# Peristaltic Motion of Non-Newtonian Fluid with Heat and Mass Transfer through a Porous Medium in Channel under Uniform Magnetic Field

Authors: Nabil T. M. Eldabe; Bothaina M. Agoor; Heba Alame
Journal: Journal of Fluids
 (2014)

---

## Abstract

This paper is devoted to the study of the peristaltic motion of non-Newtonian fluid with heat and mass transfer through a porous medium in the channel under the effect of magnetic field. A modified Casson non-Newtonian constitutive model is employed for the transport fluid. A perturbation series’ method of solution of the stream function is discussed. The effects of various parameters of interest such as the magnetic parameter, Casson parameter, and permeability parameter on the velocity, pressure rise, temperature, and concentration are discussed and illustrated graphically through a set of figures.


---

## Body

## 1. Introduction

Peristaltic motion is a phenomenon that occurs when expansion and contraction of an extensible tube in a fluid generate progressive waves which propagate along the length of the tube, mixing and transporting the fluid in the direction of wave propagation. In some biomedical instruments, such as heart-lung machines, peristaltic motion is used to pump blood and other biological fluids [1]. Peristaltic pumping is a form of fluid transport generally from a region of lower to higher pressure, by means of a progressive wave of area contraction or expansion, which propagates along the length of a tube like structure. Some electrochemical reactions are held responsible for this phenomenon. This mechanism occurs in swallowing of food through oesophagus, in the ureter, the gastro intestinal tract, the bile duct, and even in small blood vessels. It has now been accepted that most of the physiological fluids behave like a non-Newtonian fluids. The peristaltic flows have attracted a number of researchers because of wide applications in physiology and industry. The theoretical work of peristaltic transport primarily with the inertia free Newtonian flow driven by a sinusoidal transverse wave of small amplitude is investigated by Fung et al. [2]. Burns and Parkes [3] studied the peristaltic motion of a viscous fluid through a pipe and channel by considering sinusoidal variations at the walls. A mathematical study of the peristaltic transport of Casson fluid is given by Mernone and Mazumdar [4, 5]; they used the perturbation method to solve the problem. Mekheimer [6, 7] studied the peristaltic transport of MHD flow. Peristaltic transport of Casson fluid in a channel is discussed by Nagarani and Sarojamma [8, 9]. El Shehawy et al. [10] Studied the peristaltic transport in a symmetric channel through a porous medium. Finite element solutions for non-Newtonian pulsatile flow in a non-Darcian porous medium are given by Bharagava et al. [11]. Mekheirmer and Abd elmaboud [12] discussed the influence of heat transfer and magnetic field on peristaltic transport. Nadeem et al. [13] have discussed the influence of heat and mass transfer on peristaltic flow of third order fluid in a diverging tube. Abdelmaboud and Mekheimer [14] analyzed the transport of second order fluid through a porous medium. Abd Elmaboud [15] studied the heat transfer characteristics of micropolar fluid through an isotropic porous medium in a two-dimensional channel with rhythmically contracting walls. El-dabe et al. [16] studied the effects of radiation on the unsteady flow of an incompressible non-Newtonian (Jeffrey) fluid through porous medium. Mustafa et al. [17] studied the peristaltic transport of nanofluid in a channel with complaint walls. Anwr Beg and Tripathi [18] introduced a theoretical study to examine the peristaltic pumping with double-diffusive convection in nanofluids through a deformable channel. El-dabe et al. [19] have discussed the effects of heat and mass transfer on the MHD flow of an incompressible, electrically conducting couple stress fluid through a porous medium in an asymmetric flexible channel over which a traveling wave of contraction and expansion is produced, resulting in a peristaltic motion. El-dabe et al. [20] studied the peristaltic motion of incompressible micropolar fluid through a porous medium in a two-dimensional channel under the effects of heat absorption and chemical reaction in the presence of magnetic field. Ebaid and Emad Aly [21] showed the mathematical model describing the slip peristaltic flow of nanofluid application to the cancer treatment. Emad and Ebaid [22] applied two different analytical and numerical methods to solve the system describing the mixed convection boundary layer nanofluids flow along an inclined plate embedded in a porous medium. Abd Elmaboud [23] investigated the magneto thermodynamic aspects micropolar fluid (blood model) through an isotropic porous medium in a nonuniform channel with rhythmically contracting walls. Noreen et al. [24] studied the mathematical model to investigate the mixed convective heat and mass transfer effects on peristaltic flow of magnetohydrodynamic pseudoplastic fluid in a symmetric channel. Hayat et al. [25] discussed the effects of heat and mass transfer on the peristaltic flow in the presence of an induced magnetic field. Noreen [26] consider the peristaltic flow of third order nanofluid in an asymmetric channel with an induced magnetic field.The main aim of this work is to study the peristaltic motion of non-Newtonian fluid with heat and mass transfer through a porous medium in the channel under the effect of magnetic field. A modified Casson non-Newtonian constitutive model is employed for the transport fluid. A perturbation series’ method of solution of the stream function is discussed. The effects of various parameters of interest such as the magnetic parameter, Casson parameter, and permeability parameter on the velocity, pressure rise, and temperature are discussed and illustrated graphically through a set of figures.


## 2. Mathematical Analysis

Consider the peristaltic motion of non-Newtonian fluid through a porous medium in two-dimensional channel, having widthd. A rectangular coordinate system x
,
y is chosen such that x-axis lies along the direction of wave progression and y-axis normal to it. The fluid is subjected to a constant magnetic field B
_
=
(
0
,
B
0
,
0
). Let u and v be the velocity components. The vertical displacements for the upper and lower walls are ζ and -
ζ, see Figure 1, where ζ is defined by
(1)
ζ
(
x
,
t
)
=
a
cos
⁡
2
π
λ
(
x
-
c
t
)
.
λ is the wavelength, t is the time and a is the amplitude of the sinusoidal waves travelling along the channel at velocity c. The constitutive equation for the non-Newtonian Casson fluid can be written as in [27].Figure 1
Sketch of the problem.Consider(2)
τ
i
j
=
{
2
(
μ
B
+
p
y
2
π
)
e
i
j
,
π
>
π
c
,
2
(
μ
B
+
P
y
2
π
c
)
e
i
j
,
π
<
π
c
,
where τ
i
j is the components of the stress tensor, π
=
e
i
j
e
j
i and e
i
j are the (
i
,
j
)th components of the deformation rate, π is the product of the component of deformation rate by itself, π
c is a critical value of this product based on the Nakamura-Sawada model [27], μ
B is the plastic dynamic viscosity of the non-Newtonian fluid, and p
y is yield stress of slurry fluid.The equations governing the fluid motion can be written as follows.The continuity equation is(3)
∂
u
∂
x
+
∂
v
∂
y
=
0
.The momentum equations are(4)
ρ
(
∂
u
∂
t
+
u
∂
u
∂
x
+
v
∂
u
∂
y
)
=
-
∂
P
∂
x
+
(
μ
+
μ
B
)
(
∂
2
u
∂
x
2
+
∂
2
u
∂
y
2
)
-
(
σ
B
0
2
+
μ
k
)
u
,
ρ
(
∂
v
∂
t
+
u
∂
v
∂
x
+
v
∂
v
∂
y
)
=
-
∂
P
∂
y
+
(
μ
+
μ
B
)
(
∂
2
v
∂
x
2
+
∂
2
v
∂
y
2
)
-
(
σ
B
0
2
+
μ
k
)
v
.The energy equation is(5)
(
∂
T
∂
t
+
u
∂
T
∂
x
+
v
∂
T
∂
y
)
=
k
1
ρ
c
p
(
∂
2
T
∂
x
2
+
∂
2
T
∂
y
2
)
.The concentration equation is(6)
(
∂
C
∂
t
+
u
∂
C
∂
x
+
v
∂
C
∂
y
)
=
D
m
(
∂
2
C
∂
x
2
+
∂
2
C
∂
y
2
)
.Lorentz force:(7)
F
_
=
J
_
∧
B
_
,
J
_
=
σ
V
_
∧
B
_
,
where k is the permeability of the medium, J is the current density, c
p is the specific heat of the fluid, k
1 is the coefficient of heat conduction, T is the temperature of the fluid, D
m is the coefficient of mass diffusivity, C is the concentration of the fluid, μ is the coefficient of viscosity, σ is the electrical conductivity, P is pressure, and B
o is the strength of the applied magnetic field.The appropriate boundary conditions are(8)
u
=
0
,
v
=
∂
ζ
∂
t
,
T
=
T
1
,
C
=
C
1
ccccccccccccccccccccccicccccicc
at
y
=
d
+
ζ
,
u
=
0
,
v
=
-
∂
ζ
∂
t
,
T
=
T
0
,
C
=
C
0
ccccccccccccccccccccccciciccccc
at
y
=
-
d
-
ζ
.Using the following nondimensional variables:(9)
x
′
=
x
d
,
y
′
=
y
d
,
u
′
=
u
c
,
v
′
=
v
c
,
t
′
=
c
t
d
,
p
′
=
p
ρ
c
2
,
ζ
′
=
ζ
d
,
ε
=
a
d
,
T
=
(
T
1
-
T
0
)
θ
+
T
0
,
C
=
(
C
1
-
C
0
)
ϕ
+
C
0
,
equations (4)–(8) after dropping the stars mark reduce to
(10)
(
∂
u
∂
t
+
u
∂
u
∂
x
+
v
∂
u
∂
y
)
=
-
∂
P
∂
x
+
(
τ
0
+
1
R
e
)
(
∂
2
u
∂
x
2
+
∂
2
u
∂
y
2
)
-
(
M
+
1
K
)
u
,
(
∂
v
∂
t
+
u
∂
v
∂
x
+
v
∂
v
∂
y
)
=
-
∂
P
∂
y
+
(
τ
0
+
1
R
e
)
(
∂
2
v
∂
x
2
+
∂
2
v
∂
y
2
)
-
(
M
+
1
K
)
v
,
(
∂
θ
∂
t
+
u
∂
θ
∂
x
+
v
∂
θ
∂
y
)
=
k
1
ρ
c
p
(
∂
2
θ
∂
x
2
+
∂
2
θ
∂
y
2
)
,
(
∂
ϕ
∂
t
+
u
∂
ϕ
∂
x
+
v
∂
ϕ
∂
y
)
=
k
1
ρ
c
p
(
∂
2
ϕ
∂
x
2
+
∂
2
ϕ
∂
y
2
)
,with the boundary conditions(11)
u
=
0
,
v
=
α
ε
sin
α
(
x
-
t
)
,
θ
=
1
,
ϕ
=
1
cccccccccccccccccccccccccccccccccciccc
at
y
=
1
+
ζ
,
u
=
0
,
v
=
-
α
ε
sin
α
(
x
-
t
)
,
θ
=
0
,
ϕ
=
0
cccccccccccccccccccccccccccccccccicccc
at
y
=
-
1
-
ζ
,
where M
=
σ
B
0
2
d
/
ρ
c is the magnetic parameter, K
=
c
k
/
ϑ
d is the permeability parameter, α
=
2
d
π
/
μ is the wave number, τ
0
=
μ
B
/
ρ
c
d is the Casson parameter, and R
e
=
ρ
c
d
/
μ is the Reynolds number.Now, we shall define a stream functionψ as u
=
ψ
y and v
=
-
ψ
x then (10) can be written as
(12)
ψ
y
t
+
ψ
y
ψ
y
x
-
ψ
x
ψ
y
y
=
-
∂
p
∂
x
+
(
τ
0
+
1
R
e
)
(
ψ
y
y
y
+
ψ
y
x
x
)
-
(
M
+
1
K
)
ψ
y
,
-
ψ
x
t
-
ψ
y
ψ
x
x
+
ψ
x
ψ
x
y
=
-
∂
p
∂
y
-
(
τ
0
+
1
R
e
)
(
ψ
x
x
x
+
ψ
y
y
x
)
+
(
M
+
1
K
)
ψ
x
,
(
∂
θ
∂
t
+
ψ
Y
∂
θ
∂
x
-
ψ
X
∂
θ
∂
y
)
=
k
1
ρ
c
p
(
∂
2
θ
∂
x
2
+
∂
2
θ
∂
y
2
)
,
(
∂
ϕ
∂
t
+
ψ
Y
∂
ϕ
∂
x
-
ψ
X
∂
ϕ
∂
y
)
=
D
m
(
∂
2
ϕ
∂
x
2
+
∂
2
ϕ
∂
y
2
)
,
with conditions
(13)
ψ
y
=
0
,
ψ
x
=
-
α
ε
sin
α
(
x
-
t
)
,
θ
=
1
,
ϕ
=
1
cccccccccccccccccccccccccccccccciccccccccc
at
y
=
1
+
ζ
,
ψ
y
=
0
,
ψ
x
=
α
ε
sin
α
(
x
-
t
)
,
θ
=
0
,
ϕ
=
0
ccccccccccccccccccccccccccccccccccccc
ciat
y
=
-
1
-
ζ
.
Express a stream function ψ, p, θ, and ϕ as a series in terms of small amplitude ratio ε, we have
(14)
ψ
(
x
,
y
,
t
)
=
ψ
0
+
ε
ψ
1
+
⋯
,
p
(
x
,
y
)
=
p
0
+
ε
p
1
+
⋯
,
θ
(
x
,
y
,
t
)
=
θ
0
+
ε
θ
1
+
⋯
,
ϕ
(
x
,
y
,
t
)
=
ϕ
0
+
ε
ϕ
1
+
⋯
,
where ψ
0 is a function of y only. Substituting (14) in (12) and collecting the terms in ε, we get the following system of equations.Coefficient ofε
0:
(15)
∂
p
0
∂
y
=
0
,
this
leads
to
P
o
is
a
function
of
x
,
p
0
=
c
1
(
x
)
,
(16)
ψ
o
y
y
y
-
λ
1
2
ψ
0
y
+
L
=
0
,
(17)
∂
2
θ
0
∂
y
2
=
0
,
(18)
∂
2
ϕ
0
∂
y
2
=
0
,
where
(19)
λ
1
=
(
M
+
(
1
/
K
)
)
(
τ
0
+
(
1
/
R
e
)
)
,
L
=
(
∂
p
0
/
∂
x
(
τ
0
+
(
1
/
R
e
)
)
)
.
and coefficient of ε:
(20)
ψ
1
y
t
+
ψ
0
y
ψ
1
y
x
-
ψ
1
x
ψ
0
y
y
=
-
∂
p
1
∂
x
+
(
τ
0
+
1
R
e
)
(
ψ
1
y
y
y
+
ψ
1
y
x
x
)
-
(
M
+
1
K
)
ψ
1
y
,
(21)
-
ψ
1
x
t
-
ψ
0
y
ψ
1
x
x
=
-
∂
p
1
∂
y
-
(
τ
0
+
1
R
e
)
(
ψ
1
x
x
x
+
ψ
1
y
y
x
)
+
(
M
+
1
K
)
ψ
1
x
,
(22)
(
∂
θ
1
∂
t
+
ψ
0
y
∂
θ
1
∂
x
-
ψ
1
x
∂
θ
1
∂
y
)
=
k
1
ρ
c
p
(
∂
2
θ
1
∂
x
2
+
∂
2
θ
1
∂
y
2
)
,
(23)
(
∂
ϕ
1
∂
t
+
ψ
0
y
∂
ϕ
1
∂
x
-
ψ
1
x
∂
ϕ
1
∂
y
)
=
D
m
(
∂
2
ϕ
1
∂
x
2
+
∂
2
ϕ
1
∂
y
2
)
.
Also, boundary conditions (13) can be written after using the Taylor series expansions about y
=
±
1
±
ζ as follows:
(24)
ψ
0
y
(
1
)
=
0
,
θ
0
(
1
)
=
1
,
ϕ
0
(
1
)
=
1
,
ψ
1
y
(
1
)
=
-
ψ
0
y
y
(
1
)
cos
⁡
α
(
x
-
t
)
,
θ
1
(
1
)
=
-
θ
0
y
(
1
)
cos
⁡
α
(
x
-
t
)
,
ψ
0
(
-
1
)
=
0
,
θ
0
(
-
1
)
=
0
,
ϕ
0
(
-
1
)
=
0
,
ψ
1
x
(
1
)
=
-
α
sin
α
(
x
-
t
)
,
ψ
0
y
(
-
1
)
=
0
,
ϕ
1
(
1
)
=
-
ϕ
0
y
(
1
)
cos
⁡
α
(
x
-
t
)
.
Using conditions (24) with (15), the solution of (16) can be written as
(25)
ψ
0
(
y
)
=
-
L
λ
1
(
y
-
sinh
⁡
λ
1
y
λ
1
cosh
⁡
λ
1
)
.
The flow rate, q, is given by
(26)
q
=
∫
0
1
u
d
y
=
∫
0
1
∂
ψ
∂
y
d
y
=
ψ
(
1
)
-
ψ
(
0
)
=
-
1
λ
1
(
τ
0
+
(
1
/
R
e
)
)
(
1
-
sinh
⁡
λ
1
λ
1
cosh
⁡
λ
1
)
∂
p
0
∂
x
.
The pressure rise is given by
(27)
Δ
p
0
=
∫
0
1
∂
p
0
∂
x
d
x
=
-
(
τ
0
+
(
1
/
R
e
)
)
λ
1
q
(
1
-
(
sinh
⁡
λ
1
/
λ
1
cosh
⁡
λ
1
)
)
.Eliminating the pressure terms in (20) and (21), we have
(28)
∇
2
ψ
1
t
+
ψ
0
y
∇
2
ψ
1
x
-
ψ
1
x
∇
2
ψ
0
y
=
(
τ
0
+
1
R
e
)
∇
4
ψ
1
-
(
M
+
1
K
)
∇
2
ψ
1
.
From conditions (24) and (25) we can write ψ
1, θ
1, and ϕ
1 in the form
(29)
ψ
1
(
x
,
y
,
t
)
=
f
(
y
)
cos
⁡
α
(
x
-
t
)
+
g
(
y
)
sin
α
(
x
-
t
)
.
θ
1
(
x
,
y
,
t
)
=
h
1
(
y
)
cos
⁡
α
(
x
-
t
)
,
ϕ
1
(
x
,
y
,
t
)
=
h
2
(
y
)
cos
⁡
α
(
x
-
t
)
.
Equation (28) can be simplified by using (29) and assuming that the wave number α
=
2
π
d
/
λ is small, so the terms of α
2and higher can be neglected.We get(30)
α
[
(
(
1
-
L
λ
1
)
+
cosh
⁡
λ
1
y
cosh
⁡
λ
1
)
f
′′
+
L
cosh
⁡
λ
1
y
cosh
⁡
λ
1
f
(
(
1
-
L
λ
1
)
+
cosh
⁡
λ
1
y
cosh
⁡
λ
1
)
]
sin
α
(
x
-
t
)
-
α
[
(
(
1
-
L
λ
1
)
+
cosh
⁡
λ
1
y
cosh
⁡
λ
1
)
g
′′
c
c
c
c
c
c
c
c
c
-
L
cosh
⁡
λ
1
y
cosh
⁡
λ
1
g
(
(
1
-
L
λ
1
)
+
cosh
⁡
λ
1
y
cosh
⁡
λ
1
)
]
cos
⁡
α
(
x
-
t
)
=
[
(
τ
0
+
1
R
e
)
f
′
′
′
′
-
(
M
+
1
R
e
)
f
′
′
]
cos
⁡
α
(
x
-
t
)
c
c
c
c
c
c
c
+
[
(
τ
0
+
1
R
e
)
g
′
′
′
′
-
(
M
+
1
K
)
g
′
′
]
sin
α
(
x
-
t
)
.
Collecting coefficients of cos
⁡
α
(
x
-
t
) and sin
α
(
x
-
t
) on either side of (30), two differential equations for f
(
y
) and g
(
y
) are obtained as follows:
(31)
-
α
[
(
(
1
-
L
λ
1
)
+
cosh
⁡
λ
1
y
cosh
⁡
λ
1
)
g
′′
+
L
cosh
⁡
λ
1
y
cosh
⁡
λ
1
g
]
=
[
(
τ
0
+
1
R
e
)
f
′
′
′
′
-
(
M
+
1
K
)
f
′
′
]
,
α
[
(
(
1
-
L
λ
1
)
+
cosh
⁡
λ
1
y
cosh
⁡
λ
1
)
f
′′
+
L
cosh
⁡
λ
1
y
cosh
⁡
λ
1
f
]
=
[
(
τ
0
+
1
R
e
)
g
′′′′
-
(
M
+
1
K
)
g
′′
]
.
Equation (31) can be simplified by assuming that
(32)
f
(
y
)
=
f
0
+
α
f
1
+
⋯
,
g
(
y
)
=
g
0
+
α
g
1
+
⋯
.
Substituting (32) into (31), and equating terms in α, the following ordinary differential equations are obtained for f
o, f
1, g
0, and g
1, respectively:
(33)
f
0
′′′′
-
λ
1
2
f
0
′′
=
0
,
g
0
′′′′
-
λ
1
2
g
0
′′
=
0
,
-
[
(
(
1
-
L
λ
1
)
+
cosh
⁡
λ
1
y
cosh
⁡
λ
1
)
g
0
′′
-
L
cosh
⁡
λ
1
y
cosh
⁡
λ
1
g
0
]
=
[
(
τ
0
+
1
R
e
)
f
1
′
′
′
′
-
(
M
+
1
K
)
f
1
′
]
,
[
(
(
1
-
L
λ
1
)
+
cosh
⁡
λ
1
y
cosh
⁡
λ
1
)
f
0
′′
-
L
cosh
⁡
λ
1
y
cosh
⁡
λ
1
f
0
]
=
[
(
τ
0
+
1
R
e
)
g
1
′
′
′
′
-
(
M
+
1
K
)
g
1
′
′
]
with boundary conditions:
(34)
f
0
(
1
)
=
1
,
f
0
′
(
1
)
=
-
A
,
f
0
(
-
1
)
=
-
1
,
f
0
′
(
-
1
)
=
-
A
,
g
0
(
1
)
=
0
,
g
0
′
(
1
)
=
0
,
g
0
(
-
1
)
=
0
,
g
0
′
(
-
1
)
=
0
,
f
1
(
1
)
=
0
,
f
1
′
(
1
)
=
0
,
f
1
(
-
1
)
=
0
,
f
1
′
(
-
1
)
=
0
,
g
1
(
1
)
=
0
,
g
1
′
(
1
)
=
0
,
g
1
′
(
-
1
)
=
0
,
g
1
(
-
1
)
=
0
.
Solving (33) by using (34), we get
(35)
f
0
(
y
)
=
λ
2
y
-
λ
3
sinh
⁡
λ
1
y
,
(36)
f
1
(
y
)
=
0
,
g
0
(
y
)
=
0
,
(37)
g
1
(
y
)
=
λ
7
y
cosh
⁡
λ
1
y
+
(
λ
8
+
λ
9
y
2
)
sinh
⁡
λ
1
y
+
λ
10
sinh
⁡
2
λ
1
y
+
λ
11
y
.
Substituting (35) in (29), we get
(38)
ψ
1
(
x
,
y
,
t
)
=
(
λ
2
y
-
λ
3
sinh
⁡
λ
1
y
)
cos
⁡
α
(
x
-
t
)
+
α
sin
α
(
x
-
t
)
×
(
λ
7
y
cosh
⁡
λ
1
y
+
(
λ
8
+
λ
9
y
2
)
sinh
⁡
λ
1
y
c
c
c
+
λ
10
sinh
⁡
2
λ
1
y
+
λ
11
y
(
λ
8
+
λ
9
y
2
)
)
.
Substituting (25) and (38) in (14), we get
(39)
ψ
(
x
,
y
,
t
)
=
-
L
λ
1
(
y
-
sinh
⁡
λ
1
y
λ
1
cosh
⁡
λ
1
)
+
ε
(
(
(
(
λ
8
+
λ
9
y
2
)
)
)
(
λ
2
y
-
λ
3
sinh
⁡
λ
1
y
)
cos
⁡
α
(
x
-
t
)
c
c
c
c
+
α
sin
α
(
x
-
t
)
cccc
×
(
λ
7
y
cosh
⁡
λ
1
y
+
(
λ
8
+
λ
9
y
2
)
sinh
⁡
λ
1
y
c
c
c
c
c
c
c
c
c
+
λ
10
sinh
⁡
2
λ
1
y
+
λ
11
y
(
λ
8
+
λ
9
y
2
)
)
)
.
The velocity components can be written as
(40)
u
(
x
,
y
,
t
)
=
-
L
λ
1
(
1
-
cosh
⁡
λ
1
y
cosh
⁡
λ
1
)
+
ε
(
(
(
λ
8
+
λ
9
y
2
)
(
λ
2
-
λ
3
λ
1
cosh
⁡
λ
1
y
)
cos
⁡
α
(
x
-
t
)
c
c
c
+
α
sin
α
(
x
-
t
)
c
c
c
×
(
λ
7
cosh
⁡
λ
1
y
+
λ
1
λ
7
y
sinh
⁡
λ
1
y
c
c
c
c
c
c
+
λ
1
(
λ
8
+
λ
9
y
2
)
cosh
⁡
λ
1
y
c
c
c
c
c
c
+
2
λ
1
λ
10
cosh
⁡
2
λ
1
y
c
c
c
c
c
c
+
λ
11
+
2
λ
9
y
sinh
⁡
λ
1
y
)
(
(
λ
8
+
λ
9
y
2
)
)
.
v
(
x
,
y
,
t
)
=
α
ε
[
(
λ
2
y
-
λ
3
sinh
⁡
λ
1
y
)
sin
α
(
x
-
t
)
]
Substituting (39) in (17), (18), (22), and (23) and using (24) we get
(41)
θ
o
(
y
)
=
y
+
1
2
,
φ
o
(
y
)
=
y
+
1
2
,
θ
1
(
x
,
y
,
t
)
=
(
(
6
λ
1
2
)
-
1
-
6
λ
13
sinh
⁡
λ
1
y
λ
1
2
λ
12
y
3
)
(
-
λ
1
2
λ
12
y
+
λ
1
2
λ
12
y
3
+
6
λ
13
y
sinh
⁡
λ
1
c
c
-
6
λ
13
sinh
⁡
λ
1
y
λ
1
2
λ
12
y
3
)
(
6
λ
1
2
)
-
1
)
×
cos
⁡
α
(
x
-
t
)
,
φ
1
(
x
,
y
,
t
)
=
(
(
6
λ
1
2
)
-
1
(
λ
1
2
λ
12
y
3
-
λ
1
2
λ
14
y
+
λ
1
2
λ
12
y
3
+
6
λ
15
y
sinh
⁡
λ
1
c
c
c
-
6
λ
15
sinh
⁡
λ
1
y
λ
1
2
λ
12
y
3
)
(
6
λ
1
2
)
-
1
)
cos
⁡
α
(
x
-
t
)
.
Substituting (41) in (14), we get
(42)
θ
(
x
,
y
,
t
)
=
(
y
+
1
2
)
+
ε
α
(
(
6
λ
1
2
)
-
1
(
-
λ
1
2
λ
12
y
+
λ
1
2
λ
12
y
3
c
c
c
c
c
c
+
6
λ
13
y
sinh
⁡
λ
1
-
6
λ
13
sinh
⁡
λ
1
y
λ
1
2
λ
12
y
3
)
(
6
λ
1
2
)
-
1
)
×
cos
⁡
α
(
x
-
t
)
,
(43)
ϕ
(
x
,
y
,
t
)
=
y
+
1
2
+
ε
α
(
(
6
λ
1
2
)
-
1
(
-
λ
1
2
λ
14
y
+
λ
1
2
λ
14
y
3
+
6
λ
15
y
sinh
⁡
λ
1
-
6
λ
15
sinh
⁡
λ
1
y
λ
1
2
)
×
(
6
λ
1
2
)
-
1
)
cos
⁡
α
(
x
-
t
)
,
where
(44)
A
=
ψ
0
y
y
(
1
)
=
L
sinh
⁡
λ
1
λ
1
cosh
⁡
λ
1
,
λ
2
=
A
sinh
⁡
λ
1
+
λ
1
cosh
⁡
λ
1
-
sinh
⁡
λ
1
+
λ
1
cosh
⁡
λ
1
,
λ
3
=
1
+
A
-
sinh
⁡
λ
1
+
λ
1
cosh
⁡
λ
1
,
λ
4
=
λ
1
2
λ
3
(
1
-
(
L
/
λ
1
)
)
(
τ
0
+
(
1
/
R
e
)
)
,
λ
5
=
(
L
+
λ
1
2
)
λ
2
(
τ
0
+
(
1
/
R
e
)
)
cosh
⁡
λ
1
,
λ
6
=
L
λ
2
cosh
⁡
λ
1
,
λ
7
=
(
(
24
λ
1
2
λ
4
+
60
λ
1
λ
6
)
(
24
λ
1
λ
4
+
60
λ
6
)
sinh
⁡
λ
1
λ
7
=
-
(
24
λ
1
2
λ
4
+
60
λ
1
λ
6
)
cosh
⁡
λ
1
)
λ
7
=
×
(
48
λ
1
4
(
λ
1
cosh
⁡
λ
1
-
sinh
⁡
λ
1
)
)
-
1
,
λ
8
=
(
(
24
λ
1
2
λ
4
+
48
λ
1
λ
6
)
sinh
⁡
λ
1
λ
8
=
-
12
λ
1
2
λ
6
cosh
⁡
λ
1
-
2
λ
5
sinh
⁡
2
λ
1
λ
8
=
+
4
λ
1
λ
5
cosh
⁡
2
λ
1
(
24
λ
1
2
λ
4
+
48
λ
1
λ
6
)
)
λ
8
=
×
(
48
λ
1
4
(
λ
1
cosh
⁡
λ
1
-
sinh
⁡
λ
1
)
)
-
1
,
λ
9
=
(
12
λ
1
2
λ
6
)
cosh
⁡
λ
1
-
12
λ
1
λ
6
sinh
⁡
λ
1
48
λ
1
4
(
λ
1
cosh
⁡
λ
1
-
sinh
⁡
λ
1
)
,
λ
10
=
(
2
λ
5
)
sinh
⁡
λ
1
-
2
λ
1
λ
5
cosh
⁡
λ
1
48
λ
1
4
(
λ
1
cosh
⁡
λ
1
-
sinh
⁡
λ
1
)
,
λ
11
=
(
(
24
λ
1
2
λ
4
)
-
12
λ
1
λ
5
sinh
⁡
2
λ
1
+
3
λ
1
λ
5
sinh
⁡
λ
1
λ
11
=
-
λ
1
λ
5
sinh
⁡
3
λ
1
+
48
λ
1
λ
6
λ
11
=
+
(
12
λ
1
λ
6
-
30
λ
6
)
cosh
⁡
2
λ
1
(
24
λ
1
2
λ
4
)
)
λ
11
=
×
(
48
λ
1
4
(
λ
1
cosh
⁡
λ
1
-
sinh
⁡
λ
1
)
)
-
1
,
λ
12
=
λ
2
tan
α
(
x
-
t
)
C
p
,
λ
13
=
λ
3
tan
α
(
x
-
t
)
C
p
,
λ
14
=
λ
2
sin
α
(
x
-
t
)
D
m
cos
⁡
α
(
x
-
t
)
,
λ
15
=
λ
3
sin
α
(
x
-
t
)
D
m
cos
⁡
α
(
x
-
t
)
.


## 3. Results and Discussion

In this work, we have studied the effect of different parameters of the considered problem on the solutions of the momentum, heat, and mass equations. This discussion is illustrated graphically through a set of Figures2–28. Since Figures 2 and 3 illustrated the influence of the Casson parameter τ
0 on the velocity component v, hence we noticed that the velocity component v increases with the increase of the Casson parameter τ
0 for 0
≤
y
≤
1 and decreases for -
1
≤
y
≤
0. The effect of the magnetic parameter M on the velocity component v is shown in Figures 4 and 5. These figures reveal that the velocity component v decreases with the increase of M at 0
≤
y
≤
1 and increases at -
1
≤
y
≤
0. Figures 6 and 7 depicted the behavior of permeability parameter K on the velocity component v. It is noticed that the velocity component v decreases with K in the region -
1
≤
y
≤
0, and it increases in the region 0
≤
y
≤
1. Figures 8 and 9 showed the effect of τ
0 on the velocity component u; it is clear that the velocity component u increases with increasing of τ
0. Also, the velocity component u decreases when the magnetic parameter M increases, then shown through Figures 10 and 11. From Figures 12 and 13, since the motion is sinusoidal, we have seen that the longitudinal velocity u increases or decreases as the permeability parameter K increases. Figures 14 and 16 illustrated the influence of τ
0 and K on the pressure rise Δ
p
0. These figures show that the pressure riseΔ
p
0 decreases with the increase of both τ
0 and K. Figure 15 displayed the effected of the magnetic parameter M on the pressure rise Δ
p
0; it is noticed that the magnitude of Δ
p
0 increases with M. We can see from Figures 17 and 18 that the temperature θ decreases when M increases in the interval -
1
≤
y
≤
0, while it increases in the interval 0
≤
y
≤
1. We observed from Figures 19, 20, 21, and 22 that the temperature θ increases when K and τ
0 increase in the interval -
1
≤
y
≤
0, and it decreases in the interval 0
≤
y
≤
1. In Figures 23 and 24, it is seen that the concentration distribution ϕ decreases with M in the region -
1
≤
y
≤
0, but it increases in the region 0
≤
y
≤
1. The concentration distribution ϕ increases when K and τ
0 increase in the interval -
1
≤
y
≤
0, and it decreases in the interval 0
≤
y
≤
1, this is shown in Figures 25, 26, 27, and 28.Figure 2
The velocityv is plotted against the distance for different values of τ
o at α
=
0.1
,
ε
=
0.1
,
M
=
5
,
k
=
0.9.Figure 3
The velocity is plotted against the time for different values ofτ
o at α
=
0.1
,
ε
=
0.1
,
M
=
5
,
k
=
0.9.Figure 4
The velocityv is plotted against the distance for different values of M at α
=
0.1
,
ε
=
0.1
,
τ
o
=
2
,
k
=
0.9.Figure 5
The velocityv is plotted against the time t for different values of M at α
=
0.1
,
ε
=
0.1
,
τ
o
=
2
,
k
=
0.9.Figure 6
The velocityv is plotted against the distance for different values of k at α
=
0.1
,
ε
=
0.1
,
M
=
5
,
τ
o
=
2.Figure 7
The velocityv is plotted against the time t for different values of k at α
=
0.1
,
ε
=
0.1
,
M
=
5
,
τ
o
=
2.Figure 8
The velocityu is plotted against the distance for different values of τ
o at α
=
0.1
,
ε
=
0.1
,
M
=
5
,
k
=
0.9.Figure 9
The velocityu is plotted against the time t for different values of τ
o at α
=
0.1
,
ε
=
0.1
,
M
=
5
,
k
=
0.9.Figure 10
The velocityu is plotted against the distance for different values of M at α
=
0.1
,
ε
=
0.1
,
τ
o
=
2
,
k
=
0.9.Figure 11
The velocityu is plotted against the time t for different values of M at α
=
0.1
,
ε
=
0.1
,
τ
o
=
2
,
k
=
0.9.Figure 12
The velocityu is plotted against the distance for different values of k at α
=
0.1
,
ε
=
0.1
,
M
=
5
,
τ
o
=
2.Figure 13
The velocityu is plotted against the time t for different values of k at α
=
0.1
,
ε
=
0.1
,
M
=
5
,
τ
o
=
2.Figure 14
Δ
p
o is plotted against q for different values of τ
o at α
=
0.1
,
ε
=
0.1
,
M
=
5
,
k
=
0.2.Figure 15
Δ
p
o is plotted against q for different values of τ
o at α
=
0.1, ε
=
0.1, M
=
5, k
=
0.2.Figure 16
Δ
p
o is plotted against q for different values of k at α
=
0.1, ε
=
0.1, M
=
5, τ
o
=
2.Figure 17
The temperatureθ is plotted against the distance for different values of M at α
=
0.1, ε
=
0.1, τ
o
=
2, k
=
0.9.Figure 18
The temperatureθ is plotted against the time t for different values of M at α
=
0.1, ε
=
0.1, τ
o
=
2, k
=
0.9.Figure 19
The temperatureθ is plotted against the distance for different values of k at α
=
0.1, ε
=
0.1, τ
o
=
2, M
=
5.Figure 20
The temperatureθ is plotted against the time t for different values of k at α
=
0.1, ε
=
0.1, τ
o
=
2, M
=
5.Figure 21
The temperatureθ is plotted against the distance for different values of τ
o at α
=
0.1, ε
=
0.1, k
=
0.9, M
=
5.Figure 22
The temperatureθ is plotted against the time t for different values of τ
o at α
=
0.1, ε
=
0.1, k
=
0.9, M
=
5.Figure 23
The concentration distributionϕ is plotted against the distance for different values of M at α
=
0.1, ε
=
0.1, τ
o
=
2, k
=
0.9.Figure 24
The concentration distributionϕ is plotted against the time t for different values of M at α
=
0.1, ε
=
0.1, τ
o
=
2, K
=
0.9.Figure 25
The concentration distributionϕ is plotted against the distance for different values of k at α
=
0.1, ε
=
0.1, τ
o
=
2, M
=
5.Figure 26
The concentration distributionϕ is plotted against the time t for different values of k at α
=
0.1, ε
=
0.1, τ
o
=
2, M
=
5.Figure 27
The concentration distributionϕ is plotted against the distance for different values of τ
o at α
=
0.1, ε
=
0.1, M
=
5, k
=
0.9.Figure 28
The concentration distributionϕ is plotted against the time t for different values of τ
0 at α
=
0.1, ε
=
0.1, K
=
0.9, M
=
5.



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*Source: 525769-2014-04-10.xml*