119903+1

13minus 2119909119903+1

2119909119903

13minus 119909119903

1119909119903

14

+ 2119909119903

2119909119903

13) cos120595 minus Sc Sr Re Pr

1 minus Du Sc Sr(119909119903

3119909119903

3+ (1 + 119877)

sdot 119863minus1119909119903

2119909119903

2+Ha2119909119903

2119909119903

2+1199042

Pr119909119903

6119909119903

6+119877

2(119909119903

3119909119903

3

+ 4119909119903

5119909119903

5+ 4119909119903

3119909119903

5) + 2119909

119903

2119909119903

9minus 119909119903

1119909119903

10) cos120595

(20)

To solve for (119909119903+1119894

119894 = 1 2 14) the solutions to sevenseparate initial value problems denoted by 119909ℎ1

119894(120582) 119909ℎ2

119894(120582)

119909ℎ3

119894(120582) 119909ℎ4

119894(120582) 119909ℎ5

119894(120582) 119909ℎ6

119894(120582) and 119909ℎ7

119894(120582) (which are the

solutions of the homogeneous system corresponding to (20))and 1199091199011

119894(120582) (which is the particular solution of (20)) with

the following initial conditions are obtained by using the 4th-order Runge-Kutta method

119909ℎ1

3(0) = 1

119909ℎ1

119894(0) = 0 for 119894 = 3

119909ℎ2

4(0) = 1

119909ℎ2

119894(0) = 0 for 119894 = 4

119909ℎ3

6(0) = 1

119909ℎ3

119894(0) = 0 for 119894 = 6

119909ℎ4

8(0) = 1

119909ℎ4

119894(0) = 0 for 119894 = 8

119909ℎ5

10(0) = 1

119909ℎ5

119894(0) = 0 for 119894 = 10

119909ℎ6

12(0) = 1

119909ℎ6

119894(0) = 0 for 119894 = 12

119909ℎ7

14(0) = 1

119909ℎ7

119894(0) = 0 for 119894 = 14

1199091199011

1(0) = 1 minus 119886

1199091199011

2(0) = 119909

1199011

3(0) = 119909

1199011

4(0) = 119909

1199011

5(0) = 0

1199091199011

6(0) = 119909

1199011

7(0) = 119909

1199011

8(0) = 119909

1199011

9(0) = 0

1199091199011

10(0) = 119909

1199011

11(0) = 119909

1199011

12(0) = 0

1199091199011

13(0) = 119909

1199011

14(0) = 0

(21)

By using the principle of superposition the general solutioncan be written as

119909119899+1

119894(120582) = 119862

1119909ℎ1

119894(120582) + 119862

2119909ℎ2

119894(120582) + 119862

3119909ℎ3

119894(120582)

+ 1198624119909ℎ4

119894(120582) + 119862

5119909ℎ5

119894(120582) + 119862

6119909ℎ6

119894(120582)

+ 1198627119909ℎ7

119894(120582) + 119909

1199011

119894(120582)

(22)

where 1198621 1198622 1198623 1198624 1198625 1198626 and 119862

7are the unknown

constants and are determined by considering the boundaryconditions at 120582 = 1 This solution (119909119903+1

119894 119894 = 1 2 14)

is then compared with solution at the previous step (119909119903119894

119894 = 1 2 14) and further iteration is performed if theconvergence has not been achieved

4 Results and Discussion

The system of nonlinear differential equations (18) subject toboundary conditions (19) is solved numerically by the quasi-linearizationmethodThe influences of various fluid and geo-metric parameters such as Soret number Sr Dufour numberDu Hartmann number Ha chemical reaction parameter KrSchmidt number Sc Eckert number Ec and Prandtl numberPr on nondimensional velocity components microrotationtemperature distribution and concentration are analyzedthrough graphs in the domain [0 1]

Figures 2 and 3 show the influence of Sr and Du ontemperature and concentration From these figures it is

Journal of Engineering 7

0

05

1

15

2

25

3

0 02 04 06 08 1

Tem

pera

ture

Sr = 01Sr = 05

Sr = 1Sr = 15

120582

(a)

0

02

04

06

08

1

12

0 02 04 06 08 1

Con

cent

ratio

n

120582

Sr = 01Sr = 05

Sr = 1Sr = 15

(b)

Figure 2 Effect of Sr on (a) temperature and (b) concentration for Kr = 2 Gr = 4 Gm = 4 Re = 2 Du = 2 Sc = 022 Pr = 1 119886 = 02 120595 = 02119877 = 02 119869

1= 02 119904

1= 2 119904

2= 10 Ha = 1119863minus1 = 2 and Ec = 1

0 02 04 06 08 1120582

0

05

1

15

2

25

3

35

Tem

pera

ture

Du = 01Du = 1

Du = 2Du = 3

(a)

Du = 01Du = 1

Du = 2Du = 3

0

02

04

06

08

1

12C

once

ntra

tion

02 04 06 08 10120582

(b)

Figure 3 Effect of Du on (a) temperature and (b) concentration for Kr = 2 Gr = 4 Gm = 4 Re = 2 Sr = 1 Sc = 022 Pr = 1 119886 = 02 120595 = 02119877 = 02 119869

1= 02 119904

1= 2 119904

2= 10 Ha = 1119863minus1 = 2 and Ec = 1

evident that the temperature of the fluid increases whereasthe concentration decreases with the increasing of Sr and DuThis is because of the difference between the temperatures ofthe fluid and surface as well as the difference between concen-trations of the fluid and surface concentrations are increasedwith Sr and Du Figure 4 describes the behavior of the tem-perature distribution and concentration for the various valuesof Kr AsKr increases the temperature distribution of the fluidalso increases whereas the concentration decreases from thelower plate to the upper plate It is clear that the increase inthe Kr produces a decrease in the species concentrationThiscauses the concentration buoyancy effects to decrease as Krincreases The effect of Ec on velocity components micro-rotation and temperature is presented in Figure 5 As Ec

increases the radial velocity microrotation and temperatureare decreasing towards the upper plate However the axialvelocity decreases towards the center of the channel and thenincreases Since the Eckert number is the relation betweenthe kinetic energy and enthalpy as enthalpy increases thetemperature distribution decreases Figure 6 elucidates thechange in velocity components microrotation temperaturedistribution and concentration for different values of PrIt is observed that the axial velocity reaches highest valuenear the hot plate and the radial velocity microrotation andconcentration decrease whereas the temperature increaseswith the increasing value of Pr Physically if Pr increases thethermal diffusivity decreases and this leads to the decreasein the heat transfer ability at the thermal boundary layer

8 Journal of Engineering

0

02

04

06

08

1

12

14

16

18Te

mpe

ratu

re

02 04 06 08 10120582

Kr = 2Kr = 4

Kr = 6Kr = 8

(a)

0

01

02

03

04

05

06

07

08

Con

cent

ratio

n

02 04 06 08 10120582

Kr = 2Kr = 4

Kr = 6Kr = 8

(b)

Figure 4 Effect of Du on (a) temperature and (b) concentration for Du = 1 Gr = 4 Gm = 4 Re = 2 Sr = 02 Sc = 022 Pr = 1 119886 = 02120595 = 08119877 = 2 119869

1= 02 119904

1= 2 119904

2= 10 Ha = 2119863minus1 = 2 and Ec = 1

Axi

al v

eloc

ity

0

02

04

06

08

1

12

02 04 06 08 10120582

Ec = 02Ec = 06

Ec = 1Ec = 14

(a)

Radi

al v

eloc

ity

Ec = 02Ec = 06

Ec = 1Ec = 14

055

057

059

061

063

065

067

069

0 04 06 08 102120582

(b)

minus025

minus02

minus015

minus01

minus005

0

005

01

Mic

roro

tatio

n 02 04 06 08 10120582

Ec = 02Ec = 06

Ec = 1Ec = 14

(c)

Ec = 02Ec = 06

Ec = 1Ec = 14

0

05

1

15

2

25

3

35

4

Tem

pera

ture

02 04 06 08 10120582

(d)

Figure 5 Effect of Ec on (a) axial velocity (b) radial velocity (c) microrotation and (d) temperature for 119863minus1 = 2 Du = 02 Gr = 4 Re = 2Sr = 02 Ha = 2 Sc = 022 119886 = 02 120595 = 08 119877 = 2 119869

1= 2 119904

1= 2 119904

2= 1 Gm = 4 Kr = 2 and Pr = 071

Journal of Engineering 9

0

02

04

06

08

1

12A

xial

vel

ocity

02 04 06 08 10120582

Pr = 02Pr = 04

Pr = 071Pr = 1

(a)

Pr = 02Pr = 04

Pr = 071Pr = 1

055

057

059

061

063

065

067

069

071

Radi

al v

eloc

ity

02 04 06 08 10120582

(b)

0

005

01

015

0 02 04 06 08 1

Mic

roro

tatio

n

120582

Pr = 02Pr = 04

Pr = 071Pr = 1

minus02

minus015

minus01

minus005

(c)

Pr = 02Pr = 04

Pr = 071Pr = 1

0

02

04

06

08

1

12

14

16Te

mpe

ratu

re

02 04 06 08 10120582

(d)

Pr = 02Pr = 04

Pr = 071Pr = 1

0

01

02

03

04

05

06

07

Con

cent

ratio

n

02 04 06 08 10120582

(e)

Figure 6 Effect of Pr on (a) axial velocity (b) radial velocity (c) microrotation (d) temperature and (e) concentration for 119863minus1 = 2 Du =02 Gr = 4 Re = 2 Sr = 02 Ha = 2 Sc = 022 119886 = 02 120595 = 08 119877 = 2 119869

1= 02 119904

1= 2 119904

2= 1 Gm = 4 Kr = 2 and Ec = 1

10 Journal of Engineering

0 02 04 06 08 1

Axi

al v

eloc

ity

0

02

04

06

08

1

12

14

16

120582

Ha = 1Ha = 2

Ha = 3Ha = 4

(a)

058

063

068

073

078

083

088

093

098

0 02 04 06 08 1

Radi

al v

eloc

ity

120582

Ha = 1Ha = 2

Ha = 3Ha = 4

(b)

0

005

01

015

0 02 04 06 08 1

Mic

roro

tatio

n

120582

Ha = 1Ha = 2

Ha = 3Ha = 4

minus025

minus02

minus015

minus01

minus005

(c)

0

02

04

06

08

1

12

14

0 02 04 06 08 1

Tem

pera

ture

120582

Ha = 1Ha = 2

Ha = 3Ha = 4

(d)

0 02 04 06 08 1120582

Ha = 1Ha = 2

Ha = 3Ha = 4

0

02

01

03

05

07

09

04

06

08

1

Con

cent

ratio

n

(e)

Figure 7 Effect of Ha on (a) axial velocity (b) radial velocity (c) microrotation (d) temperature and (e) concentration for Kr = 2 Du = 002Gr = 4 Re = 2 Sr = 02 Sc = 1 Pr = 02 119886 = 04 120595 = 02 119877 = 2 119869

1= 2 119904

1= 2 119904

2= 2 Gm = 4119863minus1 = 02 and Ec = 1

Journal of Engineering 11

Figure 7 displays the change in the velocity componentsmicrorotation temperature distribution and concentrationfor several values of Ha From this it is observed that whenHa increases the temperature distribution also increaseswhereas the concentration decreases from the lower plate toupper plate and the axial velocity attains maximum valueat the center of the plates However the radial velocity andmicrorotation increase towards the center of the plates andthen decrease This is due to the fact that the magnetic forceretards the flow in both axial and radial directions Thevariations in the velocity components microrotation tem-perature distribution and concentration for different valuesof 119863minus1 are shown in Figure 8 From these one can deducethat the temperature distribution is increasing with 119863minus1whereas the radial velocity microrotation and concentrationare decreasing towards the upper plate and the axial velocitydecreases towards the center of the plates and then increasesbecause the resistance offered by the porosity of the mediumis more than the resistance due to the magnetic lines of force

5 Conclusions

The thermal diffusion and diffusion thermoeffects on com-bined free and forced convection magnetohydrodynamicflow of micropolar fluid in a porous medium between twoparallel plates with chemical reaction are considered Thenumerical solution of the transformed governing equations isobtained by the method of quasilinearization and the resultsare analyzed for various fluid and geometric parametersthrough graphs From the results the following is concluded

(i) The influences of Sr and Du on temperature andconcentration are similar

(ii) The temperature of the fluid is enhanced whereasthe concentration of the fluid is decreased with theincreasing of Ha and119863minus1

(iii) Kr reduces the concentration and enhances the tem-perature of the fluid

(iv) Ec and Pr exhibit similar effects on the velocitycomponents and microrotation whereas it is oppositein the case of temperature

Nomenclature

119886 Injection suction ratio 1 minus 11988111198812

119905 Timeℎ Distance between two parallel plates1198811119890119894120596119905 Injection velocity

1198812119890119894120596119905 Suction velocity

119901 Fluid pressure119902 Velocity vector119888 Specific heat at constant temperature119897 Microrotation vector119873 Microrotation componentEc Eckert number 120583119881

2120588ℎ119888(119879

2minus 1198791)

119896 Thermal conductivity

1198961 Viscosity parameter

1198962 Permeability of the medium

1198963 Chemical reaction rate

119906 Velocity component in 119909-directionV Velocity component in 119910-directionPr Prandtl number 120583119888119896Re Suction Reynolds number 120588119881

2ℎ120583

119895 Gyration parameter119869 Current density1198691 Nondimensional gyration parameter

120588119895ℎ1198812120574

119861 Total magnetic field119887 Induced magnetic field1198610 Magnetic flux density

119863 Rate of deformation tensor119864 Electric fieldHa Hartmann number 119861

0ℎradic120590120583

119863minus1 Inverse Darcy parameter ℎ2119896

2

119877 Nondimensional viscosity parameter 1198961120583

1199041 Nondimensional micropolar parameter

1198961ℎ2120574

1199042 Nondimensional micropolar parameter

120574119888ℎ2119896

119879 Temperature1198791119890119894120596119905 Temperature of the lower plate

1198792119890119894120596119905 Temperature of the upper plate

119879lowast Dimensionless temperature

(119879 minus 1198791119890119894120596119905)(1198792minus 1198791)119890119894120596119905

119862 Concentration119862lowast Nondimensional concentration

(119862 minus 1198620119890119894120596119905)(1198621minus 1198620)119890119894120596119905

1198620119890119894120596119905 Concentration of the lower plate

1198621119890119894120596119905 Concentration of the upper plate

1198631 Mass diffusivity

Kr Nondimensional chemical reaction param-eter 119896

3ℎ21198631

Sc Schmidt number 1205921198631

Gr Thermal Grashof number 120588119892120573119879(1198792minus

1198791)ℎ21205831198812

Gm Solutal Grashof number 120588119892120573119862(1198621minus

1198620)ℎ21205831198812

Sh Sherwood number 119899119860ℎ120592(119862

1minus 1198620)

119899119860 Mass transfer rate

Sr Soret number11986311198961198791205921198812119888119879119898119899119860

Du Dufour number119863111989611987911989911986012058811988812059221198812119888119904119896

119879119898 Mean temperature

119896119879 Thermal diffusion ratio

119888119904 Concentration susceptibility

Greek Letters

120582 Dimensionless 119910 coordinate 119910ℎ120572 120573 120574 Gyro viscosity parameters120577 Dimensionless axial variable (119880

01198861198812minus 119909ℎ)

120592 Kinematic viscosity120588 Fluid density120583 Fluid viscosity

12 Journal of Engineering

0

02

04

06

08

1

12

14

16

0 02 04 06 08 1

Axi

al v

eloc

ity

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

(a)

078

088

098

0 02 04 06 08 1

Radi

al v

eloc

ity

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

(b)

0

005

01

015

0 02 04 06 08 1

Mic

roro

tatio

n

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

minus025

minus02

minus03

minus015

minus01

minus005

(c)

0

05

1

15

2

25

3

35

4

45

0 02 04 06 08 1

Tem

pera

ture

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

(d)

0

02

04

06

08

1

12

0 02 04 06 08 1

Con

cent

ratio

n

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

(e)

Figure 8 Effect of 119863minus1 on (a) axial velocity (b) radial velocity (c) microrotation (d) temperature and (e) concentration for Kr = 2 Du =002 Gr = 4 Re = 2 Sr = 02 Ha = 1 Sc = 02 Pr = 02 119886 = 02 120595 = 02 119877 = 2 119869

1= 2 119904

1= 2 119904

2= 2 Gm = 4 and Ec = 1

Journal of Engineering 13

1205831015840 Magnetic permeability120590 Electric conductivity120595 Nondimensional frequency parameter 120596119905

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank referees for their suggestions which haveresulted in the improvement of the paper Also one ofthe authors (N Naresh Kumar) is grateful to the DefenceResearch and Development Organization Government ofIndia for providing senior research fellowship

References

[1] A S Berman ldquoLaminar flow in channels with porous wallsrdquoJournal of Applied Physics vol 24 pp 1232ndash1235 1953

[2] R M Terril and G M Shrestha ldquoLaminar flow throughparallel and uniformly porous walls of different permeabilityrdquoZeitschrift fur Angewandte Mathematik und Physik vol 16 pp470ndash482 1965

[3] A C Eringen ldquoTheory of micropolar fluidsrdquo Journal of Mathe-matics and Mechanics vol 16 pp 1ndash18 1966

[4] V U K Sastry and V Rama Mohan Rao ldquoNumerical solutionof micropolar fluid flow in a channel with porous wallsrdquoInternational Journal of Engineering Science vol 20 no 5 pp631ndash642 1982

[5] O Ojjela and N Naresh Kumar ldquoUnsteady MHD flow andheat transfer of micropolar fluid in a porous medium betweenparallel platesrdquo Canadian Journal of Physics vol 93 no 8 pp880ndash887 2015

[6] D Srinivasacharya J V RamanaMurthy and D VenugopalamldquoUnsteady stokes flow of micropolar fluid between two parallelporous platesrdquo International Journal of Engineering Science vol39 no 14 pp 1557ndash1563 2001

[7] G Maiti ldquoConvective heat transfer in micropolar fluid flowthrough a horizontal parallel plate channelrdquo ZAMMmdashJournalof Applied Mathematics and Mechanics vol 55 no 2 pp 105ndash111 1975

[8] R Bhargava L Kumar andH S Takhar ldquoNumerical solution offree convection MHD micropolar fluid flow between two par-allel porous vertical platesrdquo International Journal of EngineeringScience vol 41 no 2 pp 123ndash136 2003

[9] M Ashraf and A R Wehgal ldquoMHD flow and heat transfer ofmicropolar fluid between two porous disksrdquoAppliedMathemat-ics and Mechanics vol 33 no 1 pp 51ndash64 2012

[10] A IslamMd H A Biswas Md R Islam and SMMohiuddinldquoMHDmicropolar fluid flow through vertical porous mediumrdquoAcademic Research International vol 1 no 3 pp 381ndash393 2011

[11] S Nadeem M Hussain and M Naz ldquoMHD stagnation flow ofa micropolar fluid through a porous mediumrdquo Meccanica vol45 no 6 pp 869ndash880 2010

[12] O Ojjela and N Naresh Kumar ldquoUnsteady heat and masstransfer of chemically reacting micropolar fluid in a porous

channel with hall and ion slip currentsrdquo International ScholarlyResearch Notices vol 2014 Article ID 646957 11 pages 2014

[13] N S Elgazery ldquoThe effects of chemical reaction Hall andion-slip currents on MHD flow with temperature dependentviscosity and thermal diffusivityrdquoCommunications in NonlinearScience and Numerical Simulation vol 14 no 4 pp 1267ndash12832009

[14] M S Ayano ldquoMixed convection flow micropolar fluid over avertical plate subject to hall and ion-slip currentsrdquo InternationalJournal of Engineering amp Applied Sciences vol 5 no 3 pp 38ndash52 2013

[15] A R M K Aurangzaib N F Mohammad and S Shafie ldquoSoretand dufour effects on unsteadyMHD flow of a micropolar fluidin the presence of thermophoresis deposition particlerdquo WorldApplied Sciences Journal vol 21 no 5 pp 766ndash773 2013

[16] D Srinivasacharya and C RamReddy ldquoSoret and Dufoureffects on mixed convection in a non-Darcy porous mediumsaturated with micropolar fluidrdquo Nonlinear Analysis Modellingand Control vol 16 no 1 pp 100ndash115 2011

[17] A Mahdy ldquoSoret and Dufour effect on double diffusion mixedconvection from a vertical surface in a porous medium satu-rated with a non-Newtonian fluidrdquo Journal of Non-NewtonianFluid Mechanics vol 165 no 11-12 pp 568ndash575 2010

[18] T Hayat andM Nawaz ldquoSoret and Dufour effects on the mixedconvection flow of a second grade fluid subject to Hall and ion-slip currentsrdquo International Journal for Numerical Methods inFluids vol 67 no 9 pp 1073ndash1099 2011

[19] H P Rani and C N Kim ldquoA numerical study of the dufourand soret effects on unsteady natural convection flow pastan isothermal vertical cylinderrdquo Korean Journal of ChemicalEngineering vol 26 no 4 pp 946ndash954 2009

[20] D Pal and H Mondal ldquoEffects of Soret Dufour chemicalreaction and thermal radiation on MHD non-Darcy unsteadymixed convective heat and mass transfer over a stretchingsheetrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 4 pp 1942ndash1958 2011

[21] B K Sharma K Yadav N K Mishra and R C ChaudharyldquoSoret and Dufour effects on unsteady MHDmixed convectionflow past a radiative vertical porous plate embedded in a porousmedium with chemical reactionrdquo Applied Mathematics vol 3no 7 pp 717ndash723 2012

[22] E S Lee and L T Fan ldquoQuasilinearization technique forsolution of boundary layer equationsrdquoThe Canadian Journal ofChemical Engineering vol 46 no 3 pp 200ndash204 1968

[23] T Hymavathi and B Shanker ldquoA quasilinearization approachto heat transfer in MHD visco-elastic fluid flowrdquo AppliedMathematics and Computation vol 215 no 6 pp 2045ndash20542009

[24] C L Huang ldquoApplication of quasilinearization technique to thevertical channel flowandheat convectionrdquo International Journalof Non-Linear Mechanics vol 13 no 2 pp 55ndash60 1978

[25] S S Motsa P Sibanda and S Shateyi ldquoOn a new quasi-linearization method for systems of nonlinear boundary valueproblemsrdquo Mathematical Methods in the Applied Sciences vol34 no 11 pp 1406ndash1413 2011

[26] R E Bellman and R E Kalaba Quasilinearization andBoundary-Value Problems Elsevier New York NY USA 1965

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Engineering 7

0

05

1

15

2

25

3

0 02 04 06 08 1

Tem

pera

ture

Sr = 01Sr = 05

Sr = 1Sr = 15

120582

(a)

0

02

04

06

08

1

12

0 02 04 06 08 1

Con

cent

ratio

n

120582

Sr = 01Sr = 05

Sr = 1Sr = 15

(b)

Figure 2 Effect of Sr on (a) temperature and (b) concentration for Kr = 2 Gr = 4 Gm = 4 Re = 2 Du = 2 Sc = 022 Pr = 1 119886 = 02 120595 = 02119877 = 02 119869

1= 02 119904

1= 2 119904

2= 10 Ha = 1119863minus1 = 2 and Ec = 1

0 02 04 06 08 1120582

0

05

1

15

2

25

3

35

Tem

pera

ture

Du = 01Du = 1

Du = 2Du = 3

(a)

Du = 01Du = 1

Du = 2Du = 3

0

02

04

06

08

1

12C

once

ntra

tion

02 04 06 08 10120582

(b)

Figure 3 Effect of Du on (a) temperature and (b) concentration for Kr = 2 Gr = 4 Gm = 4 Re = 2 Sr = 1 Sc = 022 Pr = 1 119886 = 02 120595 = 02119877 = 02 119869

1= 02 119904

1= 2 119904

2= 10 Ha = 1119863minus1 = 2 and Ec = 1

evident that the temperature of the fluid increases whereasthe concentration decreases with the increasing of Sr and DuThis is because of the difference between the temperatures ofthe fluid and surface as well as the difference between concen-trations of the fluid and surface concentrations are increasedwith Sr and Du Figure 4 describes the behavior of the tem-perature distribution and concentration for the various valuesof Kr AsKr increases the temperature distribution of the fluidalso increases whereas the concentration decreases from thelower plate to the upper plate It is clear that the increase inthe Kr produces a decrease in the species concentrationThiscauses the concentration buoyancy effects to decrease as Krincreases The effect of Ec on velocity components micro-rotation and temperature is presented in Figure 5 As Ec

increases the radial velocity microrotation and temperatureare decreasing towards the upper plate However the axialvelocity decreases towards the center of the channel and thenincreases Since the Eckert number is the relation betweenthe kinetic energy and enthalpy as enthalpy increases thetemperature distribution decreases Figure 6 elucidates thechange in velocity components microrotation temperaturedistribution and concentration for different values of PrIt is observed that the axial velocity reaches highest valuenear the hot plate and the radial velocity microrotation andconcentration decrease whereas the temperature increaseswith the increasing value of Pr Physically if Pr increases thethermal diffusivity decreases and this leads to the decreasein the heat transfer ability at the thermal boundary layer

8 Journal of Engineering

0

02

04

06

08

1

12

14

16

18Te

mpe

ratu

re

02 04 06 08 10120582

Kr = 2Kr = 4

Kr = 6Kr = 8

(a)

0

01

02

03

04

05

06

07

08

Con

cent

ratio

n

02 04 06 08 10120582

Kr = 2Kr = 4

Kr = 6Kr = 8

(b)

Figure 4 Effect of Du on (a) temperature and (b) concentration for Du = 1 Gr = 4 Gm = 4 Re = 2 Sr = 02 Sc = 022 Pr = 1 119886 = 02120595 = 08119877 = 2 119869

1= 02 119904

1= 2 119904

2= 10 Ha = 2119863minus1 = 2 and Ec = 1

Axi

al v

eloc

ity

0

02

04

06

08

1

12

02 04 06 08 10120582

Ec = 02Ec = 06

Ec = 1Ec = 14

(a)

Radi

al v

eloc

ity

Ec = 02Ec = 06

Ec = 1Ec = 14

055

057

059

061

063

065

067

069

0 04 06 08 102120582

(b)

minus025

minus02

minus015

minus01

minus005

0

005

01

Mic

roro

tatio

n 02 04 06 08 10120582

Ec = 02Ec = 06

Ec = 1Ec = 14

(c)

Ec = 02Ec = 06

Ec = 1Ec = 14

0

05

1

15

2

25

3

35

4

Tem

pera

ture

02 04 06 08 10120582

(d)

Figure 5 Effect of Ec on (a) axial velocity (b) radial velocity (c) microrotation and (d) temperature for 119863minus1 = 2 Du = 02 Gr = 4 Re = 2Sr = 02 Ha = 2 Sc = 022 119886 = 02 120595 = 08 119877 = 2 119869

1= 2 119904

1= 2 119904

2= 1 Gm = 4 Kr = 2 and Pr = 071

Journal of Engineering 9

0

02

04

06

08

1

12A

xial

vel

ocity

02 04 06 08 10120582

Pr = 02Pr = 04

Pr = 071Pr = 1

(a)

Pr = 02Pr = 04

Pr = 071Pr = 1

055

057

059

061

063

065

067

069

071

Radi

al v

eloc

ity

02 04 06 08 10120582

(b)

0

005

01

015

0 02 04 06 08 1

Mic

roro

tatio

n

120582

Pr = 02Pr = 04

Pr = 071Pr = 1

minus02

minus015

minus01

minus005

(c)

Pr = 02Pr = 04

Pr = 071Pr = 1

0

02

04

06

08

1

12

14

16Te

mpe

ratu

re

02 04 06 08 10120582

(d)

Pr = 02Pr = 04

Pr = 071Pr = 1

0

01

02

03

04

05

06

07

Con

cent

ratio

n

02 04 06 08 10120582

(e)

Figure 6 Effect of Pr on (a) axial velocity (b) radial velocity (c) microrotation (d) temperature and (e) concentration for 119863minus1 = 2 Du =02 Gr = 4 Re = 2 Sr = 02 Ha = 2 Sc = 022 119886 = 02 120595 = 08 119877 = 2 119869

1= 02 119904

1= 2 119904

2= 1 Gm = 4 Kr = 2 and Ec = 1

10 Journal of Engineering

0 02 04 06 08 1

Axi

al v

eloc

ity

0

02

04

06

08

1

12

14

16

120582

Ha = 1Ha = 2

Ha = 3Ha = 4

(a)

058

063

068

073

078

083

088

093

098

0 02 04 06 08 1

Radi

al v

eloc

ity

120582

Ha = 1Ha = 2

Ha = 3Ha = 4

(b)

0

005

01

015

0 02 04 06 08 1

Mic

roro

tatio

n

120582

Ha = 1Ha = 2

Ha = 3Ha = 4

minus025

minus02

minus015

minus01

minus005

(c)

0

02

04

06

08

1

12

14

0 02 04 06 08 1

Tem

pera

ture

120582

Ha = 1Ha = 2

Ha = 3Ha = 4

(d)

0 02 04 06 08 1120582

Ha = 1Ha = 2

Ha = 3Ha = 4

0

02

01

03

05

07

09

04

06

08

1

Con

cent

ratio

n

(e)

Figure 7 Effect of Ha on (a) axial velocity (b) radial velocity (c) microrotation (d) temperature and (e) concentration for Kr = 2 Du = 002Gr = 4 Re = 2 Sr = 02 Sc = 1 Pr = 02 119886 = 04 120595 = 02 119877 = 2 119869

1= 2 119904

1= 2 119904

2= 2 Gm = 4119863minus1 = 02 and Ec = 1

Journal of Engineering 11

Figure 7 displays the change in the velocity componentsmicrorotation temperature distribution and concentrationfor several values of Ha From this it is observed that whenHa increases the temperature distribution also increaseswhereas the concentration decreases from the lower plate toupper plate and the axial velocity attains maximum valueat the center of the plates However the radial velocity andmicrorotation increase towards the center of the plates andthen decrease This is due to the fact that the magnetic forceretards the flow in both axial and radial directions Thevariations in the velocity components microrotation tem-perature distribution and concentration for different valuesof 119863minus1 are shown in Figure 8 From these one can deducethat the temperature distribution is increasing with 119863minus1whereas the radial velocity microrotation and concentrationare decreasing towards the upper plate and the axial velocitydecreases towards the center of the plates and then increasesbecause the resistance offered by the porosity of the mediumis more than the resistance due to the magnetic lines of force

5 Conclusions

The thermal diffusion and diffusion thermoeffects on com-bined free and forced convection magnetohydrodynamicflow of micropolar fluid in a porous medium between twoparallel plates with chemical reaction are considered Thenumerical solution of the transformed governing equations isobtained by the method of quasilinearization and the resultsare analyzed for various fluid and geometric parametersthrough graphs From the results the following is concluded

(i) The influences of Sr and Du on temperature andconcentration are similar

(ii) The temperature of the fluid is enhanced whereasthe concentration of the fluid is decreased with theincreasing of Ha and119863minus1

(iii) Kr reduces the concentration and enhances the tem-perature of the fluid

(iv) Ec and Pr exhibit similar effects on the velocitycomponents and microrotation whereas it is oppositein the case of temperature

Nomenclature

119886 Injection suction ratio 1 minus 11988111198812

119905 Timeℎ Distance between two parallel plates1198811119890119894120596119905 Injection velocity

1198812119890119894120596119905 Suction velocity

119901 Fluid pressure119902 Velocity vector119888 Specific heat at constant temperature119897 Microrotation vector119873 Microrotation componentEc Eckert number 120583119881

2120588ℎ119888(119879

2minus 1198791)

119896 Thermal conductivity

1198961 Viscosity parameter

1198962 Permeability of the medium

1198963 Chemical reaction rate

119906 Velocity component in 119909-directionV Velocity component in 119910-directionPr Prandtl number 120583119888119896Re Suction Reynolds number 120588119881

2ℎ120583

119895 Gyration parameter119869 Current density1198691 Nondimensional gyration parameter

120588119895ℎ1198812120574

119861 Total magnetic field119887 Induced magnetic field1198610 Magnetic flux density

119863 Rate of deformation tensor119864 Electric fieldHa Hartmann number 119861

0ℎradic120590120583

119863minus1 Inverse Darcy parameter ℎ2119896

2

119877 Nondimensional viscosity parameter 1198961120583

1199041 Nondimensional micropolar parameter

1198961ℎ2120574

1199042 Nondimensional micropolar parameter

120574119888ℎ2119896

119879 Temperature1198791119890119894120596119905 Temperature of the lower plate

1198792119890119894120596119905 Temperature of the upper plate

119879lowast Dimensionless temperature

(119879 minus 1198791119890119894120596119905)(1198792minus 1198791)119890119894120596119905

119862 Concentration119862lowast Nondimensional concentration

(119862 minus 1198620119890119894120596119905)(1198621minus 1198620)119890119894120596119905

1198620119890119894120596119905 Concentration of the lower plate

1198621119890119894120596119905 Concentration of the upper plate

1198631 Mass diffusivity

Kr Nondimensional chemical reaction param-eter 119896

3ℎ21198631

Sc Schmidt number 1205921198631

Gr Thermal Grashof number 120588119892120573119879(1198792minus

1198791)ℎ21205831198812

Gm Solutal Grashof number 120588119892120573119862(1198621minus

1198620)ℎ21205831198812

Sh Sherwood number 119899119860ℎ120592(119862

1minus 1198620)

119899119860 Mass transfer rate

Sr Soret number11986311198961198791205921198812119888119879119898119899119860

Du Dufour number119863111989611987911989911986012058811988812059221198812119888119904119896

119879119898 Mean temperature

119896119879 Thermal diffusion ratio

119888119904 Concentration susceptibility

Greek Letters

120582 Dimensionless 119910 coordinate 119910ℎ120572 120573 120574 Gyro viscosity parameters120577 Dimensionless axial variable (119880

01198861198812minus 119909ℎ)

120592 Kinematic viscosity120588 Fluid density120583 Fluid viscosity

12 Journal of Engineering

0

02

04

06

08

1

12

14

16

0 02 04 06 08 1

Axi

al v

eloc

ity

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

(a)

078

088

098

0 02 04 06 08 1

Radi

al v

eloc

ity

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

(b)

0

005

01

015

0 02 04 06 08 1

Mic

roro

tatio

n

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

minus025

minus02

minus03

minus015

minus01

minus005

(c)

0

05

1

15

2

25

3

35

4

45

0 02 04 06 08 1

Tem

pera

ture

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

(d)

0

02

04

06

08

1

12

0 02 04 06 08 1

Con

cent

ratio

n

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

(e)

Figure 8 Effect of 119863minus1 on (a) axial velocity (b) radial velocity (c) microrotation (d) temperature and (e) concentration for Kr = 2 Du =002 Gr = 4 Re = 2 Sr = 02 Ha = 1 Sc = 02 Pr = 02 119886 = 02 120595 = 02 119877 = 2 119869

1= 2 119904

1= 2 119904

2= 2 Gm = 4 and Ec = 1

Journal of Engineering 13

1205831015840 Magnetic permeability120590 Electric conductivity120595 Nondimensional frequency parameter 120596119905

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank referees for their suggestions which haveresulted in the improvement of the paper Also one ofthe authors (N Naresh Kumar) is grateful to the DefenceResearch and Development Organization Government ofIndia for providing senior research fellowship

References

[1] A S Berman ldquoLaminar flow in channels with porous wallsrdquoJournal of Applied Physics vol 24 pp 1232ndash1235 1953

[2] R M Terril and G M Shrestha ldquoLaminar flow throughparallel and uniformly porous walls of different permeabilityrdquoZeitschrift fur Angewandte Mathematik und Physik vol 16 pp470ndash482 1965

[3] A C Eringen ldquoTheory of micropolar fluidsrdquo Journal of Mathe-matics and Mechanics vol 16 pp 1ndash18 1966

[4] V U K Sastry and V Rama Mohan Rao ldquoNumerical solutionof micropolar fluid flow in a channel with porous wallsrdquoInternational Journal of Engineering Science vol 20 no 5 pp631ndash642 1982

[5] O Ojjela and N Naresh Kumar ldquoUnsteady MHD flow andheat transfer of micropolar fluid in a porous medium betweenparallel platesrdquo Canadian Journal of Physics vol 93 no 8 pp880ndash887 2015

[6] D Srinivasacharya J V RamanaMurthy and D VenugopalamldquoUnsteady stokes flow of micropolar fluid between two parallelporous platesrdquo International Journal of Engineering Science vol39 no 14 pp 1557ndash1563 2001

[7] G Maiti ldquoConvective heat transfer in micropolar fluid flowthrough a horizontal parallel plate channelrdquo ZAMMmdashJournalof Applied Mathematics and Mechanics vol 55 no 2 pp 105ndash111 1975

[8] R Bhargava L Kumar andH S Takhar ldquoNumerical solution offree convection MHD micropolar fluid flow between two par-allel porous vertical platesrdquo International Journal of EngineeringScience vol 41 no 2 pp 123ndash136 2003

[9] M Ashraf and A R Wehgal ldquoMHD flow and heat transfer ofmicropolar fluid between two porous disksrdquoAppliedMathemat-ics and Mechanics vol 33 no 1 pp 51ndash64 2012

[10] A IslamMd H A Biswas Md R Islam and SMMohiuddinldquoMHDmicropolar fluid flow through vertical porous mediumrdquoAcademic Research International vol 1 no 3 pp 381ndash393 2011

[11] S Nadeem M Hussain and M Naz ldquoMHD stagnation flow ofa micropolar fluid through a porous mediumrdquo Meccanica vol45 no 6 pp 869ndash880 2010

[12] O Ojjela and N Naresh Kumar ldquoUnsteady heat and masstransfer of chemically reacting micropolar fluid in a porous

channel with hall and ion slip currentsrdquo International ScholarlyResearch Notices vol 2014 Article ID 646957 11 pages 2014

[13] N S Elgazery ldquoThe effects of chemical reaction Hall andion-slip currents on MHD flow with temperature dependentviscosity and thermal diffusivityrdquoCommunications in NonlinearScience and Numerical Simulation vol 14 no 4 pp 1267ndash12832009

[14] M S Ayano ldquoMixed convection flow micropolar fluid over avertical plate subject to hall and ion-slip currentsrdquo InternationalJournal of Engineering amp Applied Sciences vol 5 no 3 pp 38ndash52 2013

[15] A R M K Aurangzaib N F Mohammad and S Shafie ldquoSoretand dufour effects on unsteadyMHD flow of a micropolar fluidin the presence of thermophoresis deposition particlerdquo WorldApplied Sciences Journal vol 21 no 5 pp 766ndash773 2013

[16] D Srinivasacharya and C RamReddy ldquoSoret and Dufoureffects on mixed convection in a non-Darcy porous mediumsaturated with micropolar fluidrdquo Nonlinear Analysis Modellingand Control vol 16 no 1 pp 100ndash115 2011

[17] A Mahdy ldquoSoret and Dufour effect on double diffusion mixedconvection from a vertical surface in a porous medium satu-rated with a non-Newtonian fluidrdquo Journal of Non-NewtonianFluid Mechanics vol 165 no 11-12 pp 568ndash575 2010

[18] T Hayat andM Nawaz ldquoSoret and Dufour effects on the mixedconvection flow of a second grade fluid subject to Hall and ion-slip currentsrdquo International Journal for Numerical Methods inFluids vol 67 no 9 pp 1073ndash1099 2011

[19] H P Rani and C N Kim ldquoA numerical study of the dufourand soret effects on unsteady natural convection flow pastan isothermal vertical cylinderrdquo Korean Journal of ChemicalEngineering vol 26 no 4 pp 946ndash954 2009

[20] D Pal and H Mondal ldquoEffects of Soret Dufour chemicalreaction and thermal radiation on MHD non-Darcy unsteadymixed convective heat and mass transfer over a stretchingsheetrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 4 pp 1942ndash1958 2011

[21] B K Sharma K Yadav N K Mishra and R C ChaudharyldquoSoret and Dufour effects on unsteady MHDmixed convectionflow past a radiative vertical porous plate embedded in a porousmedium with chemical reactionrdquo Applied Mathematics vol 3no 7 pp 717ndash723 2012

[22] E S Lee and L T Fan ldquoQuasilinearization technique forsolution of boundary layer equationsrdquoThe Canadian Journal ofChemical Engineering vol 46 no 3 pp 200ndash204 1968

[23] T Hymavathi and B Shanker ldquoA quasilinearization approachto heat transfer in MHD visco-elastic fluid flowrdquo AppliedMathematics and Computation vol 215 no 6 pp 2045ndash20542009

[24] C L Huang ldquoApplication of quasilinearization technique to thevertical channel flowandheat convectionrdquo International Journalof Non-Linear Mechanics vol 13 no 2 pp 55ndash60 1978

[25] S S Motsa P Sibanda and S Shateyi ldquoOn a new quasi-linearization method for systems of nonlinear boundary valueproblemsrdquo Mathematical Methods in the Applied Sciences vol34 no 11 pp 1406ndash1413 2011

[26] R E Bellman and R E Kalaba Quasilinearization andBoundary-Value Problems Elsevier New York NY USA 1965

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 Journal of Engineering

0

02

04

06

08

1

12

14

16

18Te

mpe

ratu

re

02 04 06 08 10120582

Kr = 2Kr = 4

Kr = 6Kr = 8

(a)

0

01

02

03

04

05

06

07

08

Con

cent

ratio

n

02 04 06 08 10120582

Kr = 2Kr = 4

Kr = 6Kr = 8

(b)

Figure 4 Effect of Du on (a) temperature and (b) concentration for Du = 1 Gr = 4 Gm = 4 Re = 2 Sr = 02 Sc = 022 Pr = 1 119886 = 02120595 = 08119877 = 2 119869

1= 02 119904

1= 2 119904

2= 10 Ha = 2119863minus1 = 2 and Ec = 1

Axi

al v

eloc

ity

0

02

04

06

08

1

12

02 04 06 08 10120582

Ec = 02Ec = 06

Ec = 1Ec = 14

(a)

Radi

al v

eloc

ity

Ec = 02Ec = 06

Ec = 1Ec = 14

055

057

059

061

063

065

067

069

0 04 06 08 102120582

(b)

minus025

minus02

minus015

minus01

minus005

0

005

01

Mic

roro

tatio

n 02 04 06 08 10120582

Ec = 02Ec = 06

Ec = 1Ec = 14

(c)

Ec = 02Ec = 06

Ec = 1Ec = 14

0

05

1

15

2

25

3

35

4

Tem

pera

ture

02 04 06 08 10120582

(d)

Figure 5 Effect of Ec on (a) axial velocity (b) radial velocity (c) microrotation and (d) temperature for 119863minus1 = 2 Du = 02 Gr = 4 Re = 2Sr = 02 Ha = 2 Sc = 022 119886 = 02 120595 = 08 119877 = 2 119869

1= 2 119904

1= 2 119904

2= 1 Gm = 4 Kr = 2 and Pr = 071

Journal of Engineering 9

0

02

04

06

08

1

12A

xial

vel

ocity

02 04 06 08 10120582

Pr = 02Pr = 04

Pr = 071Pr = 1

(a)

Pr = 02Pr = 04

Pr = 071Pr = 1

055

057

059

061

063

065

067

069

071

Radi

al v

eloc

ity

02 04 06 08 10120582

(b)

0

005

01

015

0 02 04 06 08 1

Mic

roro

tatio

n

120582

Pr = 02Pr = 04

Pr = 071Pr = 1

minus02

minus015

minus01

minus005

(c)

Pr = 02Pr = 04

Pr = 071Pr = 1

0

02

04

06

08

1

12

14

16Te

mpe

ratu

re

02 04 06 08 10120582

(d)

Pr = 02Pr = 04

Pr = 071Pr = 1

0

01

02

03

04

05

06

07

Con

cent

ratio

n

02 04 06 08 10120582

(e)

Figure 6 Effect of Pr on (a) axial velocity (b) radial velocity (c) microrotation (d) temperature and (e) concentration for 119863minus1 = 2 Du =02 Gr = 4 Re = 2 Sr = 02 Ha = 2 Sc = 022 119886 = 02 120595 = 08 119877 = 2 119869

1= 02 119904

1= 2 119904

2= 1 Gm = 4 Kr = 2 and Ec = 1

10 Journal of Engineering

0 02 04 06 08 1

Axi

al v

eloc

ity

0

02

04

06

08

1

12

14

16

120582

Ha = 1Ha = 2

Ha = 3Ha = 4

(a)

058

063

068

073

078

083

088

093

098

0 02 04 06 08 1

Radi

al v

eloc

ity

120582

Ha = 1Ha = 2

Ha = 3Ha = 4

(b)

0

005

01

015

0 02 04 06 08 1

Mic

roro

tatio

n

120582

Ha = 1Ha = 2

Ha = 3Ha = 4

minus025

minus02

minus015

minus01

minus005

(c)

0

02

04

06

08

1

12

14

0 02 04 06 08 1

Tem

pera

ture

120582

Ha = 1Ha = 2

Ha = 3Ha = 4

(d)

0 02 04 06 08 1120582

Ha = 1Ha = 2

Ha = 3Ha = 4

0

02

01

03

05

07

09

04

06

08

1

Con

cent

ratio

n

(e)

Figure 7 Effect of Ha on (a) axial velocity (b) radial velocity (c) microrotation (d) temperature and (e) concentration for Kr = 2 Du = 002Gr = 4 Re = 2 Sr = 02 Sc = 1 Pr = 02 119886 = 04 120595 = 02 119877 = 2 119869

1= 2 119904

1= 2 119904

2= 2 Gm = 4119863minus1 = 02 and Ec = 1

Journal of Engineering 11

Figure 7 displays the change in the velocity componentsmicrorotation temperature distribution and concentrationfor several values of Ha From this it is observed that whenHa increases the temperature distribution also increaseswhereas the concentration decreases from the lower plate toupper plate and the axial velocity attains maximum valueat the center of the plates However the radial velocity andmicrorotation increase towards the center of the plates andthen decrease This is due to the fact that the magnetic forceretards the flow in both axial and radial directions Thevariations in the velocity components microrotation tem-perature distribution and concentration for different valuesof 119863minus1 are shown in Figure 8 From these one can deducethat the temperature distribution is increasing with 119863minus1whereas the radial velocity microrotation and concentrationare decreasing towards the upper plate and the axial velocitydecreases towards the center of the plates and then increasesbecause the resistance offered by the porosity of the mediumis more than the resistance due to the magnetic lines of force

5 Conclusions

The thermal diffusion and diffusion thermoeffects on com-bined free and forced convection magnetohydrodynamicflow of micropolar fluid in a porous medium between twoparallel plates with chemical reaction are considered Thenumerical solution of the transformed governing equations isobtained by the method of quasilinearization and the resultsare analyzed for various fluid and geometric parametersthrough graphs From the results the following is concluded

(i) The influences of Sr and Du on temperature andconcentration are similar

(ii) The temperature of the fluid is enhanced whereasthe concentration of the fluid is decreased with theincreasing of Ha and119863minus1

(iii) Kr reduces the concentration and enhances the tem-perature of the fluid

(iv) Ec and Pr exhibit similar effects on the velocitycomponents and microrotation whereas it is oppositein the case of temperature

Nomenclature

119886 Injection suction ratio 1 minus 11988111198812

119905 Timeℎ Distance between two parallel plates1198811119890119894120596119905 Injection velocity

1198812119890119894120596119905 Suction velocity

119901 Fluid pressure119902 Velocity vector119888 Specific heat at constant temperature119897 Microrotation vector119873 Microrotation componentEc Eckert number 120583119881

2120588ℎ119888(119879

2minus 1198791)

119896 Thermal conductivity

1198961 Viscosity parameter

1198962 Permeability of the medium

1198963 Chemical reaction rate

119906 Velocity component in 119909-directionV Velocity component in 119910-directionPr Prandtl number 120583119888119896Re Suction Reynolds number 120588119881

2ℎ120583

119895 Gyration parameter119869 Current density1198691 Nondimensional gyration parameter

120588119895ℎ1198812120574

119861 Total magnetic field119887 Induced magnetic field1198610 Magnetic flux density

119863 Rate of deformation tensor119864 Electric fieldHa Hartmann number 119861

0ℎradic120590120583

119863minus1 Inverse Darcy parameter ℎ2119896

2

119877 Nondimensional viscosity parameter 1198961120583

1199041 Nondimensional micropolar parameter

1198961ℎ2120574

1199042 Nondimensional micropolar parameter

120574119888ℎ2119896

119879 Temperature1198791119890119894120596119905 Temperature of the lower plate

1198792119890119894120596119905 Temperature of the upper plate

119879lowast Dimensionless temperature

(119879 minus 1198791119890119894120596119905)(1198792minus 1198791)119890119894120596119905

119862 Concentration119862lowast Nondimensional concentration

(119862 minus 1198620119890119894120596119905)(1198621minus 1198620)119890119894120596119905

1198620119890119894120596119905 Concentration of the lower plate

1198621119890119894120596119905 Concentration of the upper plate

1198631 Mass diffusivity

Kr Nondimensional chemical reaction param-eter 119896

3ℎ21198631

Sc Schmidt number 1205921198631

Gr Thermal Grashof number 120588119892120573119879(1198792minus

1198791)ℎ21205831198812

Gm Solutal Grashof number 120588119892120573119862(1198621minus

1198620)ℎ21205831198812

Sh Sherwood number 119899119860ℎ120592(119862

1minus 1198620)

119899119860 Mass transfer rate

Sr Soret number11986311198961198791205921198812119888119879119898119899119860

Du Dufour number119863111989611987911989911986012058811988812059221198812119888119904119896

119879119898 Mean temperature

119896119879 Thermal diffusion ratio

119888119904 Concentration susceptibility

Greek Letters

120582 Dimensionless 119910 coordinate 119910ℎ120572 120573 120574 Gyro viscosity parameters120577 Dimensionless axial variable (119880

01198861198812minus 119909ℎ)

120592 Kinematic viscosity120588 Fluid density120583 Fluid viscosity

12 Journal of Engineering

0

02

04

06

08

1

12

14

16

0 02 04 06 08 1

Axi

al v

eloc

ity

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

(a)

078

088

098

0 02 04 06 08 1

Radi

al v

eloc

ity

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

(b)

0

005

01

015

0 02 04 06 08 1

Mic

roro

tatio

n

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

minus025

minus02

minus03

minus015

minus01

minus005

(c)

0

05

1

15

2

25

3

35

4

45

0 02 04 06 08 1

Tem

pera

ture

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

(d)

0

02

04

06

08

1

12

0 02 04 06 08 1

Con

cent

ratio

n

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

(e)

Figure 8 Effect of 119863minus1 on (a) axial velocity (b) radial velocity (c) microrotation (d) temperature and (e) concentration for Kr = 2 Du =002 Gr = 4 Re = 2 Sr = 02 Ha = 1 Sc = 02 Pr = 02 119886 = 02 120595 = 02 119877 = 2 119869

1= 2 119904

1= 2 119904

2= 2 Gm = 4 and Ec = 1

Journal of Engineering 13

1205831015840 Magnetic permeability120590 Electric conductivity120595 Nondimensional frequency parameter 120596119905

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank referees for their suggestions which haveresulted in the improvement of the paper Also one ofthe authors (N Naresh Kumar) is grateful to the DefenceResearch and Development Organization Government ofIndia for providing senior research fellowship

References

[1] A S Berman ldquoLaminar flow in channels with porous wallsrdquoJournal of Applied Physics vol 24 pp 1232ndash1235 1953

[2] R M Terril and G M Shrestha ldquoLaminar flow throughparallel and uniformly porous walls of different permeabilityrdquoZeitschrift fur Angewandte Mathematik und Physik vol 16 pp470ndash482 1965

[3] A C Eringen ldquoTheory of micropolar fluidsrdquo Journal of Mathe-matics and Mechanics vol 16 pp 1ndash18 1966

[4] V U K Sastry and V Rama Mohan Rao ldquoNumerical solutionof micropolar fluid flow in a channel with porous wallsrdquoInternational Journal of Engineering Science vol 20 no 5 pp631ndash642 1982

[5] O Ojjela and N Naresh Kumar ldquoUnsteady MHD flow andheat transfer of micropolar fluid in a porous medium betweenparallel platesrdquo Canadian Journal of Physics vol 93 no 8 pp880ndash887 2015

[6] D Srinivasacharya J V RamanaMurthy and D VenugopalamldquoUnsteady stokes flow of micropolar fluid between two parallelporous platesrdquo International Journal of Engineering Science vol39 no 14 pp 1557ndash1563 2001

[7] G Maiti ldquoConvective heat transfer in micropolar fluid flowthrough a horizontal parallel plate channelrdquo ZAMMmdashJournalof Applied Mathematics and Mechanics vol 55 no 2 pp 105ndash111 1975

[8] R Bhargava L Kumar andH S Takhar ldquoNumerical solution offree convection MHD micropolar fluid flow between two par-allel porous vertical platesrdquo International Journal of EngineeringScience vol 41 no 2 pp 123ndash136 2003

[9] M Ashraf and A R Wehgal ldquoMHD flow and heat transfer ofmicropolar fluid between two porous disksrdquoAppliedMathemat-ics and Mechanics vol 33 no 1 pp 51ndash64 2012

[10] A IslamMd H A Biswas Md R Islam and SMMohiuddinldquoMHDmicropolar fluid flow through vertical porous mediumrdquoAcademic Research International vol 1 no 3 pp 381ndash393 2011

[11] S Nadeem M Hussain and M Naz ldquoMHD stagnation flow ofa micropolar fluid through a porous mediumrdquo Meccanica vol45 no 6 pp 869ndash880 2010

[12] O Ojjela and N Naresh Kumar ldquoUnsteady heat and masstransfer of chemically reacting micropolar fluid in a porous

channel with hall and ion slip currentsrdquo International ScholarlyResearch Notices vol 2014 Article ID 646957 11 pages 2014

[13] N S Elgazery ldquoThe effects of chemical reaction Hall andion-slip currents on MHD flow with temperature dependentviscosity and thermal diffusivityrdquoCommunications in NonlinearScience and Numerical Simulation vol 14 no 4 pp 1267ndash12832009

[14] M S Ayano ldquoMixed convection flow micropolar fluid over avertical plate subject to hall and ion-slip currentsrdquo InternationalJournal of Engineering amp Applied Sciences vol 5 no 3 pp 38ndash52 2013

[15] A R M K Aurangzaib N F Mohammad and S Shafie ldquoSoretand dufour effects on unsteadyMHD flow of a micropolar fluidin the presence of thermophoresis deposition particlerdquo WorldApplied Sciences Journal vol 21 no 5 pp 766ndash773 2013

[16] D Srinivasacharya and C RamReddy ldquoSoret and Dufoureffects on mixed convection in a non-Darcy porous mediumsaturated with micropolar fluidrdquo Nonlinear Analysis Modellingand Control vol 16 no 1 pp 100ndash115 2011

[17] A Mahdy ldquoSoret and Dufour effect on double diffusion mixedconvection from a vertical surface in a porous medium satu-rated with a non-Newtonian fluidrdquo Journal of Non-NewtonianFluid Mechanics vol 165 no 11-12 pp 568ndash575 2010

[18] T Hayat andM Nawaz ldquoSoret and Dufour effects on the mixedconvection flow of a second grade fluid subject to Hall and ion-slip currentsrdquo International Journal for Numerical Methods inFluids vol 67 no 9 pp 1073ndash1099 2011

[19] H P Rani and C N Kim ldquoA numerical study of the dufourand soret effects on unsteady natural convection flow pastan isothermal vertical cylinderrdquo Korean Journal of ChemicalEngineering vol 26 no 4 pp 946ndash954 2009

[20] D Pal and H Mondal ldquoEffects of Soret Dufour chemicalreaction and thermal radiation on MHD non-Darcy unsteadymixed convective heat and mass transfer over a stretchingsheetrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 4 pp 1942ndash1958 2011

[21] B K Sharma K Yadav N K Mishra and R C ChaudharyldquoSoret and Dufour effects on unsteady MHDmixed convectionflow past a radiative vertical porous plate embedded in a porousmedium with chemical reactionrdquo Applied Mathematics vol 3no 7 pp 717ndash723 2012

[22] E S Lee and L T Fan ldquoQuasilinearization technique forsolution of boundary layer equationsrdquoThe Canadian Journal ofChemical Engineering vol 46 no 3 pp 200ndash204 1968

[23] T Hymavathi and B Shanker ldquoA quasilinearization approachto heat transfer in MHD visco-elastic fluid flowrdquo AppliedMathematics and Computation vol 215 no 6 pp 2045ndash20542009

[24] C L Huang ldquoApplication of quasilinearization technique to thevertical channel flowandheat convectionrdquo International Journalof Non-Linear Mechanics vol 13 no 2 pp 55ndash60 1978

[25] S S Motsa P Sibanda and S Shateyi ldquoOn a new quasi-linearization method for systems of nonlinear boundary valueproblemsrdquo Mathematical Methods in the Applied Sciences vol34 no 11 pp 1406ndash1413 2011

[26] R E Bellman and R E Kalaba Quasilinearization andBoundary-Value Problems Elsevier New York NY USA 1965

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Engineering 9

0

02

04

06

08

1

12A

xial

vel

ocity

02 04 06 08 10120582

Pr = 02Pr = 04

Pr = 071Pr = 1

(a)

Pr = 02Pr = 04

Pr = 071Pr = 1

055

057

059

061

063

065

067

069

071

Radi

al v

eloc

ity

02 04 06 08 10120582

(b)

0

005

01

015

0 02 04 06 08 1

Mic

roro

tatio

n

120582

Pr = 02Pr = 04

Pr = 071Pr = 1

minus02

minus015

minus01

minus005

(c)

Pr = 02Pr = 04

Pr = 071Pr = 1

0

02

04

06

08

1

12

14

16Te

mpe

ratu

re

02 04 06 08 10120582

(d)

Pr = 02Pr = 04

Pr = 071Pr = 1

0

01

02

03

04

05

06

07

Con

cent

ratio

n

02 04 06 08 10120582

(e)

Figure 6 Effect of Pr on (a) axial velocity (b) radial velocity (c) microrotation (d) temperature and (e) concentration for 119863minus1 = 2 Du =02 Gr = 4 Re = 2 Sr = 02 Ha = 2 Sc = 022 119886 = 02 120595 = 08 119877 = 2 119869

1= 02 119904

1= 2 119904

2= 1 Gm = 4 Kr = 2 and Ec = 1

10 Journal of Engineering

0 02 04 06 08 1

Axi

al v

eloc

ity

0

02

04

06

08

1

12

14

16

120582

Ha = 1Ha = 2

Ha = 3Ha = 4

(a)

058

063

068

073

078

083

088

093

098

0 02 04 06 08 1

Radi

al v

eloc

ity

120582

Ha = 1Ha = 2

Ha = 3Ha = 4

(b)

0

005

01

015

0 02 04 06 08 1

Mic

roro

tatio

n

120582

Ha = 1Ha = 2

Ha = 3Ha = 4

minus025

minus02

minus015

minus01

minus005

(c)

0

02

04

06

08

1

12

14

0 02 04 06 08 1

Tem

pera

ture

120582

Ha = 1Ha = 2

Ha = 3Ha = 4

(d)

0 02 04 06 08 1120582

Ha = 1Ha = 2

Ha = 3Ha = 4

0

02

01

03

05

07

09

04

06

08

1

Con

cent

ratio

n

(e)

Figure 7 Effect of Ha on (a) axial velocity (b) radial velocity (c) microrotation (d) temperature and (e) concentration for Kr = 2 Du = 002Gr = 4 Re = 2 Sr = 02 Sc = 1 Pr = 02 119886 = 04 120595 = 02 119877 = 2 119869

1= 2 119904

1= 2 119904

2= 2 Gm = 4119863minus1 = 02 and Ec = 1

Journal of Engineering 11

Figure 7 displays the change in the velocity componentsmicrorotation temperature distribution and concentrationfor several values of Ha From this it is observed that whenHa increases the temperature distribution also increaseswhereas the concentration decreases from the lower plate toupper plate and the axial velocity attains maximum valueat the center of the plates However the radial velocity andmicrorotation increase towards the center of the plates andthen decrease This is due to the fact that the magnetic forceretards the flow in both axial and radial directions Thevariations in the velocity components microrotation tem-perature distribution and concentration for different valuesof 119863minus1 are shown in Figure 8 From these one can deducethat the temperature distribution is increasing with 119863minus1whereas the radial velocity microrotation and concentrationare decreasing towards the upper plate and the axial velocitydecreases towards the center of the plates and then increasesbecause the resistance offered by the porosity of the mediumis more than the resistance due to the magnetic lines of force

5 Conclusions

The thermal diffusion and diffusion thermoeffects on com-bined free and forced convection magnetohydrodynamicflow of micropolar fluid in a porous medium between twoparallel plates with chemical reaction are considered Thenumerical solution of the transformed governing equations isobtained by the method of quasilinearization and the resultsare analyzed for various fluid and geometric parametersthrough graphs From the results the following is concluded

(i) The influences of Sr and Du on temperature andconcentration are similar

(ii) The temperature of the fluid is enhanced whereasthe concentration of the fluid is decreased with theincreasing of Ha and119863minus1

(iii) Kr reduces the concentration and enhances the tem-perature of the fluid

(iv) Ec and Pr exhibit similar effects on the velocitycomponents and microrotation whereas it is oppositein the case of temperature

Nomenclature

119886 Injection suction ratio 1 minus 11988111198812

119905 Timeℎ Distance between two parallel plates1198811119890119894120596119905 Injection velocity

1198812119890119894120596119905 Suction velocity

119901 Fluid pressure119902 Velocity vector119888 Specific heat at constant temperature119897 Microrotation vector119873 Microrotation componentEc Eckert number 120583119881

2120588ℎ119888(119879

2minus 1198791)

119896 Thermal conductivity

1198961 Viscosity parameter

1198962 Permeability of the medium

1198963 Chemical reaction rate

119906 Velocity component in 119909-directionV Velocity component in 119910-directionPr Prandtl number 120583119888119896Re Suction Reynolds number 120588119881

2ℎ120583

119895 Gyration parameter119869 Current density1198691 Nondimensional gyration parameter

120588119895ℎ1198812120574

119861 Total magnetic field119887 Induced magnetic field1198610 Magnetic flux density

119863 Rate of deformation tensor119864 Electric fieldHa Hartmann number 119861

0ℎradic120590120583

119863minus1 Inverse Darcy parameter ℎ2119896

2

119877 Nondimensional viscosity parameter 1198961120583

1199041 Nondimensional micropolar parameter

1198961ℎ2120574

1199042 Nondimensional micropolar parameter

120574119888ℎ2119896

119879 Temperature1198791119890119894120596119905 Temperature of the lower plate

1198792119890119894120596119905 Temperature of the upper plate

119879lowast Dimensionless temperature

(119879 minus 1198791119890119894120596119905)(1198792minus 1198791)119890119894120596119905

119862 Concentration119862lowast Nondimensional concentration

(119862 minus 1198620119890119894120596119905)(1198621minus 1198620)119890119894120596119905

1198620119890119894120596119905 Concentration of the lower plate

1198621119890119894120596119905 Concentration of the upper plate

1198631 Mass diffusivity

Kr Nondimensional chemical reaction param-eter 119896

3ℎ21198631

Sc Schmidt number 1205921198631

Gr Thermal Grashof number 120588119892120573119879(1198792minus

1198791)ℎ21205831198812

Gm Solutal Grashof number 120588119892120573119862(1198621minus

1198620)ℎ21205831198812

Sh Sherwood number 119899119860ℎ120592(119862

1minus 1198620)

119899119860 Mass transfer rate

Sr Soret number11986311198961198791205921198812119888119879119898119899119860

Du Dufour number119863111989611987911989911986012058811988812059221198812119888119904119896

119879119898 Mean temperature

119896119879 Thermal diffusion ratio

119888119904 Concentration susceptibility

Greek Letters

120582 Dimensionless 119910 coordinate 119910ℎ120572 120573 120574 Gyro viscosity parameters120577 Dimensionless axial variable (119880

01198861198812minus 119909ℎ)

120592 Kinematic viscosity120588 Fluid density120583 Fluid viscosity

12 Journal of Engineering

0

02

04

06

08

1

12

14

16

0 02 04 06 08 1

Axi

al v

eloc

ity

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

(a)

078

088

098

0 02 04 06 08 1

Radi

al v

eloc

ity

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

(b)

0

005

01

015

0 02 04 06 08 1

Mic

roro

tatio

n

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

minus025

minus02

minus03

minus015

minus01

minus005

(c)

0

05

1

15

2

25

3

35

4

45

0 02 04 06 08 1

Tem

pera

ture

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

(d)

0

02

04

06

08

1

12

0 02 04 06 08 1

Con

cent

ratio

n

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

(e)

Figure 8 Effect of 119863minus1 on (a) axial velocity (b) radial velocity (c) microrotation (d) temperature and (e) concentration for Kr = 2 Du =002 Gr = 4 Re = 2 Sr = 02 Ha = 1 Sc = 02 Pr = 02 119886 = 02 120595 = 02 119877 = 2 119869

1= 2 119904

1= 2 119904

2= 2 Gm = 4 and Ec = 1

Journal of Engineering 13

1205831015840 Magnetic permeability120590 Electric conductivity120595 Nondimensional frequency parameter 120596119905

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank referees for their suggestions which haveresulted in the improvement of the paper Also one ofthe authors (N Naresh Kumar) is grateful to the DefenceResearch and Development Organization Government ofIndia for providing senior research fellowship

References

[1] A S Berman ldquoLaminar flow in channels with porous wallsrdquoJournal of Applied Physics vol 24 pp 1232ndash1235 1953

[2] R M Terril and G M Shrestha ldquoLaminar flow throughparallel and uniformly porous walls of different permeabilityrdquoZeitschrift fur Angewandte Mathematik und Physik vol 16 pp470ndash482 1965

[3] A C Eringen ldquoTheory of micropolar fluidsrdquo Journal of Mathe-matics and Mechanics vol 16 pp 1ndash18 1966

[4] V U K Sastry and V Rama Mohan Rao ldquoNumerical solutionof micropolar fluid flow in a channel with porous wallsrdquoInternational Journal of Engineering Science vol 20 no 5 pp631ndash642 1982

[5] O Ojjela and N Naresh Kumar ldquoUnsteady MHD flow andheat transfer of micropolar fluid in a porous medium betweenparallel platesrdquo Canadian Journal of Physics vol 93 no 8 pp880ndash887 2015

[6] D Srinivasacharya J V RamanaMurthy and D VenugopalamldquoUnsteady stokes flow of micropolar fluid between two parallelporous platesrdquo International Journal of Engineering Science vol39 no 14 pp 1557ndash1563 2001

[7] G Maiti ldquoConvective heat transfer in micropolar fluid flowthrough a horizontal parallel plate channelrdquo ZAMMmdashJournalof Applied Mathematics and Mechanics vol 55 no 2 pp 105ndash111 1975

[8] R Bhargava L Kumar andH S Takhar ldquoNumerical solution offree convection MHD micropolar fluid flow between two par-allel porous vertical platesrdquo International Journal of EngineeringScience vol 41 no 2 pp 123ndash136 2003

[9] M Ashraf and A R Wehgal ldquoMHD flow and heat transfer ofmicropolar fluid between two porous disksrdquoAppliedMathemat-ics and Mechanics vol 33 no 1 pp 51ndash64 2012

[10] A IslamMd H A Biswas Md R Islam and SMMohiuddinldquoMHDmicropolar fluid flow through vertical porous mediumrdquoAcademic Research International vol 1 no 3 pp 381ndash393 2011

[11] S Nadeem M Hussain and M Naz ldquoMHD stagnation flow ofa micropolar fluid through a porous mediumrdquo Meccanica vol45 no 6 pp 869ndash880 2010

[12] O Ojjela and N Naresh Kumar ldquoUnsteady heat and masstransfer of chemically reacting micropolar fluid in a porous

channel with hall and ion slip currentsrdquo International ScholarlyResearch Notices vol 2014 Article ID 646957 11 pages 2014

[13] N S Elgazery ldquoThe effects of chemical reaction Hall andion-slip currents on MHD flow with temperature dependentviscosity and thermal diffusivityrdquoCommunications in NonlinearScience and Numerical Simulation vol 14 no 4 pp 1267ndash12832009

[14] M S Ayano ldquoMixed convection flow micropolar fluid over avertical plate subject to hall and ion-slip currentsrdquo InternationalJournal of Engineering amp Applied Sciences vol 5 no 3 pp 38ndash52 2013

[15] A R M K Aurangzaib N F Mohammad and S Shafie ldquoSoretand dufour effects on unsteadyMHD flow of a micropolar fluidin the presence of thermophoresis deposition particlerdquo WorldApplied Sciences Journal vol 21 no 5 pp 766ndash773 2013

[16] D Srinivasacharya and C RamReddy ldquoSoret and Dufoureffects on mixed convection in a non-Darcy porous mediumsaturated with micropolar fluidrdquo Nonlinear Analysis Modellingand Control vol 16 no 1 pp 100ndash115 2011

[17] A Mahdy ldquoSoret and Dufour effect on double diffusion mixedconvection from a vertical surface in a porous medium satu-rated with a non-Newtonian fluidrdquo Journal of Non-NewtonianFluid Mechanics vol 165 no 11-12 pp 568ndash575 2010

[18] T Hayat andM Nawaz ldquoSoret and Dufour effects on the mixedconvection flow of a second grade fluid subject to Hall and ion-slip currentsrdquo International Journal for Numerical Methods inFluids vol 67 no 9 pp 1073ndash1099 2011

[19] H P Rani and C N Kim ldquoA numerical study of the dufourand soret effects on unsteady natural convection flow pastan isothermal vertical cylinderrdquo Korean Journal of ChemicalEngineering vol 26 no 4 pp 946ndash954 2009

[20] D Pal and H Mondal ldquoEffects of Soret Dufour chemicalreaction and thermal radiation on MHD non-Darcy unsteadymixed convective heat and mass transfer over a stretchingsheetrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 4 pp 1942ndash1958 2011

[21] B K Sharma K Yadav N K Mishra and R C ChaudharyldquoSoret and Dufour effects on unsteady MHDmixed convectionflow past a radiative vertical porous plate embedded in a porousmedium with chemical reactionrdquo Applied Mathematics vol 3no 7 pp 717ndash723 2012

[22] E S Lee and L T Fan ldquoQuasilinearization technique forsolution of boundary layer equationsrdquoThe Canadian Journal ofChemical Engineering vol 46 no 3 pp 200ndash204 1968

[23] T Hymavathi and B Shanker ldquoA quasilinearization approachto heat transfer in MHD visco-elastic fluid flowrdquo AppliedMathematics and Computation vol 215 no 6 pp 2045ndash20542009

[24] C L Huang ldquoApplication of quasilinearization technique to thevertical channel flowandheat convectionrdquo International Journalof Non-Linear Mechanics vol 13 no 2 pp 55ndash60 1978

[25] S S Motsa P Sibanda and S Shateyi ldquoOn a new quasi-linearization method for systems of nonlinear boundary valueproblemsrdquo Mathematical Methods in the Applied Sciences vol34 no 11 pp 1406ndash1413 2011

[26] R E Bellman and R E Kalaba Quasilinearization andBoundary-Value Problems Elsevier New York NY USA 1965

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 Journal of Engineering

0 02 04 06 08 1

Axi

al v

eloc

ity

0

02

04

06

08

1

12

14

16

120582

Ha = 1Ha = 2

Ha = 3Ha = 4

(a)

058

063

068

073

078

083

088

093

098

0 02 04 06 08 1

Radi

al v

eloc

ity

120582

Ha = 1Ha = 2

Ha = 3Ha = 4

(b)

0

005

01

015

0 02 04 06 08 1

Mic

roro

tatio

n

120582

Ha = 1Ha = 2

Ha = 3Ha = 4

minus025

minus02

minus015

minus01

minus005

(c)

0

02

04

06

08

1

12

14

0 02 04 06 08 1

Tem

pera

ture

120582

Ha = 1Ha = 2

Ha = 3Ha = 4

(d)

0 02 04 06 08 1120582

Ha = 1Ha = 2

Ha = 3Ha = 4

0

02

01

03

05

07

09

04

06

08

1

Con

cent

ratio

n

(e)

Figure 7 Effect of Ha on (a) axial velocity (b) radial velocity (c) microrotation (d) temperature and (e) concentration for Kr = 2 Du = 002Gr = 4 Re = 2 Sr = 02 Sc = 1 Pr = 02 119886 = 04 120595 = 02 119877 = 2 119869

1= 2 119904

1= 2 119904

2= 2 Gm = 4119863minus1 = 02 and Ec = 1

Journal of Engineering 11

Figure 7 displays the change in the velocity componentsmicrorotation temperature distribution and concentrationfor several values of Ha From this it is observed that whenHa increases the temperature distribution also increaseswhereas the concentration decreases from the lower plate toupper plate and the axial velocity attains maximum valueat the center of the plates However the radial velocity andmicrorotation increase towards the center of the plates andthen decrease This is due to the fact that the magnetic forceretards the flow in both axial and radial directions Thevariations in the velocity components microrotation tem-perature distribution and concentration for different valuesof 119863minus1 are shown in Figure 8 From these one can deducethat the temperature distribution is increasing with 119863minus1whereas the radial velocity microrotation and concentrationare decreasing towards the upper plate and the axial velocitydecreases towards the center of the plates and then increasesbecause the resistance offered by the porosity of the mediumis more than the resistance due to the magnetic lines of force

5 Conclusions

The thermal diffusion and diffusion thermoeffects on com-bined free and forced convection magnetohydrodynamicflow of micropolar fluid in a porous medium between twoparallel plates with chemical reaction are considered Thenumerical solution of the transformed governing equations isobtained by the method of quasilinearization and the resultsare analyzed for various fluid and geometric parametersthrough graphs From the results the following is concluded

(i) The influences of Sr and Du on temperature andconcentration are similar

(ii) The temperature of the fluid is enhanced whereasthe concentration of the fluid is decreased with theincreasing of Ha and119863minus1

(iii) Kr reduces the concentration and enhances the tem-perature of the fluid

(iv) Ec and Pr exhibit similar effects on the velocitycomponents and microrotation whereas it is oppositein the case of temperature

Nomenclature

119886 Injection suction ratio 1 minus 11988111198812

119905 Timeℎ Distance between two parallel plates1198811119890119894120596119905 Injection velocity

1198812119890119894120596119905 Suction velocity

119901 Fluid pressure119902 Velocity vector119888 Specific heat at constant temperature119897 Microrotation vector119873 Microrotation componentEc Eckert number 120583119881

2120588ℎ119888(119879

2minus 1198791)

119896 Thermal conductivity

1198961 Viscosity parameter

1198962 Permeability of the medium

1198963 Chemical reaction rate

119906 Velocity component in 119909-directionV Velocity component in 119910-directionPr Prandtl number 120583119888119896Re Suction Reynolds number 120588119881

2ℎ120583

119895 Gyration parameter119869 Current density1198691 Nondimensional gyration parameter

120588119895ℎ1198812120574

119861 Total magnetic field119887 Induced magnetic field1198610 Magnetic flux density

119863 Rate of deformation tensor119864 Electric fieldHa Hartmann number 119861

0ℎradic120590120583

119863minus1 Inverse Darcy parameter ℎ2119896

2

119877 Nondimensional viscosity parameter 1198961120583

1199041 Nondimensional micropolar parameter

1198961ℎ2120574

1199042 Nondimensional micropolar parameter

120574119888ℎ2119896

119879 Temperature1198791119890119894120596119905 Temperature of the lower plate

1198792119890119894120596119905 Temperature of the upper plate

119879lowast Dimensionless temperature

(119879 minus 1198791119890119894120596119905)(1198792minus 1198791)119890119894120596119905

119862 Concentration119862lowast Nondimensional concentration

(119862 minus 1198620119890119894120596119905)(1198621minus 1198620)119890119894120596119905

1198620119890119894120596119905 Concentration of the lower plate

1198621119890119894120596119905 Concentration of the upper plate

1198631 Mass diffusivity

Kr Nondimensional chemical reaction param-eter 119896

3ℎ21198631

Sc Schmidt number 1205921198631

Gr Thermal Grashof number 120588119892120573119879(1198792minus

1198791)ℎ21205831198812

Gm Solutal Grashof number 120588119892120573119862(1198621minus

1198620)ℎ21205831198812

Sh Sherwood number 119899119860ℎ120592(119862

1minus 1198620)

119899119860 Mass transfer rate

Sr Soret number11986311198961198791205921198812119888119879119898119899119860

Du Dufour number119863111989611987911989911986012058811988812059221198812119888119904119896

119879119898 Mean temperature

119896119879 Thermal diffusion ratio

119888119904 Concentration susceptibility

Greek Letters

120582 Dimensionless 119910 coordinate 119910ℎ120572 120573 120574 Gyro viscosity parameters120577 Dimensionless axial variable (119880

01198861198812minus 119909ℎ)

120592 Kinematic viscosity120588 Fluid density120583 Fluid viscosity

12 Journal of Engineering

0

02

04

06

08

1

12

14

16

0 02 04 06 08 1

Axi

al v

eloc

ity

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

(a)

078

088

098

0 02 04 06 08 1

Radi

al v

eloc

ity

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

(b)

0

005

01

015

0 02 04 06 08 1

Mic

roro

tatio

n

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

minus025

minus02

minus03

minus015

minus01

minus005

(c)

0

05

1

15

2

25

3

35

4

45

0 02 04 06 08 1

Tem

pera

ture

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

(d)

0

02

04

06

08

1

12

0 02 04 06 08 1

Con

cent

ratio

n

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

(e)

Figure 8 Effect of 119863minus1 on (a) axial velocity (b) radial velocity (c) microrotation (d) temperature and (e) concentration for Kr = 2 Du =002 Gr = 4 Re = 2 Sr = 02 Ha = 1 Sc = 02 Pr = 02 119886 = 02 120595 = 02 119877 = 2 119869

1= 2 119904

1= 2 119904

2= 2 Gm = 4 and Ec = 1

Journal of Engineering 13

1205831015840 Magnetic permeability120590 Electric conductivity120595 Nondimensional frequency parameter 120596119905

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank referees for their suggestions which haveresulted in the improvement of the paper Also one ofthe authors (N Naresh Kumar) is grateful to the DefenceResearch and Development Organization Government ofIndia for providing senior research fellowship

References

[1] A S Berman ldquoLaminar flow in channels with porous wallsrdquoJournal of Applied Physics vol 24 pp 1232ndash1235 1953

[2] R M Terril and G M Shrestha ldquoLaminar flow throughparallel and uniformly porous walls of different permeabilityrdquoZeitschrift fur Angewandte Mathematik und Physik vol 16 pp470ndash482 1965

[3] A C Eringen ldquoTheory of micropolar fluidsrdquo Journal of Mathe-matics and Mechanics vol 16 pp 1ndash18 1966

[4] V U K Sastry and V Rama Mohan Rao ldquoNumerical solutionof micropolar fluid flow in a channel with porous wallsrdquoInternational Journal of Engineering Science vol 20 no 5 pp631ndash642 1982

[5] O Ojjela and N Naresh Kumar ldquoUnsteady MHD flow andheat transfer of micropolar fluid in a porous medium betweenparallel platesrdquo Canadian Journal of Physics vol 93 no 8 pp880ndash887 2015

[6] D Srinivasacharya J V RamanaMurthy and D VenugopalamldquoUnsteady stokes flow of micropolar fluid between two parallelporous platesrdquo International Journal of Engineering Science vol39 no 14 pp 1557ndash1563 2001

[7] G Maiti ldquoConvective heat transfer in micropolar fluid flowthrough a horizontal parallel plate channelrdquo ZAMMmdashJournalof Applied Mathematics and Mechanics vol 55 no 2 pp 105ndash111 1975

[8] R Bhargava L Kumar andH S Takhar ldquoNumerical solution offree convection MHD micropolar fluid flow between two par-allel porous vertical platesrdquo International Journal of EngineeringScience vol 41 no 2 pp 123ndash136 2003

[9] M Ashraf and A R Wehgal ldquoMHD flow and heat transfer ofmicropolar fluid between two porous disksrdquoAppliedMathemat-ics and Mechanics vol 33 no 1 pp 51ndash64 2012

[10] A IslamMd H A Biswas Md R Islam and SMMohiuddinldquoMHDmicropolar fluid flow through vertical porous mediumrdquoAcademic Research International vol 1 no 3 pp 381ndash393 2011

[11] S Nadeem M Hussain and M Naz ldquoMHD stagnation flow ofa micropolar fluid through a porous mediumrdquo Meccanica vol45 no 6 pp 869ndash880 2010

[12] O Ojjela and N Naresh Kumar ldquoUnsteady heat and masstransfer of chemically reacting micropolar fluid in a porous

channel with hall and ion slip currentsrdquo International ScholarlyResearch Notices vol 2014 Article ID 646957 11 pages 2014

[13] N S Elgazery ldquoThe effects of chemical reaction Hall andion-slip currents on MHD flow with temperature dependentviscosity and thermal diffusivityrdquoCommunications in NonlinearScience and Numerical Simulation vol 14 no 4 pp 1267ndash12832009

[14] M S Ayano ldquoMixed convection flow micropolar fluid over avertical plate subject to hall and ion-slip currentsrdquo InternationalJournal of Engineering amp Applied Sciences vol 5 no 3 pp 38ndash52 2013

[15] A R M K Aurangzaib N F Mohammad and S Shafie ldquoSoretand dufour effects on unsteadyMHD flow of a micropolar fluidin the presence of thermophoresis deposition particlerdquo WorldApplied Sciences Journal vol 21 no 5 pp 766ndash773 2013

[16] D Srinivasacharya and C RamReddy ldquoSoret and Dufoureffects on mixed convection in a non-Darcy porous mediumsaturated with micropolar fluidrdquo Nonlinear Analysis Modellingand Control vol 16 no 1 pp 100ndash115 2011

[17] A Mahdy ldquoSoret and Dufour effect on double diffusion mixedconvection from a vertical surface in a porous medium satu-rated with a non-Newtonian fluidrdquo Journal of Non-NewtonianFluid Mechanics vol 165 no 11-12 pp 568ndash575 2010

[18] T Hayat andM Nawaz ldquoSoret and Dufour effects on the mixedconvection flow of a second grade fluid subject to Hall and ion-slip currentsrdquo International Journal for Numerical Methods inFluids vol 67 no 9 pp 1073ndash1099 2011

[19] H P Rani and C N Kim ldquoA numerical study of the dufourand soret effects on unsteady natural convection flow pastan isothermal vertical cylinderrdquo Korean Journal of ChemicalEngineering vol 26 no 4 pp 946ndash954 2009

[20] D Pal and H Mondal ldquoEffects of Soret Dufour chemicalreaction and thermal radiation on MHD non-Darcy unsteadymixed convective heat and mass transfer over a stretchingsheetrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 4 pp 1942ndash1958 2011

[21] B K Sharma K Yadav N K Mishra and R C ChaudharyldquoSoret and Dufour effects on unsteady MHDmixed convectionflow past a radiative vertical porous plate embedded in a porousmedium with chemical reactionrdquo Applied Mathematics vol 3no 7 pp 717ndash723 2012

[22] E S Lee and L T Fan ldquoQuasilinearization technique forsolution of boundary layer equationsrdquoThe Canadian Journal ofChemical Engineering vol 46 no 3 pp 200ndash204 1968

[23] T Hymavathi and B Shanker ldquoA quasilinearization approachto heat transfer in MHD visco-elastic fluid flowrdquo AppliedMathematics and Computation vol 215 no 6 pp 2045ndash20542009

[24] C L Huang ldquoApplication of quasilinearization technique to thevertical channel flowandheat convectionrdquo International Journalof Non-Linear Mechanics vol 13 no 2 pp 55ndash60 1978

[25] S S Motsa P Sibanda and S Shateyi ldquoOn a new quasi-linearization method for systems of nonlinear boundary valueproblemsrdquo Mathematical Methods in the Applied Sciences vol34 no 11 pp 1406ndash1413 2011

[26] R E Bellman and R E Kalaba Quasilinearization andBoundary-Value Problems Elsevier New York NY USA 1965

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Engineering 11

Figure 7 displays the change in the velocity componentsmicrorotation temperature distribution and concentrationfor several values of Ha From this it is observed that whenHa increases the temperature distribution also increaseswhereas the concentration decreases from the lower plate toupper plate and the axial velocity attains maximum valueat the center of the plates However the radial velocity andmicrorotation increase towards the center of the plates andthen decrease This is due to the fact that the magnetic forceretards the flow in both axial and radial directions Thevariations in the velocity components microrotation tem-perature distribution and concentration for different valuesof 119863minus1 are shown in Figure 8 From these one can deducethat the temperature distribution is increasing with 119863minus1whereas the radial velocity microrotation and concentrationare decreasing towards the upper plate and the axial velocitydecreases towards the center of the plates and then increasesbecause the resistance offered by the porosity of the mediumis more than the resistance due to the magnetic lines of force

5 Conclusions

The thermal diffusion and diffusion thermoeffects on com-bined free and forced convection magnetohydrodynamicflow of micropolar fluid in a porous medium between twoparallel plates with chemical reaction are considered Thenumerical solution of the transformed governing equations isobtained by the method of quasilinearization and the resultsare analyzed for various fluid and geometric parametersthrough graphs From the results the following is concluded

(i) The influences of Sr and Du on temperature andconcentration are similar

(ii) The temperature of the fluid is enhanced whereasthe concentration of the fluid is decreased with theincreasing of Ha and119863minus1

(iii) Kr reduces the concentration and enhances the tem-perature of the fluid

(iv) Ec and Pr exhibit similar effects on the velocitycomponents and microrotation whereas it is oppositein the case of temperature

Nomenclature

119886 Injection suction ratio 1 minus 11988111198812

119905 Timeℎ Distance between two parallel plates1198811119890119894120596119905 Injection velocity

1198812119890119894120596119905 Suction velocity

119901 Fluid pressure119902 Velocity vector119888 Specific heat at constant temperature119897 Microrotation vector119873 Microrotation componentEc Eckert number 120583119881

2120588ℎ119888(119879

2minus 1198791)

119896 Thermal conductivity

1198961 Viscosity parameter

1198962 Permeability of the medium

1198963 Chemical reaction rate

119906 Velocity component in 119909-directionV Velocity component in 119910-directionPr Prandtl number 120583119888119896Re Suction Reynolds number 120588119881

2ℎ120583

119895 Gyration parameter119869 Current density1198691 Nondimensional gyration parameter

120588119895ℎ1198812120574

119861 Total magnetic field119887 Induced magnetic field1198610 Magnetic flux density

119863 Rate of deformation tensor119864 Electric fieldHa Hartmann number 119861

0ℎradic120590120583

119863minus1 Inverse Darcy parameter ℎ2119896

2

119877 Nondimensional viscosity parameter 1198961120583

1199041 Nondimensional micropolar parameter

1198961ℎ2120574

1199042 Nondimensional micropolar parameter

120574119888ℎ2119896

119879 Temperature1198791119890119894120596119905 Temperature of the lower plate

1198792119890119894120596119905 Temperature of the upper plate

119879lowast Dimensionless temperature

(119879 minus 1198791119890119894120596119905)(1198792minus 1198791)119890119894120596119905

119862 Concentration119862lowast Nondimensional concentration

(119862 minus 1198620119890119894120596119905)(1198621minus 1198620)119890119894120596119905

1198620119890119894120596119905 Concentration of the lower plate

1198621119890119894120596119905 Concentration of the upper plate

1198631 Mass diffusivity

Kr Nondimensional chemical reaction param-eter 119896

3ℎ21198631

Sc Schmidt number 1205921198631

Gr Thermal Grashof number 120588119892120573119879(1198792minus

1198791)ℎ21205831198812

Gm Solutal Grashof number 120588119892120573119862(1198621minus

1198620)ℎ21205831198812

Sh Sherwood number 119899119860ℎ120592(119862

1minus 1198620)

119899119860 Mass transfer rate

Sr Soret number11986311198961198791205921198812119888119879119898119899119860

Du Dufour number119863111989611987911989911986012058811988812059221198812119888119904119896

119879119898 Mean temperature

119896119879 Thermal diffusion ratio

119888119904 Concentration susceptibility

Greek Letters

120582 Dimensionless 119910 coordinate 119910ℎ120572 120573 120574 Gyro viscosity parameters120577 Dimensionless axial variable (119880

01198861198812minus 119909ℎ)

120592 Kinematic viscosity120588 Fluid density120583 Fluid viscosity

12 Journal of Engineering

0

02

04

06

08

1

12

14

16

0 02 04 06 08 1

Axi

al v

eloc

ity

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

(a)

078

088

098

0 02 04 06 08 1

Radi

al v

eloc

ity

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

(b)

0

005

01

015

0 02 04 06 08 1

Mic

roro

tatio

n

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

minus025

minus02

minus03

minus015

minus01

minus005

(c)

0

05

1

15

2

25

3

35

4

45

0 02 04 06 08 1

Tem

pera

ture

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

(d)

0

02

04

06

08

1

12

0 02 04 06 08 1

Con

cent

ratio

n

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

(e)

Figure 8 Effect of 119863minus1 on (a) axial velocity (b) radial velocity (c) microrotation (d) temperature and (e) concentration for Kr = 2 Du =002 Gr = 4 Re = 2 Sr = 02 Ha = 1 Sc = 02 Pr = 02 119886 = 02 120595 = 02 119877 = 2 119869

1= 2 119904

1= 2 119904

2= 2 Gm = 4 and Ec = 1

Journal of Engineering 13

1205831015840 Magnetic permeability120590 Electric conductivity120595 Nondimensional frequency parameter 120596119905

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank referees for their suggestions which haveresulted in the improvement of the paper Also one ofthe authors (N Naresh Kumar) is grateful to the DefenceResearch and Development Organization Government ofIndia for providing senior research fellowship

References

[1] A S Berman ldquoLaminar flow in channels with porous wallsrdquoJournal of Applied Physics vol 24 pp 1232ndash1235 1953

[2] R M Terril and G M Shrestha ldquoLaminar flow throughparallel and uniformly porous walls of different permeabilityrdquoZeitschrift fur Angewandte Mathematik und Physik vol 16 pp470ndash482 1965

[3] A C Eringen ldquoTheory of micropolar fluidsrdquo Journal of Mathe-matics and Mechanics vol 16 pp 1ndash18 1966

[4] V U K Sastry and V Rama Mohan Rao ldquoNumerical solutionof micropolar fluid flow in a channel with porous wallsrdquoInternational Journal of Engineering Science vol 20 no 5 pp631ndash642 1982

[5] O Ojjela and N Naresh Kumar ldquoUnsteady MHD flow andheat transfer of micropolar fluid in a porous medium betweenparallel platesrdquo Canadian Journal of Physics vol 93 no 8 pp880ndash887 2015

[6] D Srinivasacharya J V RamanaMurthy and D VenugopalamldquoUnsteady stokes flow of micropolar fluid between two parallelporous platesrdquo International Journal of Engineering Science vol39 no 14 pp 1557ndash1563 2001

[7] G Maiti ldquoConvective heat transfer in micropolar fluid flowthrough a horizontal parallel plate channelrdquo ZAMMmdashJournalof Applied Mathematics and Mechanics vol 55 no 2 pp 105ndash111 1975

[8] R Bhargava L Kumar andH S Takhar ldquoNumerical solution offree convection MHD micropolar fluid flow between two par-allel porous vertical platesrdquo International Journal of EngineeringScience vol 41 no 2 pp 123ndash136 2003

[9] M Ashraf and A R Wehgal ldquoMHD flow and heat transfer ofmicropolar fluid between two porous disksrdquoAppliedMathemat-ics and Mechanics vol 33 no 1 pp 51ndash64 2012

[10] A IslamMd H A Biswas Md R Islam and SMMohiuddinldquoMHDmicropolar fluid flow through vertical porous mediumrdquoAcademic Research International vol 1 no 3 pp 381ndash393 2011

[11] S Nadeem M Hussain and M Naz ldquoMHD stagnation flow ofa micropolar fluid through a porous mediumrdquo Meccanica vol45 no 6 pp 869ndash880 2010

[12] O Ojjela and N Naresh Kumar ldquoUnsteady heat and masstransfer of chemically reacting micropolar fluid in a porous

channel with hall and ion slip currentsrdquo International ScholarlyResearch Notices vol 2014 Article ID 646957 11 pages 2014

[13] N S Elgazery ldquoThe effects of chemical reaction Hall andion-slip currents on MHD flow with temperature dependentviscosity and thermal diffusivityrdquoCommunications in NonlinearScience and Numerical Simulation vol 14 no 4 pp 1267ndash12832009

[14] M S Ayano ldquoMixed convection flow micropolar fluid over avertical plate subject to hall and ion-slip currentsrdquo InternationalJournal of Engineering amp Applied Sciences vol 5 no 3 pp 38ndash52 2013

[15] A R M K Aurangzaib N F Mohammad and S Shafie ldquoSoretand dufour effects on unsteadyMHD flow of a micropolar fluidin the presence of thermophoresis deposition particlerdquo WorldApplied Sciences Journal vol 21 no 5 pp 766ndash773 2013

[16] D Srinivasacharya and C RamReddy ldquoSoret and Dufoureffects on mixed convection in a non-Darcy porous mediumsaturated with micropolar fluidrdquo Nonlinear Analysis Modellingand Control vol 16 no 1 pp 100ndash115 2011

[17] A Mahdy ldquoSoret and Dufour effect on double diffusion mixedconvection from a vertical surface in a porous medium satu-rated with a non-Newtonian fluidrdquo Journal of Non-NewtonianFluid Mechanics vol 165 no 11-12 pp 568ndash575 2010

[18] T Hayat andM Nawaz ldquoSoret and Dufour effects on the mixedconvection flow of a second grade fluid subject to Hall and ion-slip currentsrdquo International Journal for Numerical Methods inFluids vol 67 no 9 pp 1073ndash1099 2011

[19] H P Rani and C N Kim ldquoA numerical study of the dufourand soret effects on unsteady natural convection flow pastan isothermal vertical cylinderrdquo Korean Journal of ChemicalEngineering vol 26 no 4 pp 946ndash954 2009

[20] D Pal and H Mondal ldquoEffects of Soret Dufour chemicalreaction and thermal radiation on MHD non-Darcy unsteadymixed convective heat and mass transfer over a stretchingsheetrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 4 pp 1942ndash1958 2011

[21] B K Sharma K Yadav N K Mishra and R C ChaudharyldquoSoret and Dufour effects on unsteady MHDmixed convectionflow past a radiative vertical porous plate embedded in a porousmedium with chemical reactionrdquo Applied Mathematics vol 3no 7 pp 717ndash723 2012

[22] E S Lee and L T Fan ldquoQuasilinearization technique forsolution of boundary layer equationsrdquoThe Canadian Journal ofChemical Engineering vol 46 no 3 pp 200ndash204 1968

[23] T Hymavathi and B Shanker ldquoA quasilinearization approachto heat transfer in MHD visco-elastic fluid flowrdquo AppliedMathematics and Computation vol 215 no 6 pp 2045ndash20542009

[24] C L Huang ldquoApplication of quasilinearization technique to thevertical channel flowandheat convectionrdquo International Journalof Non-Linear Mechanics vol 13 no 2 pp 55ndash60 1978

[25] S S Motsa P Sibanda and S Shateyi ldquoOn a new quasi-linearization method for systems of nonlinear boundary valueproblemsrdquo Mathematical Methods in the Applied Sciences vol34 no 11 pp 1406ndash1413 2011

[26] R E Bellman and R E Kalaba Quasilinearization andBoundary-Value Problems Elsevier New York NY USA 1965

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

12 Journal of Engineering

0

02

04

06

08

1

12

14

16

0 02 04 06 08 1

Axi

al v

eloc

ity

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

(a)

078

088

098

0 02 04 06 08 1

Radi

al v

eloc

ity

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

(b)

0

005

01

015

0 02 04 06 08 1

Mic

roro

tatio

n

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

minus025

minus02

minus03

minus015

minus01

minus005

(c)

0

05

1

15

2

25

3

35

4

45

0 02 04 06 08 1

Tem

pera

ture

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

(d)

0

02

04

06

08

1

12

0 02 04 06 08 1

Con

cent

ratio

n

120582

Dminus1

= 1

Dminus1

= 10

Dminus1

= 20

Dminus1

= 30

(e)

Figure 8 Effect of 119863minus1 on (a) axial velocity (b) radial velocity (c) microrotation (d) temperature and (e) concentration for Kr = 2 Du =002 Gr = 4 Re = 2 Sr = 02 Ha = 1 Sc = 02 Pr = 02 119886 = 02 120595 = 02 119877 = 2 119869

1= 2 119904

1= 2 119904

2= 2 Gm = 4 and Ec = 1

Journal of Engineering 13

1205831015840 Magnetic permeability120590 Electric conductivity120595 Nondimensional frequency parameter 120596119905

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank referees for their suggestions which haveresulted in the improvement of the paper Also one ofthe authors (N Naresh Kumar) is grateful to the DefenceResearch and Development Organization Government ofIndia for providing senior research fellowship

References

[1] A S Berman ldquoLaminar flow in channels with porous wallsrdquoJournal of Applied Physics vol 24 pp 1232ndash1235 1953

[2] R M Terril and G M Shrestha ldquoLaminar flow throughparallel and uniformly porous walls of different permeabilityrdquoZeitschrift fur Angewandte Mathematik und Physik vol 16 pp470ndash482 1965

[3] A C Eringen ldquoTheory of micropolar fluidsrdquo Journal of Mathe-matics and Mechanics vol 16 pp 1ndash18 1966

[4] V U K Sastry and V Rama Mohan Rao ldquoNumerical solutionof micropolar fluid flow in a channel with porous wallsrdquoInternational Journal of Engineering Science vol 20 no 5 pp631ndash642 1982

[5] O Ojjela and N Naresh Kumar ldquoUnsteady MHD flow andheat transfer of micropolar fluid in a porous medium betweenparallel platesrdquo Canadian Journal of Physics vol 93 no 8 pp880ndash887 2015

[6] D Srinivasacharya J V RamanaMurthy and D VenugopalamldquoUnsteady stokes flow of micropolar fluid between two parallelporous platesrdquo International Journal of Engineering Science vol39 no 14 pp 1557ndash1563 2001

[7] G Maiti ldquoConvective heat transfer in micropolar fluid flowthrough a horizontal parallel plate channelrdquo ZAMMmdashJournalof Applied Mathematics and Mechanics vol 55 no 2 pp 105ndash111 1975

[8] R Bhargava L Kumar andH S Takhar ldquoNumerical solution offree convection MHD micropolar fluid flow between two par-allel porous vertical platesrdquo International Journal of EngineeringScience vol 41 no 2 pp 123ndash136 2003

[9] M Ashraf and A R Wehgal ldquoMHD flow and heat transfer ofmicropolar fluid between two porous disksrdquoAppliedMathemat-ics and Mechanics vol 33 no 1 pp 51ndash64 2012

[10] A IslamMd H A Biswas Md R Islam and SMMohiuddinldquoMHDmicropolar fluid flow through vertical porous mediumrdquoAcademic Research International vol 1 no 3 pp 381ndash393 2011

[11] S Nadeem M Hussain and M Naz ldquoMHD stagnation flow ofa micropolar fluid through a porous mediumrdquo Meccanica vol45 no 6 pp 869ndash880 2010

[12] O Ojjela and N Naresh Kumar ldquoUnsteady heat and masstransfer of chemically reacting micropolar fluid in a porous

channel with hall and ion slip currentsrdquo International ScholarlyResearch Notices vol 2014 Article ID 646957 11 pages 2014

[13] N S Elgazery ldquoThe effects of chemical reaction Hall andion-slip currents on MHD flow with temperature dependentviscosity and thermal diffusivityrdquoCommunications in NonlinearScience and Numerical Simulation vol 14 no 4 pp 1267ndash12832009

[14] M S Ayano ldquoMixed convection flow micropolar fluid over avertical plate subject to hall and ion-slip currentsrdquo InternationalJournal of Engineering amp Applied Sciences vol 5 no 3 pp 38ndash52 2013

[15] A R M K Aurangzaib N F Mohammad and S Shafie ldquoSoretand dufour effects on unsteadyMHD flow of a micropolar fluidin the presence of thermophoresis deposition particlerdquo WorldApplied Sciences Journal vol 21 no 5 pp 766ndash773 2013

[16] D Srinivasacharya and C RamReddy ldquoSoret and Dufoureffects on mixed convection in a non-Darcy porous mediumsaturated with micropolar fluidrdquo Nonlinear Analysis Modellingand Control vol 16 no 1 pp 100ndash115 2011

[17] A Mahdy ldquoSoret and Dufour effect on double diffusion mixedconvection from a vertical surface in a porous medium satu-rated with a non-Newtonian fluidrdquo Journal of Non-NewtonianFluid Mechanics vol 165 no 11-12 pp 568ndash575 2010

[18] T Hayat andM Nawaz ldquoSoret and Dufour effects on the mixedconvection flow of a second grade fluid subject to Hall and ion-slip currentsrdquo International Journal for Numerical Methods inFluids vol 67 no 9 pp 1073ndash1099 2011

[19] H P Rani and C N Kim ldquoA numerical study of the dufourand soret effects on unsteady natural convection flow pastan isothermal vertical cylinderrdquo Korean Journal of ChemicalEngineering vol 26 no 4 pp 946ndash954 2009

[20] D Pal and H Mondal ldquoEffects of Soret Dufour chemicalreaction and thermal radiation on MHD non-Darcy unsteadymixed convective heat and mass transfer over a stretchingsheetrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 4 pp 1942ndash1958 2011

[21] B K Sharma K Yadav N K Mishra and R C ChaudharyldquoSoret and Dufour effects on unsteady MHDmixed convectionflow past a radiative vertical porous plate embedded in a porousmedium with chemical reactionrdquo Applied Mathematics vol 3no 7 pp 717ndash723 2012

[22] E S Lee and L T Fan ldquoQuasilinearization technique forsolution of boundary layer equationsrdquoThe Canadian Journal ofChemical Engineering vol 46 no 3 pp 200ndash204 1968

[23] T Hymavathi and B Shanker ldquoA quasilinearization approachto heat transfer in MHD visco-elastic fluid flowrdquo AppliedMathematics and Computation vol 215 no 6 pp 2045ndash20542009

[24] C L Huang ldquoApplication of quasilinearization technique to thevertical channel flowandheat convectionrdquo International Journalof Non-Linear Mechanics vol 13 no 2 pp 55ndash60 1978

[25] S S Motsa P Sibanda and S Shateyi ldquoOn a new quasi-linearization method for systems of nonlinear boundary valueproblemsrdquo Mathematical Methods in the Applied Sciences vol34 no 11 pp 1406ndash1413 2011

[26] R E Bellman and R E Kalaba Quasilinearization andBoundary-Value Problems Elsevier New York NY USA 1965

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Engineering 13

1205831015840 Magnetic permeability120590 Electric conductivity120595 Nondimensional frequency parameter 120596119905

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank referees for their suggestions which haveresulted in the improvement of the paper Also one ofthe authors (N Naresh Kumar) is grateful to the DefenceResearch and Development Organization Government ofIndia for providing senior research fellowship

References

[1] A S Berman ldquoLaminar flow in channels with porous wallsrdquoJournal of Applied Physics vol 24 pp 1232ndash1235 1953

[2] R M Terril and G M Shrestha ldquoLaminar flow throughparallel and uniformly porous walls of different permeabilityrdquoZeitschrift fur Angewandte Mathematik und Physik vol 16 pp470ndash482 1965

[3] A C Eringen ldquoTheory of micropolar fluidsrdquo Journal of Mathe-matics and Mechanics vol 16 pp 1ndash18 1966

[4] V U K Sastry and V Rama Mohan Rao ldquoNumerical solutionof micropolar fluid flow in a channel with porous wallsrdquoInternational Journal of Engineering Science vol 20 no 5 pp631ndash642 1982

[5] O Ojjela and N Naresh Kumar ldquoUnsteady MHD flow andheat transfer of micropolar fluid in a porous medium betweenparallel platesrdquo Canadian Journal of Physics vol 93 no 8 pp880ndash887 2015

[6] D Srinivasacharya J V RamanaMurthy and D VenugopalamldquoUnsteady stokes flow of micropolar fluid between two parallelporous platesrdquo International Journal of Engineering Science vol39 no 14 pp 1557ndash1563 2001

[7] G Maiti ldquoConvective heat transfer in micropolar fluid flowthrough a horizontal parallel plate channelrdquo ZAMMmdashJournalof Applied Mathematics and Mechanics vol 55 no 2 pp 105ndash111 1975

[8] R Bhargava L Kumar andH S Takhar ldquoNumerical solution offree convection MHD micropolar fluid flow between two par-allel porous vertical platesrdquo International Journal of EngineeringScience vol 41 no 2 pp 123ndash136 2003

[9] M Ashraf and A R Wehgal ldquoMHD flow and heat transfer ofmicropolar fluid between two porous disksrdquoAppliedMathemat-ics and Mechanics vol 33 no 1 pp 51ndash64 2012

[10] A IslamMd H A Biswas Md R Islam and SMMohiuddinldquoMHDmicropolar fluid flow through vertical porous mediumrdquoAcademic Research International vol 1 no 3 pp 381ndash393 2011

[11] S Nadeem M Hussain and M Naz ldquoMHD stagnation flow ofa micropolar fluid through a porous mediumrdquo Meccanica vol45 no 6 pp 869ndash880 2010

[12] O Ojjela and N Naresh Kumar ldquoUnsteady heat and masstransfer of chemically reacting micropolar fluid in a porous

channel with hall and ion slip currentsrdquo International ScholarlyResearch Notices vol 2014 Article ID 646957 11 pages 2014

[13] N S Elgazery ldquoThe effects of chemical reaction Hall andion-slip currents on MHD flow with temperature dependentviscosity and thermal diffusivityrdquoCommunications in NonlinearScience and Numerical Simulation vol 14 no 4 pp 1267ndash12832009

[14] M S Ayano ldquoMixed convection flow micropolar fluid over avertical plate subject to hall and ion-slip currentsrdquo InternationalJournal of Engineering amp Applied Sciences vol 5 no 3 pp 38ndash52 2013

[15] A R M K Aurangzaib N F Mohammad and S Shafie ldquoSoretand dufour effects on unsteadyMHD flow of a micropolar fluidin the presence of thermophoresis deposition particlerdquo WorldApplied Sciences Journal vol 21 no 5 pp 766ndash773 2013

[16] D Srinivasacharya and C RamReddy ldquoSoret and Dufoureffects on mixed convection in a non-Darcy porous mediumsaturated with micropolar fluidrdquo Nonlinear Analysis Modellingand Control vol 16 no 1 pp 100ndash115 2011

[17] A Mahdy ldquoSoret and Dufour effect on double diffusion mixedconvection from a vertical surface in a porous medium satu-rated with a non-Newtonian fluidrdquo Journal of Non-NewtonianFluid Mechanics vol 165 no 11-12 pp 568ndash575 2010

[18] T Hayat andM Nawaz ldquoSoret and Dufour effects on the mixedconvection flow of a second grade fluid subject to Hall and ion-slip currentsrdquo International Journal for Numerical Methods inFluids vol 67 no 9 pp 1073ndash1099 2011

[19] H P Rani and C N Kim ldquoA numerical study of the dufourand soret effects on unsteady natural convection flow pastan isothermal vertical cylinderrdquo Korean Journal of ChemicalEngineering vol 26 no 4 pp 946ndash954 2009

[20] D Pal and H Mondal ldquoEffects of Soret Dufour chemicalreaction and thermal radiation on MHD non-Darcy unsteadymixed convective heat and mass transfer over a stretchingsheetrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 4 pp 1942ndash1958 2011

[21] B K Sharma K Yadav N K Mishra and R C ChaudharyldquoSoret and Dufour effects on unsteady MHDmixed convectionflow past a radiative vertical porous plate embedded in a porousmedium with chemical reactionrdquo Applied Mathematics vol 3no 7 pp 717ndash723 2012

[22] E S Lee and L T Fan ldquoQuasilinearization technique forsolution of boundary layer equationsrdquoThe Canadian Journal ofChemical Engineering vol 46 no 3 pp 200ndash204 1968

[23] T Hymavathi and B Shanker ldquoA quasilinearization approachto heat transfer in MHD visco-elastic fluid flowrdquo AppliedMathematics and Computation vol 215 no 6 pp 2045ndash20542009

[24] C L Huang ldquoApplication of quasilinearization technique to thevertical channel flowandheat convectionrdquo International Journalof Non-Linear Mechanics vol 13 no 2 pp 55ndash60 1978

[25] S S Motsa P Sibanda and S Shateyi ldquoOn a new quasi-linearization method for systems of nonlinear boundary valueproblemsrdquo Mathematical Methods in the Applied Sciences vol34 no 11 pp 1406ndash1413 2011

[26] R E Bellman and R E Kalaba Quasilinearization andBoundary-Value Problems Elsevier New York NY USA 1965

International Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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