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The Peristaltic Transport of MHD Eyring-Powell

Fluid through Porous Medium in a Three

Dimensional Rectangular Duct 1Hayat A. Ali and

2Ahmed M. Abdulhadi

1Department of Mathematics,

Baghdad University, Baghdad, Iraq.

[email protected] 2Department of Mathematics,

Baghdad University, Baghdad, Iraq.

[email protected]

Abstract This paper devotedly study the peristaltic transport of MHD Eyring-

Powell fluid flow through a porous medium in a three dimensional

rectangular duct with the effect of no slip condition and complaint wall.

The relevant equations(mass, motion) are first modelled and then

simplified under the assumptions of long wavelength and low Reynolds

number approximation. Homotopy perturbation method (HPM)is applied

to obtain an approximate analytic solution for the velocity distribution.

Physical behaviors of different parameter of interest have been presented

graphically for velocity, and trapping phenomena.

International Journal of Pure and Applied MathematicsVolume 119 No. 18 2018, 273-286ISSN: 1314-3395 (on-line version)url: http://www.acadpubl.eu/hub/Special Issue http://www.acadpubl.eu/hub/

273

1. Introduction

In the history of fluid dynamics, the area of peristaltic transportation has

obtained significant attraction because of its considerable contribution in the

fields of engineering and biomechanics as this process remains vital in many

biological mechanism and biomedical industry. Peristalsis is a mechanism of

pumping fluids in ducts when a progressive wave of area contraction or

expansion propagates along the length of a distensible tube containing fluid. It

includes the transportation of urine from the kidney to the bladder, food through

the digestive tract, bile from the gall-bladder into the duodenum, movement of

ovum in the fallopian tube, the design of roller pumps, etc.[14-16].Beside that

non-Newtonian fluids get more attention because of their vigorous use in

industrial applications, particularly in chemical engineering processes and

polymer processing. Due to this importance of non-Newtonian fluids, in 1944,

Powell and Eyring proposed a new fluid model known as Eyring-Powell fluid.

Even though this model is mathematically more complex, but deserves more

consideration because of its distinct advantages over the non-Newtonian fluid

models. Recently, some effective researches have been done on peristaltic flow

of non-Newtonian fluid in three dimensional rectangular channel with

compliant walls with different flow conditions. Reddy et al.[11] have recently

disclosed the theory that the sagittal cross- section of the uterus may be better

analyzed by a channel of rectangular cross- section than a two dimensional

channel. After that Ellahi et al.[13] examined the mathematical analysis of

peristaltic transport of an Eyring-Powell fluid through a porous rectangular

duct. Also Elliahi et al. [15] investigated the peristaltic flow in non- uniform

rectangular duct under the effects of heat and mass transfer. Effect of lateral

walls on peristaltic flow through an a symmetric rectangular duct discussed by

Mekeimer et al. [12]. Akram et al.[14] studied the same effect on peristaltic

flow of a couple stress fluids in a non- uniform rectangular duct. As well as

Abbas et al.[4] examined three dimensional peristaltic flow of hyperbolic

tangent fluid in non-uniform channel having flexible walls. In the same topic

Riaz et al.[10] detected the effect of wall properties on unsteady peristaltic flow

of an Eyring- Powell fluid in three dimensional rectangular duct .

Motivated by the above studies, in this paper the peristaltic flow of MHD

Eyring-Powell fluid in a rectangular duct through a porous medium under the

effect of compliant walls was analyzed. The flow is taken to be unsteady and

then in wave frame under the assumptions of long wavelength and low Renolds

number, the reduced non- dimensional partial differential equation is solved by

employing homotopy perturbation method (HPM) for velocity field function.

Finally the physical feature of the pertinent parameters is discussed by plotting

velocity and stream function.

2. Mathematical Formulation

Consider the unsteady peristaltic transport of an incompressible conductive

Eyring-Powell fluid in a duct of rectangular cross- section through a porous

International Journal of Pure and Applied Mathematics Special Issue

274

medium having the channel width 2𝑑 and height2𝑎. The Cartesian coordinate

system is considered in such a way that 𝑋 − axis is taken along the axial

direction, 𝑌 − axis along the lateral direction, and 𝑍 −axis is along the vertical

direction of rectangular channel. The flow is observed by the peristaltic

sinusoidal waves travelling with velocity 𝑐, propagating along the walls of the

channel.

The peristaltic waves through a duct of rectangular cross- section is prescribed

as

𝑍 = ± 𝑎 + 𝑏 𝑐𝑜𝑠 2𝜋

𝜆 𝑋 − 𝑐𝑡 (1)

In which 𝑎 and 𝑏 are the amplitudes of the waves, 𝜆 is the wavelength, 𝑡 is the

time, and 𝑋 is the direction of wave propagation. The walls parallel to 𝑋𝑍-

plane. The lateral velocity is considered here to be zero since there is no change

in lateral direction of the duct cross- section. The velocity fieldfor unsteady

flows is 𝑉 = 𝑈 𝑋, 𝑌, 𝑍, 𝑡 , 0, 𝑊 𝑋, 𝑌, 𝑍, 𝑡 where 𝑈 and 𝑊 are the axial and

vertical component of the velocity respectively. A uniform magnetic field

𝐵 = (0,0, 𝛽0) is applied normal to the flow, with this assumption the magnetic

Renolds number is small so that the induced magnetic field is neglected.

The governing equations for the flow problem are stated as:

The continuity equation 𝜕𝑈

𝜕𝑋+

𝜕𝑊

𝜕𝑍= 0, (2)

𝑋 − component momentum equation

𝜌 𝜕𝑈

𝜕𝑡+ 𝑈

𝜕𝑈

𝜕𝑋+ 𝑊

𝜕𝑈

𝜕𝑍 = −

𝜕𝑃

𝜕𝑋+

𝜕𝑆 𝑋𝑋

𝜕𝑋+

𝜕𝑆 𝑋𝑌

𝜕𝑌+

𝜕𝑆 𝑋𝑍

𝜕𝑍− 𝜍𝛽0

2𝑈 −𝜇

𝑘0𝑈, (3)

𝑌 − component momentum equation

0 = −𝜕𝑃

𝜕𝑌+

𝜕𝑆 𝑌𝑋

𝜕𝑋+

𝜕𝑆 𝑌𝑌

𝜕𝑌+

𝜕𝑆 𝑌𝑍

𝜕𝑍, (4)

𝑍 − component momentum equation

𝜌 𝜕𝑊

𝜕𝑡+ 𝑈

𝜕𝑊

𝜕𝑋+ 𝑊

𝜕𝑊

𝜕𝑍 = −

𝜕𝑃

𝜕𝑋+

𝜕𝑆 𝑍𝑋

𝜕𝑋+

𝜕𝑆 𝑍𝑌

𝜕𝑌+

𝜕𝑆 𝑍𝑍

𝜕𝑍, (5)

where 𝑆 𝑖𝑗 , 𝑖, 𝑗 = 𝑋, 𝑌, 𝑍 designates the extra stress tensor for Eyring- Powell

fluid on a face whose normal is the 𝑖-axis and act on the face 𝑗-axis,it is

described as[10]

𝑆 𝑖𝑗 = 𝜇𝜕𝑈𝑖𝜕𝑋𝑗

+1

𝛽sinh−1(

1

𝑐1

𝜕𝑈𝑖

𝜕𝑋𝑗 ) , (6)

Where 𝑈𝑖 = 𝑈, 𝑉, 𝑊 , 𝑋𝑗 = 𝑋, 𝑌, 𝑍 .

𝑐1 and 𝛽 represent the material parameters of Eyring- Powell fluid, The term

sinh−1 is approximated using the second order approximation of the hyperbolic


275

sine function as bellows:

sinh−1 1

𝑐1

𝜕𝑈𝑖

𝜕𝑋𝑗 ≅

1

𝑐1

𝜕𝑈𝑖

𝜕𝑋𝑗−

1

6

1

𝑐1

𝜕𝑈𝑖

𝜕𝑋𝑗

3

1

𝑐1

𝜕𝑈𝑖

𝜕𝑋𝑗 ≪ 1 (7)

Then Eq.(6) will be written as

𝑆 𝑖𝑗 = 𝜇𝜕𝑈𝑖𝜕𝑋𝑗

+1

𝛽𝑐1

𝜕𝑈𝑖𝜕𝑋𝑗

−1

6𝛽𝑐13 𝜕𝑈𝑖

𝜕𝑋𝑗

3

(8)

The corresponding complaint boundary conditions are

No- slip condition at the walls

𝑈 = 0 , 𝑎𝑡 𝑌 = ±𝑑 𝑈 = 0, 𝑎𝑡 𝑍 = ±𝐻(𝑋, 𝑡)

(9)

The governing equation for the flexible wall may be presented as [10]

𝐿(𝐻(𝑋, 𝑡)) = 𝑃 − 𝑃0, (10)

Where𝑃0is pressure on outside surface of the wall due to tension in muscle,

which is assumed to be zero here.L is an operator used to represent the motion

of stretched membrane with viscosity damping forces such that

𝐿 = 𝑚𝜕2

𝜕𝑡2 + 𝐷𝜕

𝜕𝑡+ 𝐵1

𝜕4

𝜕𝑋4 − 𝑇𝜕2

𝜕𝑋2 + 𝑘1, (11)

In the above equation, 𝑚 is mass per unit area, 𝐷 is coefficient of the viscosity

damping membrane, 𝐵1 is flexural rigidity of the plate, 𝑇is elastic tension in the

membrane, and 𝑘1 is spring stiffness.

Defining a wave frame 𝑥, 𝑦, 𝑧 moving with the velocity 𝑐 away from the fixed

frame 𝑋, 𝑌, 𝑍, 𝑡 and non- dimensional quantities by the following

transformations

𝑥 = 𝑋 − 𝑐𝑡 , 𝑦 = 𝑌 , 𝑢 = 𝑈 − 𝑐 , 𝑣 = 𝑉, 𝑤 = 𝑊, 𝑝 𝑥, 𝑦, 𝑧 = 𝑃 𝑋, 𝑌, 𝑍, 𝑡 (12)

where 𝑢, 𝑣, 𝑤 are the velocity components in the wave frame 𝑥, 𝑦, 𝑧 .

Introducing the non- dimensional quantities that used to simplify the governing

equations as follows:

𝑥 =𝑥

𝜆 , 𝑦 =

𝑦

𝑑 , 𝑧 =

𝑧

𝑎, 𝑢 =

𝑢

𝑐 , 𝑤 =

𝑤

𝑐𝛿 , 𝑡 =

𝑐𝑡

𝜆, ℎ =

𝐻 𝑋 ,𝑡

𝑎 , 𝑃 =

𝑎2𝑝 𝑥

𝜆𝜇𝑐, 𝛿 =

𝑎

𝜆 , 𝑆 =

𝑎𝑆 𝑥

𝜇𝑐 , 𝑅𝑒 =

𝜌𝑐𝑎

𝜇,

𝐵0

=

1

𝜇𝛽 𝑐1, 𝐴 =

𝐵0

6

𝑐

𝑐1𝑎

2, 𝑘 =

𝑘0

𝑎2 , 𝑀2 =𝜍𝛽0

2𝑎2

𝜇, 𝜙 =

𝑎

𝑏 , 𝛼2 =

1

𝑘+ 𝑀2 , 𝛽1 =

𝑎

𝑑, 𝑢 =

𝜕𝜓

𝜕𝑧, 𝑤 = −

𝜕𝜓

𝜕𝑥 (13)

where 𝑥 , 𝑦 , 𝑧 , 𝑢 , 𝑤 , 𝑡 , ℎ, 𝜓, 𝑃 , 𝑅𝑒, 𝛿, 𝜙, 𝛽1, 𝐵0, 𝐴, 𝑀, 𝑘, describe the dimensionless

axial coordinate, lateral coordinate, transverse coordinate, axial , transverse

velocity components, time, stream function, pressure, Renolds number, wave

number, amplitude ratio, the Eyring-Powell parameters, Hartman number, the

porosity parameter.

Under the assumption of long wavelength 𝜆 → ∞ and low Reynolds

number 𝑅𝑒 → 0 , the governing Eqs. (2)– (5) for the Eyring–Powell fluid in


276

non-dimensional form are achieved as: 𝜕𝑢

𝜕𝑥+

𝜕𝑤

𝜕𝑧= 0, (14)

𝜕𝑝

𝜕𝑥= 𝛽1

2 𝜕𝑆𝑥𝑦

𝜕𝑦+

𝜕𝑆𝑥𝑧

𝜕𝑧− 𝛼2 𝑢 + 1 , (15)

𝜕𝑝

𝜕𝑦= 0 , (16)

𝜕𝑝

𝜕𝑧= 0 , (17)

While the non- dimensional of extra stress components are

𝑆𝑥𝑦 = 1 + 𝑊1 𝜕𝑢

𝜕𝑦 − 𝛽1

2𝐴 𝜕𝑢

𝜕𝑦

3

, (18)

𝑆𝑥𝑧 = 1 + 𝑊1 𝜕𝑢

𝜕𝑧 − 𝐴

𝜕𝑢

𝜕𝑧

3

, (19)

The corresponding compliant boundary conditions in non-dimensional form are

described as:

𝑢 = 0 , 𝑎𝑡 𝑦 = ±1 𝑢 = 0, 𝑎𝑡 𝑧 = ±ℎ(𝑥, 𝑡)

(20)

where ℎ 𝑥, 𝑡 = ±1 ± ∅ 𝑐𝑜𝑠 2𝜋(𝑥 − 𝑡) and 0 ≤ ∅ ≤ 1.

and the dimensionless of wall properties is given as

𝐸1𝜕2

𝜕𝑡2 + 𝐸2𝜕

𝜕𝑡+ 𝐸3

𝜕4

𝜕𝑋4 − 𝐸4𝜕2

𝜕𝑋2 + 𝐸5 ℎ 𝑥, 𝑡 = 𝑝, (21)

In above relation,𝐸1 = 𝑚𝑎3𝑐/𝜆3𝜇, 𝐸2 = 𝐷𝑎3/𝜆2𝜇 , 𝐸3 = 𝐵1𝑎3/𝑐𝜆5𝜇,𝐸4 =

𝑇𝑎3/𝑐𝜆3𝜇 ,and 𝐸5 = 𝑘1𝑎3/𝜆𝜇𝑐 ,are the non-dimensional elasticity parameters.

From Eqs.(16) and (17) one can deduce that the pressure 𝑝 is not a function of

𝑦 and 𝑧.

Differentiating Eq.(15) w.r.t 𝑥 and making use of Eqs.(16),(17),and(21) yields 1

1+𝑊1

𝜕

𝜕𝑥 𝐸1

𝜕2

𝜕𝑡2 + 𝐸2𝜕

𝜕𝑡+ 𝐸3

𝜕4

𝜕𝑋4 − 𝐸4𝜕2

𝜕𝑋2 + 𝐸5 ℎ 𝑥, 𝑡 =

𝛽12 𝜕2𝑢

𝜕𝑦2 +𝜕2𝑢

𝜕𝑧2 − 𝛼2

1+𝑊1 𝑢 + 1 −

3𝐴

1+𝑊1 𝛽1

4 𝜕𝑢

𝜕𝑦

2 𝜕2𝑢

𝜕𝑦2 + 𝜕𝑢

𝜕𝑧

2 𝜕2𝑢

𝜕𝑧2 , (22)

It is noticed here when 𝛽1 = 0 the problem convert to two- dimensional

channel and 𝛽1 = 1 gives the square duct.It is also observed that 𝑊1 = 𝐴 = 0

reproduce the viscous fluid case in rectangular duct. However when 𝛼2 → 0

the solution of Riaz et al. [10] is recovered.

3. Solution of the Problem

The solution of the previously mentioned nonlinear partial differential equation

has beencalculated by homotopy perturbation method [HPM], which is defined

as []

𝐻 𝑢 𝑥, 𝑦, 𝑧, 𝑡 ;𝑞 = 𝐿1 𝑢 𝑥, 𝑦, 𝑧, 𝑡 − 𝐿1 𝑢 0 𝑥, 𝑦, 𝑧, 𝑡 + 𝑞𝐿1 𝑢 0 𝑥, 𝑦, 𝑧, 𝑡 + 𝑞 −3𝐴

1+𝑊1 𝛽1

4 𝜕𝑢

𝜕𝑦

2 𝜕2𝑢

𝜕𝑦2+


277

𝜕𝑢

𝜕𝑧

2 𝜕2𝑢

𝜕𝑧2 −

𝛼2

1+𝑊1 − 𝐶 𝑥, 𝑡 = 0 (23)

where∈ 0,1 , is called homotopy parameter which under the condition that

for 𝑞 = 0 ,the initial guess achieved and for 𝑞 = 1 we seek the final

solution.

Here𝑢 0is an initial guess for the solution of Eq. (22), which satisfies the

boundary conditions Eq.(20). 𝐿1is the linear operator which defined as

𝐿1 𝑢 = 𝛽12 𝜕2𝑢

𝜕𝑦2 +𝜕2𝑢

𝜕𝑧2 − 𝛼2

1+𝑊1 𝑢, (24)

and 𝐶 𝑥, 𝑡 =1

1+𝑊1

𝜕

𝜕𝑥 𝐸1

𝜕2

𝜕𝑡2 + 𝐸2𝜕

𝜕𝑡+ 𝐸3

𝜕4

𝜕𝑋4 − 𝐸4𝜕2

𝜕𝑋2 + 𝐸5 ℎ 𝑥, 𝑡 , (25)

We choose the following initial guess

𝑢 0 𝑥, 𝑦, 𝑧, 𝑡 =(1 −𝑦2)

𝛼𝛽 2 + 𝑧2−ℎ2

𝛼2 , (26)

Let us expand the solution of Eq. (23)as a power series in 𝑞 as follows:

𝑢 = 𝑢0 + 𝑞𝑢1 + 𝑞2𝑢2 + ⋯, (27)

Substituting Eq.(27) into Eq.(23) and then comparing the same power of 𝑞 and

choosing powers of 𝑞 from zero to one, we obtain the following problems

together with the corresponding boundary conditions.

Zeroth- Order Deformation

The differential equation of the zero- order with the boundary conditions is

obtained as follows:

𝐿1 𝑢0 𝑥, 𝑦, 𝑧, 𝑡 − 𝐿1 𝑢 0 𝑥, 𝑦, 𝑧, 𝑡 = 0, (28)

𝑢0 = 0 , 𝑎𝑡 𝑦 = ±1 𝑢0 = 0, 𝑎𝑡 𝑧 = ±ℎ(𝑥, 𝑡)

(29)

From Eq.(23), the zero- order solution is

𝑢0 𝑥, 𝑦, 𝑧, 𝑡 = 𝑢 0 𝑥, 𝑦, 𝑧, 𝑡 =(1 −𝑦2)

𝛼𝛽 2 + 𝑧2−ℎ2

𝛼2 , (30)

First- Order Deformation

The first order equation is constructed as

𝛽12 𝜕2𝑢1

𝜕𝑦2+

𝜕2𝑢1

𝜕𝑧2−

𝛼2

1+𝑊1 𝑢1 −

3𝐴

1+𝑊1 𝛽1

4 𝜕𝑢0

𝜕𝑦

2 𝜕2𝑢0

𝜕𝑦2+

𝜕𝑢0

𝜕𝑧

2 𝜕2𝑢0

𝜕𝑧2 −

𝛼2

1+𝑊1 − 𝐶 𝑥, 𝑡 = 0, (31)

Associated with the boundary conditions

𝑢1 = 0 , 𝑎𝑡 𝑦 = ±1 𝑢1 = 0, 𝑎𝑡 𝑧 = ±ℎ(𝑥, 𝑡)

(32)

by substituting the zero- order solution 𝑢0 𝑥, 𝑦, 𝑧, 𝑡 into Eq.(31) and with some

simplification along with boundary conditions, the resulting partial differential

equation is derived as below:


278

𝛽12 𝜕2𝑢1

𝜕𝑦2 +𝜕2𝑢1

𝜕𝑧2 − 𝛼2

1+𝑊1 𝑢1 =

24𝐴

1+𝑊1 𝑧2

𝛼6 −𝑦2

𝛽12𝛼6 +

𝛼2

1+𝑊1 + 𝐶 𝑥, 𝑡 = 0, (33)

𝑢1 = 0 , 𝑎𝑡 𝑦 = ±1 𝑢1 = 0, 𝑎𝑡 𝑧 = ±ℎ(𝑥, 𝑡)

(34)

Incorporating with the method of eigen function expansion, the final closed

form of solution to Eq.(33).

𝑢1 𝑥, 𝑦, ℎ, 𝑡 = 𝑎𝑛 𝑦 𝐶𝑜𝑠𝜆𝑛𝑧,∞𝑛=1 (35)

where

𝑎𝑛 𝑦 = 𝑐1𝐶𝑜𝑠ℎ 𝜃𝑛𝑦 + 𝑐2 + 𝑐3𝑦2, (36)

𝜃𝑛 = 𝜆𝑛 2

𝛽12 +

𝛼2

𝛽12 1+𝑊1

, (37)

𝑐1 =1

𝐶𝑜𝑠ℎ𝜃𝑛

48𝐴𝐶𝑜𝑠𝑛𝜋

𝜃2𝑛 1+𝑊1 𝜆𝑛𝛽1

4ℎ𝛼6 +96𝐴𝐶𝑜𝑠𝑛𝜋


4ℎ𝛼6 +1

𝜃2𝑛𝛽1

2 48𝐴

1+𝑊1 ℎ𝛼6 2𝐶𝑜𝑠𝑛𝜋

𝜆𝑛 3 −

ℎ2𝐶𝑜𝑠𝑛𝜋

𝜆𝑛 −

2𝐶𝑜𝑠𝑛𝜋

𝜆𝑛ℎ

𝛼2

1+𝑊1 + 𝐶 𝑥, 𝑡 , (38)

𝑐2 =−96𝐴𝐶𝑜𝑠𝑛𝜋


4ℎ𝛼6 −1

𝜃2𝑛𝛽1

2 48𝐴

1+𝑊1 ℎ𝛼6 2𝐶𝑜𝑠𝑛𝜋

𝜆𝑛 3 − ℎ2𝐶𝑜𝑠𝑛𝜋

𝜆𝑛 −

2𝐶𝑜𝑠𝑛𝜋

𝜆𝑛ℎ

𝛼2

1+𝑊1 +

𝐶 𝑥, 𝑡 , (39)

𝑐3 =−48𝐴𝐶𝑜𝑠𝑛𝜋


4ℎ𝛼6 , (40)

The final solution of Eq.(31) will be obtained at 𝑞 → 1, which can be defined as

𝑢 𝑥, 𝑦, 𝑧, 𝑡 =(1 −𝑦2)

𝛼𝛽 2 + 𝑧2−ℎ2

𝛼2 + 𝑎𝑛 𝑦 𝐶𝑜𝑠 2𝑛−1 𝜋

2ℎ 𝑧,∞

𝑛=1 (41)

4. Graphical Results and discussion

In this section, we inspect physical effect of various parameters on the axial

velocity and the stream function are recorded graphically.

Velocity Profile

The velocity profile is sketched for both three and two dimensional channel

upon the variation of important parameters involved in the expression of

velocity field via Figures. (1)-(6). Figs. (1), and (2) show the variation of

velocity axial against the wall tension parameter𝐸1, and viscous damping𝐸3

respectively. It is depicted that the velocity profile decreases for ascending

values of 𝐸1but An increasing function of the velocity field is seen for

ascending values of 𝐸3. Completely opposite influence for Eyring-Powell fluid

parameters 𝐴 and 𝑊1on the axial velocity is noticed through Figs.(3) and (4)

respectively, i.e. the velocity axial enhances for greater values of 𝐴and

decreasing against𝑊1enhancement. One can observed from Fig. (5) a decreasing

function for velocity field with the rise of Hartman number 𝑀. Fig.(6) describes

that when the porosity parameter 𝑘 increases the velocity profile increases very


279

rapidly.

(a) (b)

Fig. 1: Axial velocity 𝑢 versus 𝑧- direction for different values of wall

tension𝐸1 and with fixed values of parameters {𝐸2 = 0.2, 𝐸3 = 0.1, 𝐸4 =0.1, 𝐸5 = 0.3, 𝑊 = 0.9, 𝐴 = 2, 𝑀 = 0.8, 𝑘 = 1, 𝛽1 = 0.02, 𝜙 = 0.09, 𝑥 =0.02, 𝑡 = 0.01, 𝑦 = 0} in (a) Two dimension (b) Three dimension.

(a) (b)

Fig. 2: Axial velocity 𝑢 versus 𝑧- direction for different values of viscous

damping𝐸3 and with fixed values of parameters {𝐸1 = 0.1, 𝐸2 = 0.2, 𝐸4 =0.1, 𝐸5 = 0.3, 𝑊 = 0.9, 𝐴 = 2, 𝑀 = 0.8, 𝑘 = 1, 𝛽1 = 0.02, 𝜙 = 0.09, 𝑥 =0.02, 𝑡 = 0.01, 𝑦 = 0} in (a) Two dimension (b) Three dimension.

(a) (b)

Fig. 3: Axial velocity 𝑢 versus 𝑧- direction for different values of Eyring-

Powell fluid parameter𝐴,and with fixed values of parameters {𝐸1 = 0.1, 𝐸2 =0.2, 𝐸3 = 0.3, 𝐸4 = 0.1, 𝐸5 = 0.2, 𝑊1 = 0.9, 𝑀 = 0.8, 𝑘 = 1, 𝛽1 = 0.02, 𝜙 =0.09, 𝑥 = 0.02, 𝑡 = 0.01, 𝑦 = 0} in (a) Two dimension (b) Three dimension.

"E10.1 "

"E10.5 "

"E10.9 "

1.0 0.5 0.0 0.5 1.0

1524.6

1524.8

1525.0

1525.2

1525.4

1525.6

z

u

"E30.1 "

"E30.15 "

E30.2 "

1.0 0.5 0.0 0.5 1.0

1524.5

1525.0

1525.5

1526.0

1526.5

z

u

A2 "

A4"

A6"

1.0 0.5 0.0 0.5 1.0

1525

1526

1527

1528

1529

z

u


280

(a) (b)

Fig. 4: Axial velocity 𝑢 versus 𝑧- direction for different values of Eyring-

Powell fluid parameter𝑊1,and with fixed values of parameters {𝐸1 =0.1, 𝐸2 = 0.2, 𝐸3 = 0.3, 𝐸4 = 0.1, 𝐸5 = 0.2, 𝐴 = 0.9, 𝑀 = 0.8, 𝑘 = 1, 𝛽1 =0.02, 𝜙 = 0.09, 𝑥 = 0.02, 𝑡 = 0.01, 𝑦 = 0} in (a) Two dimension (b) Three

dimension.

(a) (b)

Fig. 5: Axial velocity 𝑢 versus 𝑧- direction for different values of Hartman

number𝑀,and with fixed values of parameters {𝐸1 = 0.1, 𝐸2 = 0.2, 𝐸3 =0.3, 𝐸4 = 0.1, 𝐸5 = 0.2, 𝐴 = 0.9, 𝑊1 = 0.8, 𝑘 = 1, 𝛽1 = 0.02, 𝜙 = 0.09, 𝑥 =0.02, 𝑡 = 0.01, 𝑦 = 0} in (a) Two dimension (b) Three dimension.

(a) (b)

Fig. 6: Axial velocity 𝑢 versus 𝑧- direction for different values of porosity

parameter 𝑘,and with fixed values of parameters {𝐸1 = 0.1, 𝐸2 = 0.2, 𝐸3 =0.3, 𝐸4 = 0.1, 𝐸5 = 0.2, 𝐴 = 0.9, 𝑊1 = 0.8, 𝑀 = 0.8, 𝛽1 = 0.02, 𝜙 =0.09, 𝑥 = 0.02, 𝑡 = 0.01, 𝑦 = 0} in (a) Two dimension (b) Three dimension.

Trapping Mechanism

Trapping phenomena is also an important part of peristaltic motion which is a

formation of an inside moving bolus that are bounded by multiple streamlines.

For this purpose, Figs.(7)-(10) are drawn for multiple values of important

"W10.9 "

"W11.9 "

"W12.9 "

1.0 0.5 0.0 0.5 1.0

1524.6

1524.8

1525.0

1525.2

1525.4

1525.6

1525.8

z

u

M0.01 "

M0.012 "

M0.013 "

1.0 0.5 0.0 0.5 1.0

2500.0

2500.5

2501.0

z

u

k0.1 "

k0.2 "

k0.3 "

1.0 0.5 0.0 0.5 1.0

806

808

810

812

z

u


281

parameters that impact on the trapping streamlines. From Fig.(7) we deduced

that grown up of𝐴magnitude causes two opposite effect on the trapping bolus

i.e. the magnitude of the bolus in the left side of the plane −0.7 < 𝑥 ≤−0.4 increase while the right side of the plane for −0.2 < 𝑥 ≤ 0.27 decreases

in diameter and number. The streamlines pattern for multiple values of

Hartman number 𝑀 are plotted in Fig.(8). It is found that for ascending values

of 𝑀, the trapping bolus reduces in size and more bolus create. The influence of

porosity parameter 𝑘 on the trapping bolus is seen in Fig.(9). It can be observed

that the diameter of trapped bolus rapidly increases with arising of 𝑘 magnitude.

Fig.(10) depicted that the volume of the bolus decrease but more boluscreate

when the values of aspect ratio 𝛽1 increase.

(a) (b)

Fig. 7: Streamlines for different values of Eyring-Powell parameter𝐴,and with

fixed values of parameters 𝐸1 = 0.1, 𝐸2 = 0.3𝐸3 = 0.1, 𝐸4 = 0.2, 𝐸5 =

0.3, 𝑊1 = 0.8, 𝑀 = 0.8, 𝛽1 = 0.2, 𝑘 = 1, 𝜙 =π

9, 𝑡 = 0.25, 𝑦 = 0.1 (a) 𝐴 =

0.9(b)𝐴 = 1.9 .

(a) (b)

Fig. 8: Streamlines for different values of Hartman number𝑀,and with fixed

values of parameters 𝐸1 = 0.1, 𝐸2 = 0.3𝐸3 = 0.1, 𝐸4 = 0.2, 𝐸5 = 0.3, 𝐴 =

0.8, 𝑊1 = 0.8, 𝛽1 = 0.2, 𝑘 = 1, 𝜙 =π

9, 𝑡 = 0.25, 𝑦 = 0.1 (a) 𝑀 = 0.2(b)𝑀 =

1 .


282

(a) (b)

Fig. 9: Streamlines for different values of porosity parameter 𝑘,and with fixed

values of parameters 𝐸1 = 0.1, 𝐸2 = 0.3𝐸3 = 0.1, 𝐸4 = 0.2, 𝐸5 = 0.3, 𝐴 =

0.8, 𝑊1 = 0.8, 𝛽1 = 0.2, 𝑀 = 0.9, 𝜙 =π

9, 𝑡 = 0.25, 𝑦 = 0.1 (a) 𝑘 = 0.2(b)𝑘 =

1 .

(a) (b)

Fig. 10: Streamlines for different values of aspect ratio parameter𝛽1,and with

fixed values of parameters 𝐸1 = 0.1, 𝐸2 = 0.3𝐸3 = 0.1, 𝐸4 = 0.2, 𝐸5 =

0.3, 𝐴 = 0.8, 𝑊1 = 0.8, 𝑘 = 0.2, 𝑀 = 0.9, 𝜙 =π

9, 𝑡 = 0.25, 𝑦 = 0.1 (a)𝛽1 =

0.7(b)𝛽1 = 0.9 .

5. Conclusion

In this paper, a mathematical model of the peristaltic transport of Eyring-Powell

fluid flows in a three dimensional rectangular duct with porous medium under

the influence of magnetic field has been investigated. No- slip velocity


283

condition and wall properties are taken into account. A long- wavelength and

low-Renolds number approximations are employed to simplify the governing

equation into a high non-linear partial differential equation and by solving the

resulted equation using homotopy perturbation method (HPM), the series

solution of velocity field is found. In this chapter the scope and applicability of

earlier results given by Riaz et al.[5] was extended to effect of magnetic field

and porous medium. The main results and conclusions of this investigation are

summarized as follows:

The velocity axial is an increasing function with ascending values of

viscos damping 𝐸3, and decreasing function with wall rigidity 𝐸1.

Totally opposite impact for the Eyring-Powell parameters 𝐴 and 𝑊1 on

the velocity profile is noticed, enhancement of the first one increase the

velocity profile while for ascending values of the second parameter

reduces the velocity.

The trapped bolus decreases in magnitude but enclosed by more

streamlines due enhancement of aspect ratio𝛽1values.

Enhancement of the Hartman number has a reduction effect on the

trapped bolus magnitude however bolus' number increases in the right

side of the plane only.

The flow characterization (velocity profile and streamlines) are very

sensitive against a very little variation of porosity and aspect ratio

parameters 𝑘, 𝛽1respectively.

References

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[2] J.H. He, A coupling method of homotopy technique and a perturbation technique for nonlinear problems, Int. J. Non -linear Mech. 35 (2000) 37_43.

[3] E. Babolian, A. Azizi, J. Saeidian"Some notes on using the homotopy perturbation method for solving time-dependent differential equations"J.Mathematical and Computer Modelling,50, (2009),pp. 213-224.

[4] M. Ali Abbas , Y.Q. Bai , M.M. Bhatti , M.M. Rashidi ,"Three dimensional peristaltic flow of hyperbolic tangent fluid in non-uniform channel having flexible walls"Alexandria Engineering Journal, (2016), 55, 653–662.

[5] Arshad Riaz, R. Ellahi , S. Nadeem"Peristaltic transport of a Carreau fluid in a compliant rectangular duct "Alexandria Engineering Journal, (2014) ,53, 475–484.

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phenomenon through a rectangular duct"Appl Nanosci J, (2014) 4,613–624.

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[11] MVS. Reddy, M. Mishra, S. Sreenadh, AR. Rao," Influence of lateral walls on peristaltic flow in a rectangular duct", J. Fluids Eng, 127, PP. 824-827,(2005).

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[13] R. Ellahi, M.M. Bhatti, A. A. Khan," Mathematical analysis of peristaltic transport of an Eyring-Powell fluid through a porous rectangular duct", J. Wulfenia, 22(1),PP. 266-283,(2015).

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[15] R. Ellahi, M. M. Bhatti, K. Vafai," Effects of heat and mass transfer on peristaltic flow in a non- uniform rectangular duct", Int.J. Heat Mass Transfer, 71, PP. 706-719,(2014).

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