Effect of an inclined magnetic field on peristaltic flow of Williamson fluid in an inclined channel with convective conditions

Author’s Accepted Manuscript Effect of an inclined magnetic field on peristaltic flow of Williamson fluid in an inclined channel with convective conditions T. Hayat, Shahida Bibi, M. Rafiq, A. Alsaedi, F.M. Abbasi www.elsevier.com/locate/jmmm

PII: DOI: Reference:

S0304-8853(15)30742-3 http://dx.doi.org/10.1016/j.jmmm.2015.10.107 MAGMA60800

To appear in: Journal of Magnetism and Magnetic Materials Received date: 13 June 2015 Revised date: 11 October 2015 Accepted date: 28 October 2015 Cite this article as: T. Hayat, Shahida Bibi, M. Rafiq, A. Alsaedi and F.M. Abbasi, Effect of an inclined magnetic field on peristaltic flow of Williamson fluid in an inclined channel with convective conditions, Journal of Magnetism and Magnetic Materials, http://dx.doi.org/10.1016/j.jmmm.2015.10.107 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Effect of an inclined magnetic field on peristaltic flow of Williamson fluid in an inclined channel with convective conditions T. Hayata, b,1 , Shahida Bibia , M. Rafiqa , A.Alsaedib and F. M. Abbasic a

Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan. b

Nonlinear Analysis and Applied Mathematics (NAAM) Research group,Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia. c

Department of Mathematics, CIIT, Islamabad, Pakistan.

Abstract: This paper deals with the influence of inclined magnetic field on peristaltic flow of an incompressible Williamson fluid in an inclined channel with heat and mass transfer. Convective conditions of heat and mass transfer are employed. Viscous dissipation and Joule heating are taken into consideration. Mathematical modeling also includes Soret and Dufour effects. Channel walls have compliant properties. Analysis has been carried out through long wavelength and low Reynolds number approach. Resulting problems are solved for small Weissenberg number. Impacts of variables reflecting the salient features of wall properties, Biot numbers and Soret and Dufour on the velocity, temperature, concentration and heat transfer coefficient are pointed out. Trapping phenomenon is also analyzed. Keywords: Williamson fluid, compliant walls, convective conditions, Soret and Dufour effects , Joule heating.

1

Introduction

Peristalsis is the form of fluid transport due to wave travelling along the walls of an inextensible tube/channel. This type of rhythmic contraction provides the basis of peristaltic pumps to transport fluid through different tubular parts without any direct contact. It has main advantage in the biological/medical applications where the transported matter needs not to be in direct contact with another part except the inner surface of tube. Such mechanism is quite common in the transport of urine from kidney to bladder, chyme movement in the gastrointestinal tract, vasomotion of small blood vessels, swallowing of food through 1

Corresponding author. Tel.: + 92 51 90642172.

e-mail address: [email protected]

1

esophagus, locomotion of worms, roller and finger pumps, blood pump in heart-lung machine etc. Peristalsis is also useful in noxious and corrosive fluid transport (see studies [1 − 7]). Further the peristalsis subject to magnetic field effects are significant in magnetotherapy, hyperthermia, arterial flow, cancer therapy etc. The controlled application of low intensity and frequency pulsating fields modify the cell and tissue. Magnetic susceptible of chyme is also satisfied from the heat generated by magnetic field or the ions contained in the chyme. The magnets could heat inflammations, ulceration and several diseases of bowel (intestine) and uterus. Also biomechanical engineer has proved now that rheological properties are important in the industrial and physiological processes. The non-Newtonian fluids deviate from the classical Newtonian linear relationship between the shear stress and shear rate. Due to complex rheological properties it is difficult to suggest a single model which exhibits all properties of non-Newtonian fluids. Various models of non-Newtonian fluids are proposed. Abd-Alla et al. [8] discussed the effect of radial magnetic field on peristaltic transport of Jeffrey fluid in cylindrical geometry. Abd-Alla et al. [9] also discussed the influence of rotation and initial stresses on peristaltic flow of fourth grade fluid in an asymmetric channel. The peristaltic flow of nanofluid through a porous medium with mixed convection is analyzed by Nowar [10]. Kothandapani and Prakash [11] studied the combined effects of radiation and magnetic field on peristaltic transport of nanofluid. The studies related to peristalsis with heat and mass transfer for Newtonian fluid Have been structured in refs. [12, 13] . Shaban and Abou-Zeid [14] investigated the peristalsis between two coaxial cylinders when inner tube is rigid and outer flexible. Heat transfer is also considered in the study. Kothandapani and Prakash [15] modeled the peristaltic flow of hyperbolic tangent nanofluid in an asymmetric channel with inclined magnetic field. Influence of convective conditions on peristaltic transport of non-Newtonian fluids under different flow situations are discussed in the refs. [16, 17] . Ellahi et. al. [18] studied the peristalsis in non-uniform rectangular duct with heat and mass transfer effects. Hayat et. al. [19] discussed the peristaltic flow in a vertical channel filled with nanofluid by considering Soret and Dufour effects. Peristaltic transport of fourth grade fluid with convective conditions is numerically discussed by Mustafa et. al. [20] . Effects of variable viscosity on peristalsis in the presence of magnetic field is studied by Abbasi et. al. [21] . Kothandapani and Prakash [22] analyzed the peristaltic transport of nanofluid with thermal radiation and magnetic field in a tapered channel. Mehmood et al. [23] considered the partial slip effects on peristaltic transport in a channel with heat and 2

mass transfer effects. Mathematical analysis has been carried out in the presence of magnetic field. In all of the above mentioned studies lubrication approachhas been utilized to simplify the problems. It is seen that heat transfer in peristalsis has pivoted role. Specifically such effect is useful in the cancer therapy, hyperthermia, oxygenation and hemodialysis. It is noted that simultaneous effects of heat and mass transfer are important in chemical industry for example in reservoir engineering about thermal recovery process, catalytic reactors, hot springs analysis in the sea and medicine diffusion in blood veins. Such simultaneous features also lead to the Soret and Dufour effects. On the other hand heat and mass transfer effects mostly in the past have been discussed either through prescribed surface temperature/concentration or prescribed heat/mass flux. Convective conditions for heat and mass transfer are even scarcely attended in the peristaltic flows of viscous fluids when Soret and Dufour effects are present. Also both magnetic field and channel are not taken inclined [24 − 27]. Having all such aspects in mind the purpose of present communication is to discuss the inclined magnetic field effects in peristaltic flow of Williamson fluid in an inclined channel. The channel walls have compliant properties. The Williamson fluid shows the properties of viscoelastic shear thinning fluids. It exhibits the behavior of pseudoplastic materials whose apparent viscosity or consistency decreases instantaneously with an increase in shear rate [28]. It is also worthmentioning that the transfer of energy/mass for flowing fluids does not only take place through simple conduction/diffusion, instead, bulk motion of the fluid also plays an important role in the transfer phenomenon. Bulk motion of the fluid can produce considerable alteration in the results of transfer phenomenon. Stream function formulation is adopted. Resulting problems are computed after invoking long wavelength and low Reynolds number. Graphical results for different parameters of interest are displayed and examined. Important observations have been given concluding remarks. The novelty of present attempt is specifically based upon considerations of Williamson fluid, convective heat and mass conditions and inclined nature of both channel and applied magnetic field.

2

Basic equations

The continuity, momentum, energy and concentration equations in the presence of viscous dissipation and Joule heating are given by div V = 0, 3

(1)

dV = div τ  + ρf , dt dT DKT 2 1 = k∇2 T + Ω + ∇ C + (J.J), ρCp dt Cs σ dC DK T = D∇2 C + ∇2 T, dt Tm ρ

(2) (3) (4)

where V is the velocity vector, τ  the Cauchy stress tensor, ρ the body force, d/dt the material time derivative, Cp the specific heat at constant volume, J the current density, T the temperature of fluid, D the coefficient of mass diffusivity, Ω the viscous dissipation, k the thermal diffusion ratio, Cs the concentration susceptibility, C the concentration of fluid and Tm the mean temperature. Note that the equations (3) and (4) include the Soret and Dufour effects. The constitutive equations for Williamson fluid are τ  = −pI + S,

(5)

˙ −1 ]A1 , S = [μ∞ + (μ0 + μ∞ )(1 − Γγ)

(6)

where p is the pressure, S the extra stress tensor, μ∞ the infinite shear rate viscosity, μ0 the zero shear rate viscosity, Γ the time constant and γ˙ is given by  1 Π, γ˙ = 2

(7)

in which Π = tr(A21 ) where A1 = grad V + (grad V)∗ . In the present study we have chosen μ∞ = 0 and Γγ˙ < 1. Therefore extra stress tensor can be written as follows: S = μ0 [(1 + Γγ)]A ˙ 1.

(8)

Here Eq. (8) corresponds to the viscous fluid when Γ = 0.

3

Mathematical formulation

Consider an incompressible magnetohydrodynamic (MHD) flow of Williamson fluid in a symmetric channel of width 2d1 . Both the magnetic field and channel are inclined at angles Θ and α. Here x − axis is taken along the length of channel and y − axis transverse to it (see F ig.1). A uniform magnetic field B = (Bo sin Θ, Bo cos Θ, 0) is applied. The induced magnetic field is neglected by assuming a very small magnetic Reynolds number. Also the electric field is taken absent. Heat and mass transfer is examined through convective 4

conditions. The flow is generated by sinusoidal waves propagating along the compliant walls of channel:



 2π y = ±η(x, t) = ± d1 + asin (x − ct) , λ

(9)

where a is the wave amplitude, λ the wavelength, c the wave speed and t the time.

Fig. 1: Geometry of the problem The governing equations for present flow can be expressed as follows: ∂u ∂v + = 0, ∂x ∂y

(10)

     ∂p ∂ ∂ ∂u ∂u ∂v du = − + 2μ0 (1 + Γγ) + μ (1 + Γγ) + ˙ ˙ ρ dt ∂x ∂x ∂x ∂y 0 ∂y ∂x +σB02 cos Θ(u cos Θ − v sin Θ) + ρg sin α,

(11)

     ∂p ∂ ∂ dv ∂u ∂v ∂v = − + μ (1 + Γγ) + + 2μ0 (1 + Γγ) ρ ˙ ˙ dt ∂y ∂x 0 ∂y ∂x ∂y ∂y 2 +σB0 sin Θ(u cos Θ − v sin Θ) − ρg cos α,

(12)

dT ρCp dt

  2  2  2  ∂u ∂v ∂ 2T ∂ 2T ∂u ∂v + μ0 (1 + Γγ) = k + + ˙ +2 +2 2 2 ∂x ∂y ∂y ∂x ∂x ∂y  2  DKT ∂ C ∂ 2 C + + σB02 (u2 cos2 Θ + v 2 sin2 Θ − 2uv sin Θ cos Θ), (13) + Cs ∂x2 ∂y 2  2    ∂ C ∂ 2C DKT ∂ 2 T ∂ 2T dC =D + , (14) + + dt ∂x2 ∂y 2 Tm ∂x2 ∂y 2 

in which (u, v) are the components of V in the x and y directions respectively, g the gravity effects and σ the electrical conductivity. The corresponding boundary conditions are given by u = 0,

at y = ±η, 5

(15)



 ∂5 ∂ ∂3 ∂3 ∂2 ∂ ∂u ˜ +B 5 +H η= 2μ0 (1 + Γγ) −τ 3 + m +d ˙ 2 ∂x ∂x∂t ∂x∂t ∂x ∂x ∂x ∂x    ∂u ∂v ∂ μ0 (1 + Γγ) + − σB02 cos Θ(u cos Θ − v sin Θ) ˙ + ∂y ∂y ∂x du +ρg sin α − ρ at y = ±η, dt ∂T ∂y ∂T k ∂y

= −h1 (T − T0 )

at

y = η,

= −h1 (T0 − T )

at

y = −η,

∂C ∂y ∂T D ∂y

= −h2 (C − C0 )

at

y = η,

= −h2 (C0 − C)

at

y = −η,

k

D

(16)

(17)

(18)

where η = 1 +  sin[2π(x − t)]. In above equations τ is the elastic tension in the membrane, m the mass per unit area, d˜ the coefficient of viscous damping, B the flexural rigidity of the plate, H the spring stiffness, h1 the heat transfer coefficient. Analog to the convective conditions at the wall, we have applied the mixed condition for mass transfer as well. Therefore h2 indicates the mass transfer coefficient in the similar way as the heat transfer (it is a parameter that is used to describe the ratio between actual mass flux of a specie into or out of the flowing fluid and the driving force that causes that flux). T0 and C0 represent the temperature and concentration at the upper and lower walls respectively. Defining velocity components u and v in terms of stream function and dimensionless variables by ∂ψ ∂ψ c ψ x y ct d2 p η , v = − , ψ∗ = , x∗ = , y ∗ = , t ∗ = , p∗ = 1 , W e = Γ , η ∗ = , ∂y ∂x cd1 λ d1 λ cμλ d1 d1 d1 ∗ h1 ∗ h2 T − T0 C−C0 = γ˙ , h1 = , h2 = , θ = ,φ= , c d1 d1 T0 C0

u = γ˙ ∗

the Eq. (10) is identically satisfied and Eqs. (11 − 18) in terms of stream function ψ can be

6

presented into the following forms:       ∂ψ ∂ ∂ψ ∂p ∂ ∂ 2ψ ∂ψ ∂ − = − + 2δ (1 + W eγ) ˙ Re δ ∂y ∂x ∂x ∂y ∂y ∂x ∂x ∂x∂y    2 2 ∂ ψ ∂ 2∂ ψ (1 + W eγ) ˙ −δ + ∂y ∂y 2 ∂x2   ∂ψ ∂ψ 2 cos Θ + δ sin Θ −M cos Θ ∂y ∂x Re sin α + , Fr

Re δ

3



dθ = δ Re Pr dt

∂ψ ∂ ∂ψ ∂ − ∂y ∂x ∂x ∂y



∂ψ ∂x



(20)

   2 2  2 2  2 2 ∂ θ ψ ψ ∂ ψ ∂ ∂ 2θ ∂ ˙ 4δ 2 + δ 2 2 + Br (1 + W eγ) + − δ2 2 ∂y 2 ∂x ∂x∂y ∂y 2 ∂x   2 2 ∂ φ 2∂ φ + δ + Pr Du ∂y 2 ∂x2

 2  2    ∂ψ ∂ψ ∂ψ ∂ψ +BrM 2 cos2 Θ sin Θ cos Θ (21), − δ 2 sin2 Θ −δ ∂y ∂x ∂y ∂x 

1 dφ = δ Re dt Sc



2 ∂ 2φ 2∂ φ + δ ∂y 2 ∂x2

∂ψ =0 ∂y 

  ∂p ∂ 2ψ 2 ∂ = − − 2δ (1 + W eγ) ˙ ∂y ∂y ∂x∂y   2  2 ∂ ψ 2 ∂ 2∂ ψ (1 + W eγ) ˙ −δ +δ ∂x ∂y 2 ∂x2   ∂ψ ∂ψ 2 cos Θ + δ sin Θ +M δ sin Θ ∂y ∂x Re cos α −δ , Fr

(19)

at



 + Sr

2 ∂ 2θ 2∂ θ + δ ∂y 2 ∂x2

 ,

y = ±η ,

   ∂3 ∂3 ∂2 ∂5 ∂ ∂ 2ψ 2 ∂ E1 3 + E2 + E4 5 + E5 η = 2δ (1 + W eγ) ˙ + E3 ∂x ∂x∂t2 ∂x∂t ∂x ∂x ∂x ∂x∂y    2   2 ∂ ψ ∂ψ ∂ψ ∂ 2∂ ψ 2 − M cos Θ (1 + W eγ) ˙ cos Θ + δ sin Θ −δ + ∂y ∂y 2 ∂x2 ∂y ∂x ∂ψ Re sin α − δ Re at y = ±η, + Fr ∂y ∂θ + B i1 θ = 0 ∂y ∂θ − B i1 θ = 0 ∂y 7

(22)

at

y = +η,

at

y = −η,

(23)

(24)

∂φ + χi1 φ = 0 at y = +η, ∂y ∂φ − χi1 φ = 0 at y = −η, ∂y   2 2  2 2 1/2 2 ψ ψ ∂ ψ ∂ ∂ γ˙ = 4δ 2 + − δ2 2 , ∂x∂y ∂y 2 ∂x

(25)

(26)

η = (1 +  sin 2π (x − t)) . 1 In the above Eqs.  = a/d1 is the geometric parameter, δ = dλ1 the wave number, Re = ρcd μ  c2 the Reynolds number, F r = gd the Froude number, M = σ/μB0 d1 the Hartman number, 1

Br = Pr Ec the Brinkman number, Pr = respectively, Sc =

μ ρD

μcp k

and Ec =

the Schmidt number, Sr =

c2 T 0 cp

ρT0 DKT μTm C0

the Prandtl and Eckert numbers

the Soret number, Du =

Dufour number, Bi1 = h1 d1 /k the heat transfer Biot number, χi1 =

h2 d 1 D

D m kT νCs Cp

the

the mass transfer

˜ 3 μ, E4 = Bd3 /λ3 cμ and Biot number and E1 = −τ d31 /λ3 μc, E2 = mcd31 /λ3 μ, E3 = d31 d/λ 1 E5 = Hd31 /λcμ are the non-dimensional elasticity parameters. Here asterisks have been omitted for simplicity. The reduced forms of Eqs.(19 − 26) after invoking long wavelength and low Reynolds number are

  ∂2 ∂ 2ψ ∂ 2ψ 2 2 (1 + W e γ) ˙ − M cos Θ = 0, ∂y 2 ∂y 2 ∂y 2  2 2 ∂ 2θ ∂ ψ ∂ 2φ ∂ψ 2 2 2 ) = 0, + Br (1 + W e γ) ˙ + Pr Du + BrM cos Θ( ∂y 2 ∂y 2 ∂y 2 ∂y 1 ∂ 2φ ∂ 2θ + Sr = 0, Sc ∂y 2 ∂y 2 ∂ψ =0 ∂y



at

y = ±η,

(27) (28) (29) (30)

   ∂3 ∂3 ∂2 ∂5 ∂ ∂ ∂ 2ψ ∂ψ + E4 5 + E5 η= (1 + W eγ) ˙ E1 3 + E2 − M 2 cos2 Θ + E3 2 2 ∂x ∂x∂t ∂x∂t ∂x ∂x ∂y ∂y ∂y Re sin α + at y = ±η,(31) Fr ∂θ + B i1 θ = 0 ∂y ∂θ − B i1 θ = 0 ∂y ∂φ + χi 1 φ = 0 ∂y ∂φ − χi1 φ = 0 ∂y 8

at

y = +η,

at

y = −η,

at

y = +η,

at

y = −η,

(32)

(33)

with γ˙ =

4

∂ 2ψ . ∂y 2

(34)

Solution procedure

It seems difficult to solve the Eqs. (27 − 33) in closed form. Thus we aim to find the series solutions for small Weissenberg number and write

ψ = ψ 0 + W eψ 1 + O W e2 ,

θ = θ0 + W eθ1 + O W e2 ,

φ = φ0 + W eφ1 + O W e2 ,

(36)

Z = Z0 + W eZ1 + O(W e2 ).

(38)

(35)

(37)

Here we have considered the zeroth and first order systems only due to the fact that the perturbation parameter is taken small (W e  1) so that the contribution of higher order terms are negligible due to order analysis. Such contribution for higher order systems is very small. The zeroth order term gives major contribution. First order terms have less value when compared with the zeroth order term. It is noted that subsequent terms are smaller. This fact ensures the convergence. In order to avoid length we only consider the solutions upto first order. This fact is also in fact followed from the previous studies (see refs. [6, 9, 16 and 21]).

4.1

Zeroth-order system and solution

At this order the systems through Eqs. (27 − 38) are ∂ 4ψ0 ∂ 2ψ0 2 2 − M cos Θ = 0, ∂y 4 ∂y 2 2  2 2  ∂ 2 θ0 ∂ ψ0 ∂ 2 φ0 ∂ψ 0 2 2 + Br + Pr Du 2 + BrM cos Θ = 0, ∂y 2 ∂y 2 ∂y ∂y 1 ∂ 2 φ0 ∂ 2 θ0 + Sr = 0, Sc ∂y 2 ∂y 2 ∂ψ 0 = 0, at y = ±η, ∂y     ∂3 ∂ ∂ 2ψ0 ∂ψ ∂3 ∂2 ∂5 ∂ + E4 5 + E5 η= − M 2 cos2 Θ 0 + E3 E1 3 + E2 2 2 ∂x ∂x∂t ∂x∂t ∂x ∂x ∂y ∂y ∂y Re sin α + at y = ±η, Fr 9

(39) (40) (41)

(42)

∂θ0 ∂y ∂θ0 ∂y ∂φ0 ∂y ∂φ0 ∂y

+ B i1 θ 0 = 0

at

y = +η,

− B i1 θ 0 = 0

at

y = −η,

+ χi1 φ 0 = 0

at

y = +η,

− χi1 φ0 = 0

at

y = −η.

(43)

(44)

The solutions of Eqs. (39 − 42) are given by ψ 0 = A1 y + A2 sinh[M y cos[Θ]],

(45)

θ0 = D1 + D2 y + D3 y 2 + D4 cosh[2M y cos[Θ]] + D5 cosh[M y cos[Θ]],

(46)

φ0 = C5 + C1 y + C2 y 2 + C3 cosh[M y cos[Θ]] + C4 cosh[2M y cos[Θ]].

(47)

The heat transfer coefficient can be expressed in the form Z0 = η x θ0y (η) = 2π cos[2(−t + x)](D2 + 2D3 η + 2M cos[Θ]D4 sinh[2M η cos[Θ]] +M cos[Θ]D5 sinh[M η cos[Θ]]).

4.2

(48) (49)

First order system and solution

The coefficients of O(W e) yield the following problems: ∂ 4ψ1 ∂ 2ψ1 ∂ 3ψ0 ∂ 2ψ0 ∂ 4ψ0 2 2 − M cos Θ + 2 + 2 = 0, (50) ∂y 4 ∂y 2 ∂y 3 ∂y 2 ∂y 4    2 2

 ∂ 2ψ1 ∂ 2 φ1 ∂ψ 1 ∂ 2ψ0 ∂ ψ0 ∂ψ 0 ∂ 2 θ1 2 2 + Pr Du 2 + 2BrM cos Θ = 0, + Br 2 2 2 + ∂y 2 ∂y ∂y ∂y 2 ∂y ∂y ∂y (51) 2 2 1 ∂ φ1 ∂ θ1 + Sr 2 = 0, (52) 2 Sc ∂y ∂y ∂ψ 1 = 0 ∂y ∂ 3ψ1 ∂ψ 1 ∂ 2ψ0 ∂ 3ψ0 2 2 − M cos Θ = 0 + 2 ∂y 3 ∂y ∂y 2 ∂y 3

10

at at

y = ±η, y = ±η,

(53) (54)

∂θ1 ∂y ∂θ1 ∂y ∂φ1 ∂y ∂φ1 ∂y

+ B i1 θ 1 = 0

at

y = +η,

− B i1 θ 1 = 0

at

y = −η,

+ χi1 φ 0 = 0

at

y = +η,

− χi1 φ1 = 0

at

y = −η.

(55)

(56)

Upon making use of zeroth-order solution into the first-order system and then solving the resulting problems we obtain ψ 1 = A3 (cosh M y cos[Θ]]2 − 4 cosh[M y cos[Θ]] cosh[M η cos[Θ]] + sinh[M y cos[Θ]]2 ),

(57)

θ1 = D6 + D7 y + D8 sinh[M y cos[Θ]] + D9 sinh[2M y cos[Θ]] + D10 sinh[3M y cos[Θ]], (58) φ1 = C10 + C6 y + C7 sinh[M y cos[Θ]] + C8 sinh[2M y cos[Θ]] + C9 sinh[3M y cos[Θ]]

(59)

The heat transfer coefficient is given by Z1 = η x θ1y (η) , = 2π cos[2(−t + x)](D7 + M cos[Θ]D8 cosh[M η cos[Θ]] + 2M cos[Θ]D9 cosh[2M η cos[Θ]] +3M cos[Θ]D10 cosh[3M η cos[Θ]]).

(60)

Here the constants A1 → A3 , D1 → D10 and C1 → C10 can be evaluated using MATHEMATICA.

5

Graphical results and discussion

In this section , we have presented a set of Figures to observe the behavior of sundry param eters involved in the expressions of longitudinal velocity u = ψ 0y + W eψ 1y , temperature θ, concentration φ, heat transfer coefficient Z and stream function ψ. Figures (1 − 6) display the effects of various physical parameters on the velocity profile u(y). Fig.1(a) depicts that the velocity increases when E1 and E2 enhanced. It is due to the fact that less resistance is offered to the flow because of the wall elastance and thus velocity increases. However reverse effect is observed for E3 , E4 and E5 . As E3 represents the damping which is resistive force so velocity decreases when E3 is increased. Similar behavior is observed for the velocity in case of rigidity and stiffness due to presence of damping force. Fig.1(b) shows that the velocity decreases by increasing Hartman number M . It is due to 11

the fact that magnetic field applied in the transverse direction shows damping effect on the flow. Fig.1(c) illustrates that velocity profile decreases for larger W e in the region [−1, 0] whereas it has opposite behavior in the region [0, 1]. It is observed from Fig.1(d) that for subcritical flow (F r < 1) the velocity profile u(y) has greater effect than that for critical (F r = 1) and supercritical (F r > 1) flow cases. The velocity profile u(y) decreases with an increase of Froude number F r. Fig .1(e) indicates that the velocity profile u(y) is increasing function of the inclination angle Θ of magnetic field. Clearly the fluid velocity enhances for an increase in angle of inclination α of the channel (see Fig.1(f )). It is in view of the reason fact that gravity effects become more prominent by increasing angle of inclination. Effects of pertinent parameters on temperature profile can be visualized through Fig.2. The variation of compliant wall parameters is studied in Fig.2(a). As temperature is the average kinetic energy of the particles and kinetic energy depends upon the velocity. Therefore increase in velocity by E1 and E2 leads to temperature enhancement. Similarly decrease in velocity by E3 , E4 and E5 shows decay in temperature. Fig.2(b) reveals that the temperature profile θ decreases when Hartman number M is increased. Fig.2(c) shows that when angle of inclination α of channel increases then temperature profile increases. Fig.2(d) shows that the temperature decreases for increasing Froude number F r. Fig. 2(e) depicts that the temperature profile increases when the inclination angle Θ of magnetic field is increased. Effect of Br on temperature can be observed through Fig.2(f ). The Brinkman number Br is the product of the Prandtl number P r and the Eckert number Ec. Here Eckert number occurs due to the viscous dissipation effects and the temperature enhances. Fig.2(g) depicts that the temperature increases when Du is increased. Also the temperature increases for larger Sc and Sr.(see F igs.2(h) and (i)). In fact for increasing Sr and Du the thermal diffusion is more. It results for rise in temperature. Whereas for increasing Sc the viscous effects become more prominent which results in the increase of temperature. Fig.2(j) reveals that temperature distribution decreases by increasing W e in the lower half of channel whereas it has opposite behavior in the upper half of channel. Fig.2(k) discloses that by increasing the Bi1 the temperature decreases. Here we have considered the values of Biot numbers much larger than 0.1 due to non-uniform temperature fields within the fluid. However problems involving small Biot numbers are thermally simple due to uniform temperature field within the fluid. As temperature is related to average kinetic energy of the particles therefore velocity and temperature shows similar behavior for the involved parameters. 12

Variations of different parameters on concentration field are displayed in Fig.3. Fig.3(a) illustrates that the concentration decreases by increasing E1 and E2 . However it increases through the increase in E3 , E4 and E5 . Fig.3(b) portrays the decreasing behavior for φ(y) when Dufour number Du increases. Fig.3(c) indicates decrease of concentration via increase in angle of inclination of channel α. Figs.(3(d) and 3(e)) elucidate that the concentration profile φ(y) decreases for larger Sr and Sc. The concentration profile increases in lower half of channel and it decreases in upper half of channel when values of Weissenberg number W e are increased (see F ig. 3(f )). Fig.2(g) shows that when we increase F r then concentration profile decreases. Fig.3(h) ensures that by increasing the χi1 the concentration increases. Fig. 2(i) depicts that the concentration field decreases when the inclination angle Θ of magnetic field is increased. From Fig.3(j) it is noted that the concentration field decreases through increasing the values of Br. In Fig.4 the variation of heat transfer coefficient Z(x) for various values of emerging parameters is analyzed. The heat transfer coefficient is denoted by Z(x) = η x θy (η) which actually defines the rate of heat transfer or heat flux at the upper wall. It is found that the nature of heat transfer coefficient is oscillatory. This is expected due to propagation of sinusoidal waves along the channel walls. Fig.4(a) potrays that magnitude of heat transfer coefficient Z(x) enhances for larger E1 , E2 and E3 and it decreases for E4 and E5 . Fig.4(b) shows that Z(x) increases when Brinkman number Br is increased. Here Br = 0 shows the absence of viscous dissipation effects. It is also noteworthy that heat transfer coefficient Z(x) is higher when we consider the viscous dissipation effects (i.e. when Br = 0). Further Z(x) decreases for increase in Bi1 (see F ig.4(c). F ig.4(d) indicates that Z(x) enhances for increase in P r. Magnitude of heat transfer coefficient Z(x) decreases when Hartman number M is increased (see F ig.4(e)). Fig.4(f ) displayed that when angle of inclination of the channel α increases then Z(x) decreases. Fig.4(g) shows that increase in the angle of inclination of magnetic field Θ yields enhancement in Z(x). When we increase the values of Soret Sr and Schmidt Sc numbers then the magnitude of heat transfer coefficient Z(x) increases (see F igs. 4(h) and (i)). Fig. 4(j) studies the effect of Du on Z(x). It reveals that Z(x) increases via larger Dufour number Du. Fig. 4(k) shows that magnitude of heat transfer coefficient Z(x) is increased when F r increases. Fig. 4(l) shows that the magnitude of heat transfer coefficient Z(x) decreases via increase in Weissenberg number W e. Also heat transfer coefficient Z(x) is higher for Newtonian material (W e = 0) when compared 13

with Williamson fluid (W e = 0). The formation of an internally circulating bolus of fluid by closed stream lines is shown in F igs.5 − 8. Figs. 5(a, b) depict that the size of trapped bolus increases when there is an increase in the Hartman number M . Figs. 6(a, b) shows the streamlines pattern for different values of the inclination angle α. We observed that the size of trapped bolus enhances by increasing α. We have analyzed from Figs.7(a − e) that the size of trapped bolus increases when E1 and E2 are enhanced. However it decreases when E3 is increased. Also when we decrease the values of E4 and E5 then the trapped bolus size increases. Figs.8(a, b) depict the streamlines pattern for different values of Weissenberg number W e. It is noticed that the size of trapped bolus decreases by increasing W e.

Fig. 2(a)

Fig. 2(b)

F ig. 2(a) Variation of parameters of wall properties on u(y) when  = 0.2, x = 0.3, t = 0.1, W e = 0.01, M = 2, Re = 0.2, F r = 0.8, α = Θ = π/3. (b) Variation of M on u(y) when E1 = 1, E2 = E3 = 0.3, E4 = E5 = 0.01,  = 0.2, x = 0.3, t = 0.1, W e = 0.01, Re = 0.2, F r = 0.8, α = Θ = π/3.

14

Fig. 2(c)

Fig. 2(d)

(c) Variation of W e on u(y) when E1 = 1, E2 = E3 = 0.3, E4 = E5 = 0.01,  = 0.2, x = 0.3, t = 0.1, M = 3, Re = 0.2, F r = 0.8, α = π/3, Θ = π/8. (d) Variation of F r on u(y) when E1 = 1, E2 = E3 = 0.3, E4 = E5 = 0.01,  = 0.2, x = 0.3, t = 0.1, W e = 0.03, Re = 0.2, M = 2, α = π/3, Θ = π/8.

Fig. 2(e)

Fig. 2(f)

(e) Variation of Θ on u(y) when E1 = 1, E2 = E3 = 0.3, E4 = E5 = 0.01,  = 0.2, x = 0.3, t = 0.1, W e = 0.01, Re = 0.2, F r = 0.8, α = π/3, M = 2. (f ) Variation of α on u(y) when E1 = 1, E2 = E3 = 0.3, E4 = E5 = 0.01,  = 0.2, x = 0.3, t = 0.1, W e = 0.01, Re = 0.2, F r = 0.8, M = 2, Θ = π/8.

15

Fig. 3(a)

Fig. 3(b)

F ig. 3 (a) Variation of compliant wall parameters on θ when  = 0.2, x = 0.3, t = 0.1, W e = 0.03, P r = Br = M = 2, F r = 1.2, Du = Sc = Sr = 0.5, Re = 0.2, α = Θ = π/3, Bi1 = 5. (b) Variation of M on θ when E1 = 1, E2 = E3 = 0.5, E4 = E5 = 0.01,  = 0.2, x = 0.3, t = 0.1, W e = 0.03, Re = 0.2, P r = Br = 2, Du = Sc = Sr = 0.5, F r = 1.2, α = Θ = π/3, Bi1 = 5.

Fig. 3(c)

Fig. 3(d)

(c) Variation of α on θ when E1 = 1, E2 = E3 = 0.3, E4 = E5 = 0.01,  = 0.2, x = 0.3, t = 0.1, W e = 0.03, P r = Br = 2, Du = Sc = Sr = 0.5, F r = 0.8, Re = 0.2, M = 3, Θ = π/3, Bi1 = 5. (d) Variation of F r on θ when E1 = 1, E2 = E3 = 0.3, E4 = E5 = 0.01,  = 0.2, x = 0.3, t = 0.1, W e = 0.01, P r = Br = 2, Du = Sc = Sr = 0.5, α = π/4, Re = 0.8, M = 3, Θ = π/3, Bi1 = 8. 16

Fig. 3(e)

Fig. 3(f)

(e) Variation of Θ on θ when E1 = 1, E2 = E3 = 0.3, E4 = E5 = 0.01,  = 0.2, x = 0.3, t = 0.1, W e = 0.01, P r = Br = 2, Du = Sc = Sr = 0.5, α = π/4, Re = 0.8, M = 3, F r = 1.2, Bi1 = 8. (f ) Variation of Br on θ when E1 = 1, E2 = E3 = 0.3, E4 = E5 = 0.01,  = 0.2, x = 0.3, t = 0.1, Re = 0.8, W e = 0.01, P r = 2, Du = Sc = Sr = 0.5, α = π/4, M = 3, Θ = π/3, F r = 1.2, Bi1 = 8.

Fig. 3(g)

Fig. 3(h)

(g) Variation of Du on θ when E1 = 1, E2 = E3 = 0.3, E4 = E5 = 0.01,  = 0.2, x = 0.3, t = 0.1, W e = 0.01, P r = Br = 2, Sc = Sr = 0.5, α = π/4, Re = 0.8, M = 3, Θ = π/3, F r = 1.2, Bi1 = 8. (h) Variation of Sc on θ when E1 = 1, E2 = E3 = 0.3, E4 = E5 = 0.01,  = 0.2, x = 0.3, t = 0.1, W e = 0.01, P r = Br = 2, Du = Sr = 0.5, α = π/4, Re = 0.8, M = 3, Θ = π/3, F r = 1.2, Bi1 = 8. 17

Fig. 3(i)

Fig. 3(j)

(i) Variation of Sr on θ when E1 = 1, E2 = E3 = 0.3, E4 = E5 = 0.01,  = 0.2, x = 0.3, t = 0.1, W e = 0.01, P r = Br = 2, Sc = Du = 0.5, α = π/4, Re = 0.8, M = 3, Θ = π/3, F r = 1.2, Bi1 = 8. (j) Variation of W e on θ when E1 = 1, E2 = E3 = 0.3, E4 = E5 = 0.01,  = 0.2, x = 0.3, t = .1, Br = Pr = 2, Sc = Du = Sr = 0.5, α = π/4, Re = 0.8, M = 3, Θ = π/3, F r = 1.2, Bi1 = 8.

Fig. 3(k) (k) Variation of Bi1 on θ when E1 = 1, E2 = E3 = 0.3, E4 = E5 = 0.01,  = 0.2, x = 0.3, t = 0.1, W e = 0.01, P r = Br = 2, Sc = Du = Sr = 0.5, α = π/4, Re = 0.8, M = 3, Θ = π/3, F r = 1.2.

18

Fig. 4(a)

Fig. 4(b)

F ig. 4(a) Variation of compliant wall parameters on φ when  = 0.2, x = .3, t = 0.1, W e = .03, P r = Br = M = 2, Du = Sc = Sr = 0.5, F r = 0.8, Re = 0.2, α = Θ = π/3, χi1 = 5. (b) Variation of Du on φ when E1 = 1, E2 = E3 = 0.3, E4 = E5 = 0.01,  = 0.2, x = 0.3, t = 0.1, W e = 0.01, P r = Br = 2, Sc = Sr = 0.5, α = π/4, Re = 0.8, M = 3, Θ = π/3, F r = 1.2, χi1 = 5.

Fig. 4(c)

Fig. 4(d)

(c) Variation of α on φ when E1 = 1, E2 = E3 = 0.3, E4 = E5 = 0.01,  = 0.2, x = 0.3, t = 0.1, W e = 0.03, P r = M = Br = 2, Sc = Sr = Du = 0.5, Re = 0.8, Θ = π/3, F r = 1.2, χi1 = 5. (d) Variation of Sr on φ when E1 = 1, E2 = E3 = 0.3, E4 = E5 = 0.01,  = 0.2, x = 0.3, t = 0.1, W e = 0.01, P r = Br = 2, Sc = Du = 0.5, α = π/4, Re = 0.8, M = 3, Θ = π/3, F r = 1.2, χi1 = 5. 19

Fig. 4(e)

Fig. 4(f)

(e) Variation of Sc on φ when E1 = 1, E2 = E3 = 0.3, E4 = E5 = 0.01,  = 0.2, x = 0.3, t = 0.1, W e = 0.01, P r = Br = 2, Sr = Du = 0.5, α = π/4, Re = 0.8, M = 3, Θ = π/3, F r = 1.2, χi1 = 5. (f ) Variation of W e on φ when E1 = 1, E2 = E3 = .3, E4 = E5 = 0.01,  = 0.2, x = 0.3, t = 0.1, P r = Br = 2.0, Re = 0.8, Sc = Sr = Du = 0.5, α = π/4, M = 3, Θ = π/3, F r = 1.2, χi1 = 5.

Fig. 4(g)

Fig. 4(h)

(g) Variation of F r on φ when E1 = 1, E2 = E3 = 0.3, E4 = E5 = 0.01,  = 0.2, x = 0.3, t = 0.1, W e = 0.03, P r = M = Br = 2, Sc = Sr = Du = 0.5, α = π/4, Re = 0.8, Θ = π/3, χi1 = 5. (h) Variation of χi1 on φ when E1 = 1, E2 = E3 = 0.3, E4 = E5 = 0.01,  = 0.2, x = 0.3, t = 0.1, W e = 0.01, P r = Br = 2, Sc = Sr = Du = 0.5, α = π/4, Re = 0.8, M = 3, Θ = π/3, F r = 1.2. 20

Fig. 4(i)

Fig. 4(j)

(i) Variation of Θ on φ when E1 = 1, E2 = E3 = 0.3, E4 = E5 = 0.01,  = 0.2, x = 0.3, t = 0.1, W e = .01, P r = Br = 2, Sc = Sr = Du = 0.5, α = π/4, Re = 0.8, M = 3, F r = 1.2. (j) Variation of Br on φ when E1 = 1, E2 = E3 = 0.3, E4 = E5 = 0.01,  = 0.2, x = 0.3, t = 0.1, W e = 0.03, P r = M = 2, Sc = Sr = Du = 0.5, α = π/4, Re = 0.8, Θ = π/3, F r = 1.2, χi1 = 5.

Fig. 5(a)

Fig. 5(b)

F ig. 5(a) Variation of compliant wall parameters on Z when  = 0.2, x = 0.3, t = 0.1, W e = 0.03, P r = Br = M = 2, Du = Sc = Sr = 0.5, F r = 1.2, Re = 0.8, α = Θ = π/3, Bi1 = 5. (b) Variation of Br on Z when E1 = 1, E2 = E3 = 0.5, E4 = E5 = 0.01,  = 0.2, x = 0.3, t = 0.1, W e = 0.03, P r = M = 2, Du = Sc = Sr = 0.5, F r = 1.2, Re = 0.2, α = π/3, Θ = π/3, Bi1 = 5. 21

Fig. 5(c)

Fig. 5(d)

(c) Variation of Bi1 on Z when E1 = 1, E2 = E3 = 0.3, E4 = E5 = 0.01,  = 0.2, x = 0.3, t = 0.1, W e = 0.03, P r = Br = 2, Du = Sc = Sr = 0.5, F r = 0.8, Re = 0.2, M = 3, α = Θ = π/3. (d) Variation of Pr on Z when E1 = 1, E2 = E3 = 0.3, E4 = E5 = 0.01,  = 0.2, x = 0.3, t = 0.1, W e = 0.01, Br = 2, Du = Sc = Sr = 0.5, α = π/4, Re = 0.8, M = 3, F r = 0.8, Θ = π/3, Bi1 = 8.

Fig. 5(e)

Fig. 5(f)

(e) Variation of M on Z when E1 = 1, E2 = E3 = 0.3, E4 = E5 = 0.01,  = 0.2, x = 0.3, t = 0.1, W e = 0.01, P r = Br = 2, Du = Sc = Sr = 0.5, α = Θ = π/4, Re = 0.8, F r = 1.2, Bi1 = 8. (f ) Variation of α on Z when E1 = 1, E2 = E3 = 0.3, E4 = E5 = 0.01,  = 0.2, x = 0.3, t = 0.1, W e = 0.01, P r = Br = 2, Sc = Du = Sr = 0.5, Re = 0.8, M = 3, Θ = π/3, F r = 1.2, Bi1 = 8. 22

Fig. 5(g)

Fig. 5(h)

(g) Variation of Θ on Z when E1 = 1, E2 = E3 = 0.3, E4 = E5 = 0.01,  = 0.2, x = 0.3, t = 0.1, Br = Pr = 2, Sc = Du = Sr = 0.5, W e = 0.01, α = π/4, Re = 0.8, M = 3, F r = 1.2, Bi1 = 8. (h) Variation of Sr on Z when E1 = 1, E2 = E3 = 0.3, E4 = E5 = 0.01,  = 0.2, x = 0.3, t = 0.1, W e = 0.01, P r = Br = 2, Du = Sc = 0.5, α = π/4, Re = 0.8, M = 3, Θ = π/3, F r = 1.2, Bi1 = 8.

Fig. 5(i)

Fig. 5(j)

(i) Variation of Sc on Z when E1 = 1, E2 = E3 = 0.3, E4 = E5 = 0.01,  = 0.2, x = 0.3, t = 0.1, W e = 0.01, P r = Br = 2, Sr = Du = 0.5, α = π/4, Re = 0.8, M = 3, Θ = π/3, F r = 1.2, Bi1 = 8. (j) Variation of Du on Z when E1 = 1, E2 = E3 = 0.3, E4 = E5 = 0.01,  = 0.2, x = 0.3, t = 0.1, W e = 0.01, P r = Br = 2, Sc = Sr = 0.5, α = π/4, Re = 0.8, M = 3, Θ = π/3, F r = 1.2, Bi1 = 8. 23

Fig. 5(k)

Fig. 5(l)

(k) Variation of F r on Z when E1 = 1, E2 = E3 = 0.3, E4 = E5 = 0.01,  = 0.2, x = 0.3, t = 0.1, W e = 0.01, P r = Br = 2, Sc = Du = Sr = 0.5, Re = 0.8, M = 3, α = Θ = π/3, Bi1 = 8. (l) Variation of W e on Z when E1 = 1, E2 = E3 = 0.5, E4 = E5 = 0.01,  = 0.2, x = 0.3, Re = 0.8, t = 0.1, P r = Br = 2, Sc = Du = Sr = 0.5, α = π/4, M = 3, Θ = π/3, F r = 1.2, Bi1 = 8.

(a)

(b)

F ig. 6 Variation of M on ψ for E1 = 0.5, E2 = 0.2, E3 = 0.1, E4 = 0.05, E5 = 0.3, t = 0, W e = 0.03, R = 0.2, F r = 0.8, α = Θ = π/3,  = 0.2 when (a) M = 1.5 (b) M = 2.

24

(a)

(b)

F ig. 7 Variation of α on ψ for E1 = 0.7, E2 = 0.2, E3 = 0.1, E4 = 0.01, E5 = 0.3, M = 2, t = 0, W e = 0.03, R = 0.2, F r = 0.8, Θ = π/3,  = 0.2 when (a) α = π/4 (b) α = π/3.

(a)

(b)

F ig. 8 Variation of W e on ψ for E1 = 0.7, E2 = 0.2, E3 = 0.1, E4 = 0.01, E5 = 0.3, M = 2, t = 0, R = 0.2, F r = 0.8, α = Θ = π/3,  = 0.2 when (a) W e = 0.01 (b) W e = 0.02.

25

(a)

(b)

(c)

(d)

26

(e)

(f)

F ig. 9 Variation of wall properties on ψ for (a) E1 = 0.7, E2 = 0.4, E3 = 0.2, E4 = 0.02, E5 = 0.5, M = 2, t = 0, W e = 0.03, R = 0.2, F r = 0.8, α = Θ = π/3,  = 0.2 (b) E1 = 0.9 (c) E2 = 0.6 (d) E3 = 0.5 (e) E4 = 0.01 (f ) E5 = 0.2.

6

Concluding remarks

Effects of Soret and Dufour on peristaltic transport of Williamson fluid in an inclined channel with convective conditions are examined. The main points of presented analysis are listed below: • Velocity profile decays for increasing values of Hartman number M . • Velocity in lower and upper halves of channel has opposite behavior for the Weissenberg number. • Velocity enhances for increase in the inclinations of magnetic field and channel. • Temperature profile is a decreasing function of heat transfer Biot number Bi1 . • Concentration of fluid increases when mass transfer Biot number χi1 is increased. • Concentration profile is decreasing function of F r and α . • Temperature and concentration have similar effect qualitatively for the Schmidt, Soret and Dufour numbers. 27

• Heat transfer coefficient Z(x) decreases for Bi1 . • Size of trapped bolus decreases when Weissenberg number W e is increased. • The presented results can be reduced to the case of hydrodynamic fluid for M = 0. • It is noticed that for M = E4 = E5 = W e = α = 0, the present results are identical to that of Radhakrishnamcharya and Srinivasulu [12] . • In the absence of Joule heating and W e = Du = 0, the results of temperature and concentration fields in present study are reduced to Newtonian case analyzed by Srinivas and Kothandapani [13].

28

References [1] T. Hayat, S. Hina, S. Asghar and S. Obaidat, Peristaltic flow of Maxwell fluid in an asymmetric channel with wall properties, Int. J. Phys. Sci., 7 (2012) 2145-2155. [2] D. Tripathi, Numerical study on peristaltic transport of fractional bio-fluids, J. Mech. Med. Biol., 11 (2011) 1045-1058. [3] Kh. S. Mekheimer, Y. Abd elmaboud and A. I. Abdellateef, Peristaltic transport through eccentric cylinders: Mathematical model, Appl. Bio. Biomech., 10 (2013) 19-27. [4] K. Yazdanpanth-Ardakani and H. Niroomand-Oscii, New approach in modelling peristaltic transport of non-Newtonian fluid, J. Mech. Med. Biol., 13 (2013) 1350052. [5] M. Lachiheb, Effect of coupled radial and axial variability of viscosity on the peristaltic transport of Newtonian fluid, Appl. Math. Comput., 244 (2014) 761–771. [6] Y. V. K. Ravi Kumar, P. V. Kumar, and S. Bathul, Effect of slip on peristaltic pumping of a hyperbolic tangent fluid in an inclined asymmetric channel, Adv. Appl. Sci. Res., 5 (2014) 91-108. [7] S. Hina, M. Mustafa, T. Hayat and F. E. Alsaadi, Peristaltic motion of third grade fluid in curved channel, Appl. Math. Mech., 35 (2014) 73-84. [8] A.M.Abd-Alla, S.M.Abo-Dahab and A.Kilicman, Peristaltic flow of a Jeffrey fluid under the effect of radially varying magnetic field in a tube with an endoscope, J. Magn. Magn. Mater. 384 (2015) 79-86. [9] A.M. Abd-Alla, S. M. Abo-Dahab and H.D.El-Shahrany, Effects of rotation and initial stress on peristaltic transport of fourth grade fluid with heat transfer and induced magnetic field, J. Magn. Mater., 349 (2014) 268-280. [10] K. Nowar, Peristaltic flow of a nanofluid under the effect of Hall current and porous medium, Math. Prob. Eng. (2014). [11] M. Kothandapani and J. Prakash, Effect of radiation and magnetic field on peristaltic transport of nanofluids through a porous space in a tapered asymmetric channel, J. Magn. Magn. Mater. 378 (2015) 152-163. 29

Figure 1:

30

Figure 2: [12] G. Radhakrishnamacharya, Ch. Srinivasulu, Influence of wall properties on peristaltic transport with heat transfer, C. R. Mecanique 335 (2007) 369-373. [13] S. Srinivas, M. Kothandapani, The influence of heat and mass transfer on MHD peristaltic flow through a porous space with compliant walls, Appl. Math. Comput. 213 (2009) 197-208 [14] A. A. Shaaban and Mohamed Y. Abou-Zeid, Effects of heat and mass transfer on MHD peristaltic flow of a non-Newtonian fluid through a porous medium between two coaxial cylinders, Math. Prob., Eng. 2013. [15] M. Kothandapani and J. Prakash, Influence of heat source, thermal radiation, and inclined magnetic field on peristaltic flow of a hyperbolic tangent nanofluid in a tapered asymmetric channel, IEEE Trans. Nanobioscience. 14 (2015) 385-392. [16] H. Yasmin, T. Hayat, N. Alotaib and H. Gao, Convective heat and mass transfer analysis on peristaltic flow of Williamson fluid with Hall effects and Joule heating, Int. J. Biomath., DOI: 10.1142/S1793524514500582. 7 (2014). 31

Figure 3:

32

Figure 4:

33

Figure 5:

34

Figure 6:

Figure 7:

35

Figure 8:

Figure 9: 36

[17] M. Kothandapani and J. Prakash, Convective boundary conditions effect on peristaltic flow of a MHD Jeffrey nanofluid, Appl. Nanosci. (2015) DOI 10.1007/s13204-015-0431-9 [18] R. Ellahi, M. M. Bhatti and K. Vafai, Effects of heat and mass transfer on peristaltic flow in a non-uniform rectangular duct, Int. J. Heat Mass Transfer, 71 (2014) 706-719. [19] T. Hayat, F. M. Abbasi, Maryem Al-Yami and Shatha Monaquel, Slip and Joule heating effects in mixed convection peristaltic transport of nanofluid with Soret and Dufour effects, J. Mol. Liq., 194 (2014) 93-99. [20] M. Mustafa, S. Abbasbandy, S. Hina and T. Hayat, Numerical investigation on mixed convective peristaltic flow of fourth grade fluid with Dufour and Soret effect, J. Taiwan. Ins. Chem. Eng., 45 (2014) 308-316. [21] F. M. Abbasi, T. Hayat, A. Alsaedi and B. Ahmed, Soret and Dufour effects on peristaltic transport of MHD fluid with variable viscosity, Appl. Math. Inf. Sci., 8 (2014) 211-219. [22] M. Kothandapani and J. Prakash, Effects of thermal radiation parameter and magnetic field on the peristaltic motion of Williamson nanofluids in a tapered asymmetric channel, Int. J. Heat Mass Transfer, 81 (2015) 234-245. [23] Obaid ullah mehmood, N. mustapha, S. Shafie and T. Hayat, Partial slip effect on heat and mass transfer of MHD peristaltic transport in a porous medium, Sains Malays. 43 (2014) 1109-1118. [24] V. P. Rathod and L. Devindrappa, Peristaltic transport of a conducting fluid in an asymmetric vertical channel with heat and mass transfer, J. Chem. Biol. Phys. Sci. 4 (2014) 1452-1470. [25] A. A. Khan, R. Ellahi and M. Usman, The effects of variable viscosity on the peristaltic flow of non-Newtonian fluid through a porous medium in an inclined channel with slip boundary conditions, J. Porous Media. 16 (2013) 59-67. [26] V. P. Rathod and N. G. Sridhar, Peristaltic Flow of A Couple Stress Fluids through a Porous Medium in an Inclined Channel, J. Chem. Biol. Phys. Sci. 5 (2015) 1760-1770.

37

[27] Lika. Z. Hummady, Ahmed. M. abdulhadi, A numerical study of peristaltic flow generalized Maxwell viscoelastic fluids through a porous medium in an inclined channel, Adv. Life Sci. Tech, 32 (2015) 59-72. [28] R. Ellahi, A. Riaz and S. Nadeem, Three-dimensional peristaltic flow of Williamson fluid in a rectangular duct, Indian J. Phys. 87 (2013) 1275-1281.

38