Convective heat transfer analysis on Prandtl fluid model ... · 200 A. Alsaedi et al. / Convective heat transfer analysis on Prandtl fluid model with peristalsis flow due to myomaterial

Applied Bionics and Biomechanics 10 (2013) 197–208DOI 10.3233/ABB-140084IOS Press

197

Convective heat transfer analysis on Prandtlfluid model with peristalsis

A. Alsaedia, Naheed Batoolb, H. Yasminb,∗ and T. Hayata,baDepartment of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi ArabiabDepartment of Mathematics, Quaid-I-Azam University, Islamabad, Pakistan

Abstract. The effects of magnetohydrodynamic (MHD) on peristaltic transport of Prandtl fluid in a symmetric channel havebeen studied under the assumptions of long wave length and low-Reynolds number. Channel walls are considered compliantin nature. Series solutions of axial velocity, stream function and temperature are given by using regular perturbation techniquefor small values of Prandtl fluid parameter. The effects of physical parameters on the velocity, streamlines and temperature areexamined by plotting graphs.

Keywords: Prandtl fluid, magnetic field, compliant walls, convective conditions

1. Introduction

Considerable attention has been focused on theperistaltic transport of Newtonian and non-Newtonianfluids through tubes/channels in past few decades. Thisprocess is quite important for fluid transport in viewof theoretical and industrial perspectives. Peristaltictransport has great importance in many biological sys-tems such as food swallowing through the esophagus,circulation of blood in small blood vessels and theflows of many other glandular ducts etc. In additionperistaltic pumping has many industrial applicationsinvolving biomechanical systems such as heart-lungmachine, finger and roller pumps etc. Some significantliterature illustrating Newtonian and non-Newtonianfluid with/without peristaltic transport are given inRefs. [1–6]. Recently it is a well-accepted fact thatthe peristaltic flows of magnetohydrodynamic (MHD)fluids are important in medical sciences and bioengi-neering. The MHD characteristics are useful in thedevelopment of magnetic devices, hyperthermia and

∗Corresponding author: H. Yasmin, Department of Mathematics,Quaid-I-Azam University 45320, Islamabad 44000, Pakistan. Tel.:+92 51 90642172; E-mail: [email protected].

blood reduction during surgeries and cancer tumortreatment. Also the effect of magnetic field on viscousfluid has been reported for treatment of the pathologiese.g. Gastroenric pathologies, rheumatisms, constipa-tion and hypertension that can be treated by placingone electrode either on the back or on the stomachand the other on the sole of the foot; this locationwill induce a better blood circulation. Hence severalresearchers have discussed the peristalsis with mag-netic field effects [7–11].

The idea of complaint nature of channel walls inperistalsis was initiated by Kramer [12], where hecovered an underwater object with rubber and foundconsiderable reductions in the drag. Experiments havebeen conducted to study other flows past compliantboundaries, blood flow in arteries, dolphin propulsionetc. After that Mittra and Prasad [13] examined theinfluence of compliant walls on Poiseulli flow withperistalsis. They discussed the peristaltic flow in atwo-dimensional channel. The relationship betweenstress and strain for viscous-inelastic fluids is dis-cussed by Patel and Timol [14]. Heat transfer analysishas been used to obtain information about the prop-erties of tissues and have many applications in the

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198 A. Alsaedi et al. / Convective heat transfer analysis on Prandtl fluid model with peristalsis

biomedical sciences with knowledge of initial thermalconditions. The influence of wall properties on peri-staltic transport with heat transfer is presented byRadhakrishnamacharya and Srinivasulu [15]. FurtherHayat et al. [16, 17] and Hina et al. [18] elaborated theconcept of compliant walls with heat transfer for peri-staltic transport. Rashidi et al. [19] examined the heattransfer effects in the flow by a non-isothermal wedge.Heat transfer with convective boundary conditions isinvolved in processes such as thermal energy storage,gas turbines, nuclear plants etc. Hayat et al. [20-22]presented the study of a non-Newtonian fluid withconvective conditions. Recently many researchers dis-cussed about the peristaltic transport of Prandtl fluid.Akbar et al. [23] examined the Prandtl fluid model in anasymmetric channel. Sucharitha et al. [24] and Nava-neeswara et al. [25] discussed the conducting Prandtlfluid in a porous channel. Jothia et al. [26] furtherexplained the peristaltic transport of same model underthe effects of magnetic field.

The above mentioned studies show that no attempthas been made to investigate the heat transfer onthe MHD peristaltic transport of Prandtl fluid withcomplaint walls and convective conditions. The pre-sented research article is based on the consideration ofsuch effects in peristaltic flows. The governing equa-tions of Prandtl fluid model are simplified under theassumptions of long wavelength and low Reynoldsnumber approximations and solved by using pertur-bation technique. The expressions for stream function,temperature and velocity are obtained. The effects ofemerging parameters on the velocity and temperaturerise are discussed physically and shown via graphs.

2. Mathematical formulation

We consider an incompressible MHD Prandtl fluidin a symmetric channel of width 2d. A uniform mag-netic field B0 is applied in the transverse direction tothe flow. Consider the coordinate system (x, y) wherex-axis and y-axis are taken parallel and transverse to thedirection of wave propagation respectively. The flow isinduced by periodic peristaltic waves of length λ andamplitude a with constant speed c along the channelwalls (see Fig. 1). The geometry of the wall is givenby

y = ±η(x, t) = ±[d + asin 2π

λ(x− ct)

], (1)

Fig. 1. Geometry of the problem.

The governing equations in the laboratory frame are

∂u

∂x+ ∂v∂y

= 0, (2)

ρdu

dt= −∂p

∂x+ ∂Sxx

∂x+ ∂Sxy

∂y− σB20u, (3)

ρdv

dt= −∂p

∂y+ ∂Syx

∂x+ ∂Syy

∂y, (4)

ρCpdT

dt= k

[∂2T

∂x2+ ∂

2T

∂y2

]+ (Sxx − Syy) ∂u

∂x

+ Sxy(∂v

∂x+ ∂u∂y

), (5)

where (u, v) are the components of velocity field V inthe x and y directions respectively, B0 is the strengthof the constant applied magnetic field, σ the electricalconductivity of fluid, Sij (i, j = x, y) the componentsof extra stress tensor, d/dt the material time derivative,Cp the specific heat at constant volume and T the tem-perature of fluid. The extra stress tensor S for Prandtlfluid is given by [14]

Sxy =A sin−1

[1C

[(∂u∂y

)2 + ( ∂v∂x

)2]1/2][(

∂u∂y

)2 + ( ∂v∂x

)2]1/2∂u

∂y, (6)

in whichA andC are material constants of Prandtl fluidmodel.

A. Alsaedi et al. / Convective heat transfer analysis on Prandtl fluid model with peristalsis 199

The corresponding boundary conditions are givenby

u = 0, at y = ±η, (7)

[−τ ∂

3

∂x3+m ∂

3

∂x∂t2+ d ∂

2

∂x∂t+ B ∂

5

∂x5+H ∂

∂x

]η

= + ∂Sxx∂x

+ ∂Sxy∂y

− σB20u− ρdu

dtat y = ±η, (8)

k∂T

∂y= −h1 (T − T0) at y = η, (9)

k∂T

∂y= −h2 (T0 − T ) at y = −η, (10)

In above equations τ is the elastic tension in themembrane,m the mass per unit area,d the coefficient ofviscous damping,B flexural rigidity of the plate,H thespring stiffness. h1 and h2 the heat transfer coefficientsandT0 the ambient temperature at both upper and lowerwalls of the channel.

Defining velocity components u and v in terms ofstream function and dimensionless variables as

u = ∂ψ∂y, v = −∂ψ

∂x, .ψ∗ = ψ

cd, x∗ = x

λ,

y∗ = yd, t∗ = ct

λ, p∗ = d

2p

cµλ, Sij = dSij

cµ,

η∗ = ηd, h∗1 =

h1

d, h∗2 =

h2

d, θ = T − T0

T0, (11)

the Equation (2) is identically satisfied and Equations(3–10) in terms of stream function ψ can be expressedas follows:

δRe

[∂

∂t+ ψy ∂

∂x− ψx ∂

∂y

]ψy

= −∂p∂x

+ δ∂Sxx∂x

+ ∂Sxy∂y

−M2ψy, (12)

δ3Re

[∂

∂t+ ψy ∂

∂x− ψx ∂

∂y

]ψx

= −∂p∂y

+ δ2 ∂Sxy∂x

+ δ∂Syy∂y

, (13)

δRe Pr

[∂

∂t+ ψy ∂

∂x− ψx ∂

∂y

]

= ∂2θ

∂y2+ δ2 ∂

2θ

∂x2+ Br [δ (Sxx − Syy)ψxy

+ Sxy(ψyy − δ2ψxx

)], (14)

∂ψ

∂y= 0 at y = ±η, (15)

[E1

∂3

∂x3+ E2 ∂

3

∂x∂t2+ E3 ∂

2

∂x∂t+ E4 ∂

5

∂x5+ E5 ∂

∂x

]η

= −M2ψy − δRe[∂

∂t+ ψy ∂

∂x− ψx ∂

∂y

]ψy

+ δ∂Sxx∂x

+ ∂Sxy∂y

at y = ±η, (16)

∂θ

∂y+ Bi1θ = 0 at y = +η, (17)

∂θ

∂y− Bi2θ = 0 at y = −η, (18)

where η = (1 + � sin 2π (x− t)), � = a/d the geomet-ric parameter, δ = d

λthe wave number, Re = ρcd

µ

the Reynolds number, M = √σ/µB0d the magneticparameter, Br = PrEc the Brinkman number, Pr =µcpk

and Ec = c2T0cp

the Prandtl and Eckert numbers,Bi1 = h1d/k, and Bi2 = h2d/k the Biot numbers,E1 = −τd3/λ3µc, E2 = mcd3/λ3µ, E3 = d4/λ3µ,E4 = Bd3/λ3cµ and E5 = Hd3/λcµ are the non-dimensional elasticity parameters (here asterisks havebeen omitted for simplicity).

In the limit Re → 0, the inertialess flow correspondsto Poiseuille-like longitudinal velocity profile. Thepressure gradient depends upon x and t only in lab-oratory frame. It does not depend on y. Such featurescan be expected because there is no streamline cur-vature to produce transverse pressure gradient whenδ = 0. The assumptions of long wavelength and smallReynolds number give δ = 0 and Re = 0. It should bepointed out that the theory of long wavelength and zeroReynolds number remains applicable for case of chymetransport in small intestine [27]. In this case c = 2cm/min, a = 1.25 cm and λ = 8.01 cm. Here halfwidth of intestine is small in comparison to wavelengthi.e. a/λ = 0.156. It is also declared experimentally byLew et al. [28] that Reynolds number in small intes-tine is small. Further, the situation of intrauterine fluid


flow due to myomaterial contractions is a peristaltictype fluid motion in a cavity. The sagittal cross sectionof the uterus reveals a narrow channel enclosed by twofairly parallel walls [29]. The 1 − 3 mm width of thischannel is very small compared with its 50 mm length[30], defining an opening angle from cervix to fundusof about 0.04 rad. Analysis of dynamics parameters ofthe uterus revealed frequency, wavelength, amplitudeand velocity of the fluid-wall interface during a typ-ical contractile wave were found to be 0.01 − 0.057Hz, 10 − 30 mm, 0.05 − 0.2 mm and 0.5 − 1.9 mm/srespectively. Therefore adopting low Reynolds numberand long wavelength analysis [1], Equations (12–18)reduce to the following forms:

∂p

∂x= ∂Sxy

∂y−M2ψy, (19)

∂p

∂y= 0, (20)

∂2θ

∂y2+ BrSxyψyy = 0, (21)

ψy = 0 at y = ±η, (22)[E1

∂3

∂x3+ E2 ∂

3

∂x∂t2+ E3 ∂

2

∂x∂t+ E4 ∂

5

∂x5+ E5 ∂

∂x

]η

= ∂Sxy∂y

−M2ψy at y = ±η, (23)

∂θ

∂y+ Bi1θ = 0 at y = +η, (24)

∂θ

∂y− Bi2θ = 0 at y = −η. (25)

From Equation (20), note that p /= p(y). Also theextra stress component from Equation (6) is given by

Sxy = αψyy + β6

(ψyy)3,

where α = AµC, β = αc2

C2d2.

The expression for heat transfer coefficient is givenby

Z = ηxθy(η).

3. Solution methodology

As the governing equations are highly nonlinear andexact solution seems to be impossible therefore pertur-bation method for small parameter β is used to find the

analytic solution. Thus we expand the flow quantitiesψ, Sxy, θ and Z as follows:

ψ = ψ0 + βψ1 +O(β2), (26)

Sxy = S0xy + βS1xy +O(β2), (27)

θ = θ0 + βθ1 +O(β2), (28)

Z = Z0 + βZ1 +O(β2). (29)

3.1. Zeroth order system and its solution

Using Equations (26–29) into Equations (19–25)and then collecting the coefficients of like powers ofβ0, we get the zeroth order system as follows:

∂4ψ0

∂y4− M

2

α

∂2ψ0

∂y2= 0, (30)

∂2θ0

∂y2+ αBr

(∂2ψ0

∂y2

)2= 0, (31)

∂ψ0

∂y= 0, at y = ±η, (32)

[E1

∂3

∂x3+ E2 ∂

3

∂x∂t2+ E3 ∂

2

∂x∂t+ E4 ∂

5

∂x5+ E5 ∂

∂x

]η

= ∂∂y

[∂2ψ0

∂y2

]−M2 ∂ψ0

∂yat y = ±η, (33)

∂θ0

∂y+ Bi1θ0 = 0 at y = +η, (34)

∂θ0

∂y− Bi2θ0 = 0 at y = −η, (35)

The solutions of Equations (30–35) are given by

ψ0 = A1y + A2 sin h[My√α

], (36)

θ0 = L1y + L2y2 + L3 cosh[2My√α

] + L4, (37)

Heat transfer coefficient is


Z0 = ηxθ0y (η) ,

= ηxBi1 (1 + ηBi2)M3 (Bi2 + Bi1 (1 + 2ηBi2))(

2A20√B0Brπ

2ε2(2Mη− √α sin h(

2Mη√α

)).

(38)

3.2. First order system and its solution

The coefficients of likes powers of β yield the fol-lowing system of equations:

α∂4ψ1

∂y4− M2 ∂

2ψ1

∂y2+ ∂

2ψ0

∂y2

(∂3ψ0

∂y3

)2

+ ∂4ψ0

∂y4

(∂2ψ0

∂y2

)2= 0, (39)

∂2θ1

∂y2+ Br

(α

(∂2ψ0

∂y2

∂2ψ1

∂y2

)2

+ 16

(∂2ψ0

∂y2

)4)= 0, (40)

∂ψ1

∂y= 0, at y = ±η, (41)

α∂3ψ1

∂y3−M2 ∂ψ1

∂y+ 1

2

∂3ψ0

∂y3

(∂2ψ0

∂y2

)2= 0

at y = ±η, (42)

∂θ1

∂y+ Bi1θ1 = 0 at y = +η, (43)

∂θ1

∂y− Bi2θ1 = 0 at y = −η. (44)

Invoking the zeroth order solution into the first ordersystem and then solving the resulting system we have

ψ1 = A30B0π3ε3[3 (9 + 8B1 − B2)My (45)+ 36B3My cosh(My/

√α)

+ 9(4B5M+ (−10B3 + B4) √α) sinh(My/√α)−B3

√α sinh(My/

√α)],

θ1 = D11y4 + y(D7 sinh(2My/√α) +D0(D10

−D8 sinh(4My/√α) +D9 sinh(6My/

√α)))

+D0(D2 + α(D3 cosh(2My/√α)

−D4 cosh(4My/√α) +D5 cosh(6My/

√α)

−D6 cosh(8My/√α) +D6 cosh(8My/

√α))

(Bi2+Bi1 (1 + 2ηBi2))) + y2 cosh(4My/√α)

× (D12 −D13 cosh(4My/√α)), (46)

Z1 = ηxθ1y (η) ,= ηx(4D11η3 + (2Mη(D7 cosh(2Mη/

√α)

− 2D0D8 cosh(4Mη/√α) + 3D0D9 cosh(6Mη

/√α)))/

√α+D7 sinh(Mη/

√α)

+D0(D10 −D8 sinh(4Mη/

√α)

+D9 sinh(6Mη/√α))

+2D0M√α(D3 sinh(2Mη/

√α)

− 2D4 sinh(4Mη/√α) + 3D5 sinh(6Mη/

√α)

− 4D6 sinh(8Mη/√α)) (Bi2 + Bi1 (1 + 2ηBi2))

+ 2η cosh(4Mη/√α) × (D12 −D13 cosh(4Mη/√α)) + (η2/√α)(4M sinh(4Mη/√α)

× (D12 −D13 cosh(4Mη/√α))

− (1/√α)4D13M sinh(4Mη/√α) cosh(4Mη

/√α) × (D12 −D13 cosh(4Mη/

√α))) (47)

The constants A′is (i = 0 − 2), Bj′s(j = 0 − 3),Dk′s(k = 0 − 13) and L′ms (m = 1, 4) are given inAppendix A.

4. Graphical results and discussion

Figs. (1–7) are presented to observe the behavior ofemerging parameters involved in the solution expres-sions of longitudinal velocity

(u = ψ0y + βψ1y

),

temperature θ, heat transfer coefficient Z and streamfunction ψ.

4.1. Velocity profile

Velocity profile is plotted in Fig. 2 to study theeffects of various values of Hartman number (M), Fluidparameters (α and β) and wall parameters (E1, E2,E3,E4,E5). Fig. (2a) shows that the velocity decreaseswith an increase in the fluid parameterα. While Prandtl


a1.4

8

6

4

2

0

1.2

1.0

0.8

0.6

u(y)

y

u(y)

81.5

1.0

0.5

0.0

6

4

2

0

u(y)

u(y)

0.4

0.2

0.0–1.0 –0.5 0.0 0.5 1.0

y–1.0 –0.5 0.0 0.5 1.0

–1.0 –0.5 0.0 0.5 1.0

y

y

–1.0 –0.5 0.0 0.5 1.0

b

c d

Fig. 2. (a): Variation of α on u when E1 = 1, E2 = 0.5, E3 = 0.5, E4 = 0.01, E5 = 0.3, � = 0.2, x = 0.3, t = 0.1, M = 4 and β = 0.1. (b):Variation of β on u when E1 = 1, E2 = 0.5, E3 = 0.5, E4 = 0.01, E5 = 0.3, � = 0.2, x = 0.3, t = 0.1, M = 2 and α = 1. (c): Variation ofM on u when E1 = 1, E2 = 0.5, E3 = 0.5, E4 = 0.01, E5 = 0.3, � = 0.2, x = 0.3, t = 0.1, β = 0.1 and α = 1. (d): Variation of E1, E2,E3, E4, E5 on u when M = 2, � = 0.2, x = 0.3, t = 0.1, β = 0.1 and α = 1.

fluid parameter β has opposite effect on velocity pro-file i.e. velocity increases with increase in Prandtl fluidparameter β (see Fig. (2b)). Fig. (2c) depicts that thevelocity profile decreases with an increase in the Hart-man numberM, because when the magnetic fieldB0 isapplied in transverse direction, it gives a resistance tothe flow. Fig. (2d) depicts that with an increase in themass per unit area E2 and the coefficient of viscousdamping E3 the velocity increases and it decreaseswhen the elastic tension in the membrane E1 the flex-ural rigidity of the plate E4 and the spring stiffness E5are increased. It is also worth to highlight that whenthe parameters are assigned a fixed value the velocityprofile is parabolic and has maximum magnitude nearthe center of the channel.

4.2. Temperature profile

Figure 3 represents θ(y) for different values of Bi1,Bi2, M, E1, E2, E3, E4, E5, α and β. Fig. (3a) illus-trates that the temperature decreases with an increase in

the fluid parameters α. While temperature has oppositeeffect for the Prandtl fluid parameter β. Temperatureincreases with increase in Prandtl fluid parameter β(see Fig. (3b)). Fig. (3c) shows that by increasing theBi1 the temperature decreases near the upper wall ofchannel but it has no effect near lower wall of channel.Similarly Fig. (3d) reveals that by increasing Bi2 thetemperature profile decreases near lower wall of chan-nel and it has no meaningful effect near upper wallof the channel. Fig. (3e) depicts that θ decreases byincreasing M. The effects of compliant wall param-eters in Fig. (3f) indicates that by increasing E1 andE2, the temperature enhances whereas by increasingvalues of E3, E4 and E5 temperature is decaying.

4.3. Heat transfer coefficient

In Fig. 4 we observed the variation in heat transfercoefficient Z(x) for different parameters Bi1, Bi2, M,E1, E2, E3, E4, E5, α and β appeared in the solution.The nature of the heat transfer coefficient is oscilla-


a b

c d

e f

5025

20

15

10

5

40

30

20

10

14

12

10

30

25

20

15

10

5

0

8

6

4

2

0–1.0 –0.5 0.0

y0.5 1.0 –1.0 –0.5 0.0

y0.5 1.0

–1.0 –0.5 0.0y

0.5 1.0 –1.0 –0.5 0.0y

0.5 1.0

–1.0 –0.5 0.0y

0.5 1.0 –1.0 –0.5 0.0y

0.5 1.0

(y)

θ(y

)θ

14

12

10

8

6

4

2

2.5

2.0

1.5

1.0

0.5

0.0

(y)

θ(y

)θ

(y)

θ(y)

θ

Fig. 3. (a): Variation of α on θ when E1 = 1, E2 = E3 = 0.5, E4 = 0.01, E5 = 0.3, � = 0.2, x = 0.3, t = 0.1, Br = M = 2, β = 0.02and Bi2 = 10. (b): Variation of β on θ when E1 = 1, E2 = E3 = 0.5, E4 = 0.01, E5 = 0.3, � = 0.2, x = 0.3, t = 0.1, α = Br = M = 2and Bi1 = 8. (c): Variation of Bi1 on θ when E1 = 1, E2 = E3 = 0.5, E4 = 0.01, E5 = 0.3, � = 0.2, x = 0.3, t = 0.1, α = 1.5, β = 0.02,Br = M = 2 andBi2 = 10. (d): Variation ofBi2 on θwhenE1 = 1, E2 = E3 = 0.5, E4 = 0.01, E5 = 0.3, � = 0.2, x = 0.3, t = 0.1, α = 1.5,β = 0.02, Br = M = 2 andBi1 = 8. (e): Variation ofM on θ whenE1 = 1, E2 = E3 = 0.5, E4 = 0.01, E5 = 0.3, � = 0.2, x = 0.3, t = 0.1,α = 2, β = 0.02, Br = 2, Bi1 = 8 and Bi2 = 10. (f): Variation of compliant wall parameters on θ when � = 0.2, x = 0.3, t = 0.1, Bi1 = 8,α = 2, β = 0.02 and Bi2 = 10.

tory in nature due to peristalsis motion of the waves.Magnitude of heat transfer coefficientZ(x) has increas-ing behavior for different values of α (see Fig. (4a)).For increasing β, magnitude of heat transfer coeffi-cient Z(x) also increases (see Fig. (4b)). Further forincreasing Bi1 the magnitude of heat transfer coeffi-

cient Z(x) increases and it decreases for Bi2 (see Figs.(4c) and (4d)). Magnitude of heat transfer coefficientZ(x) decreases when Hartman numberM is increased(see Fig. (4e)). Fig. (4f) depicts that magnitude of heattransfer coefficient Z(x) increases with the increase inE1, E2 and E3 and it decreases for E4 and E5.


a b

c d

e f

200300

200

100

0

–100

–200

–300

100

0

–100

z z

z

x

z

–200

50

0

–50

zz

50

0

–50

50

6

4

2

0

–2

–4

–6

0

–50

0.0 0.2 0.4 0.6 0.8 1.0

x0.0 0.2 0.4 0.6 0.8 1.0

x0.0 0.2 0.4 0.6 0.8 1.0

x0.0 0.2 0.4 0.6 0.8 1.0

x0.0 0.2 0.4 0.6 0.8 1.0

x0.0 0.2 0.4 0.6 0.8 1.0

Fig. 4. (a): Variation of α onZwhenM = 5, E1 = 1, E2 = E3 = 0.5, E4 = 0.01, E5 = 0.3, � = 0.2, t = 0.1, β = 0.01, Br = 2, Bi1 = 8 andBi2 = 10. (b): Variation of β onZwhenM = 5, E1 = 1, E2 = E3 = 0.5, E4 = 0.01, E5 = 0.3, � = 0.2, t = 0.1, α = 2, Br = 2, Bi1 = 8 andBi2 = 10. (c): Variation ofBi1 onZwhenM = 5, E1 = 1, E2 = E3 = 0.5, E4 = 0.01, E5 = 0.3, � = 0.2, t = 0.1, β = 0.01, α = 2.5, Br = 2and Bi2 = 15. (d): Variation of Bi2 on Z when M = 5, E1 = 1, E2 = E3 = 0.5, E4 = 0.01, E5 = 0.3, � = 0.2, t = 0.1, β = 0.01, Br = 2and Bi1 = 10. (e): Variation of M on Z when E1 = 1, E2 = E3 = 0.5, E4 = 0.01, E5 = 0.3, � = 0.2, t = 0.1, β = 0.01, α = 2.5,.Br = 2,Bi1 = 8 andBi2 = 10. (f): Variation of parameters of wall properties onZwhenM = 5, � = 0.2, x = 0.3, t = 0.1, β = 0.01, Br = 2, α = 2.5,Bi1 = 8 and Bi2 = 10.

4.4. Trapping

The formulation of an internally circulating bolusof fluid by closed streamlines is called trapping. Weobserved in Fig. 5 (a–c) that the size of trapped bolusincreases with increase in Hartman number M. Fig. 6

(a–c) depicts that the size of trapped bolus increaseswhen we increase the values of fluid parameter α. It isalso observed that the number of streamlines increasestoo. Fig. 7 (a–c) shows that the size of trapped bolusincreases when we increase the values of fluid param-eter β. We analyzed that the size of trapped bolus


a b c

Fig. 5. Variation ofM on ψ for E1 = 0.5, E2 = 0.2, E3 = 0.1, E4 = 0.05, E5 = 0.3, � = 0.2, t = 0.1, α = 1.5, β = 0.02, when (a):M = 2(b): M = 3 (c): M = 4.

a b c

Fig. 6. Variation of α on ψ for E1 = 0.7, E2 = 0.2, E3 = 0.1, E4 = 0.01, E5 = 0.3, � = 0.2, M = 2, t = 0.1, β = 0.02, when (a): α = 0.5,(b): α = 1, (c): α = 1.5.

a b c

Fig. 7. Variation of β on ψ for E1 = 0.7, E2 = 0.2, E3 = 0.1, E4 = 0.01, E5 = 0.3, � = 0.2, M = 2, t = 0.1, α = 2, when (a): β = 0.01,(b): β = 0.03, (c): β = 0.05.


a b c

d e f

Fig. 8. Variation of wall properties on ψ forM = 2, t = 1, β = 0.02, α = 1.5, � = 0.2,when (a):E1 = 0.7, E2 = 0.4, E3 = 0.2, E4 = 0.02,E5 = 0.5, (b): E1 = 1, E2 = 0.4, E3 = 0.2, E4 = 0.02, E5 = 0.5, (c): E1 = 0.7, E2 = 0.6, E3 = 0.2, E4 = 0.02, E5 = 0.5, (d): E1 = 0.7,E2 = 0.4, E3 = 0.5, E4 = 0.02, E5 = 0.5, (e): E1 = 0.2, E2 = 0.4, E3 = 0.2, E4 = 0, E5 = 0.2, (f): E1 = 0.2, E2 = 0.4, E3 = 0.2,E4 = 0.01, E5 = 0.2.

increases with an increase in the mass per unit area E2and the coefficient of viscous damping E3. However itdecreases when the elastic tension in the membraneE1is increased. Also when we decrease the values of theflexural rigidity of the plateE4 and the spring stiffnessE5 the trapped bolus size decreases (see Fig. 8 (a–f)).

5. Concluding remarks

In the conducted study peristaltic transport ofPrandtl fluid in a symmetric channel with convectiveconditions is examined in the presence of complaintwall properties. The main points of the above analysisare as follows:

� Prandtl fluid parameters α and β have oppositeeffects on velocity and temperature profiles andheat transfer coefficient.

� The velocity profile has decreasing behavior forincreasing values of Hartman number M.

� Temperature profile is a decreasing function ofBiot numbers Bi1 and Bi2.

� Heat transfer coefficient Z(x) increases for Bi1and it decreases when Bi2 is increased.

� Size of trapped bolus increases for both fluidparameters α and β.

References

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Appendix A

We describe here the quantities appeared in thesolutions.

A0 =((E5 − 4π2

(E1 + E2 − 4E4π2

))cos [2π (x− t)] + 2E3π sin [2π (x− t)]) ,

A1 = −2A0πε/M2,A2 = (2A0

√απε sech [M +Mε sin [2π (x− t)]])/M3,

B0 = sech(Mη/√α)4,

B1 = cosh(2M (−1 + ε sin [2π (t − x)]) /√α),

B2 = cosh(4M (−1 + ε sin [2π (t − x)]) /√α),

B3 = cosh(Mη/√α),


B4 = cosh(3M (−1 + ε sin [2π (t − x)]) /√α),

B5 = (−1 + ε sin [2π (t − x)]) sinh(Mη/√α),

L1 =A20Brπ

2�2√B0(2Mη− √α sinh(Mη/√α)) (Bi1 − Bi2)

M3 (Bi2 + Bi1 (1 + 2ηBi2)) ,

L2 = 1M2

× A20Brπ2�2 sech(Mη/√α)2,

L3 = −12M4

× A20Brπ2α�2 sech(Mη/√α)2,

L4 =A20Brπ

2�2 sech[Mη/

√α]2

2M4 (Bi2 + Bi1 (1 + 2ηBi2))(2M

√α sinh(Mη/

√α) (2 + η (Bi1 + Bi2))

+α cosh(Mη/√α) (Bi2 + Bi1 (1 + 2ηBi2))− 2M2η (4 + 3ηBi2 + ηBi1 (3 + 2ηBi2))).

Similarly the constants Dk (k = 1 − 13) in Equa-tions (45–47) can be obtained by using boundaryconditions (32-35) and (41-44) through algebraic com-putations.

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