effect of magnetic field and wall properties on

International Mathematical Forum, Vol. 6, 2011, no. 27, 1345 - 1356

Effect of Magnetic Field and Wall Properties on

Peristaltic Motion of Micropolar Fluid

N. A. S. Afifia, S. R. Mahmoudb1*and H. M. Al-Isedec

aDepartment of Mathematics, Faculty of Education Ain Shams University, Egypt

bMathematics Department, Faculty of Education King Abdul Aziz Univ., Saudi Arabia.

*E-mail: [email protected]

cDepartment of Mathematics, Faculty of Sciences Umm Al-Qura University, Saudi Arabia

Abstract

In this paper, the effect of magnetic field on wall properties for peristaltic motion of micropolar fluid in a flexible tubes is studied. The governing equation are nonlinear in nature, a regular perturbation method is used to obtain linearized system of coupled differential equations which are then solved numerically using Rung Kutta method. Results have been discussed for time mean velocity profile to observe the effect of magnetic field on wall properties in the presence of micropolarity effects. In the case of no magnetic field, our result is in agreement with previous investigations. Keywords: Micropolar fluid; Magnetic Field; Peristalsis; Free pumping; Time mean flow. I. Introduction

Several theoretical and experimental studies have been conducted to understand peristaltic action since the first investigation made an experimental study of the mechanics of peristaltic transport. M. Abd-Alla and Mahmoud [1] solved magneto-thermoelastic problem in rotating non-homogeneous orthotropic hollow cylindrical

1Permanent address: Mathematics Department, Faculty of Science, University of Sohag, 82524, Egypt.

constant magnetic field

1346 N. A. S. Afifi, S. R. Mahmoud and H. M. Al-Isede

under the hyperbolic heat conduction model. Influences of Rotation, Magnetic Field, Initial Stress and Gravity on Rayleigh Waves in a Homogeneous Orthotropic Elastic Half-Space is investigated by Abd-Alla et al. [2-3]. The effect of the rotation on wave motion through cylindrical bore in a micropolar porous cubic crystal and the effect of the rotation on waves in a cylindrical borehole filled with micropolar fluid is studied by Abd-Alla et al. [4]. Interaction of peristaltic flow with pulsatile magneto-fluid through a porous medium and aspects of a magneto-fluid with suspended particles analysed by Afifi et al [5-6]. A subclass of these microfluids is known as micropolar fluids where the fluid microelements are considered to be rigid discussed by Eringen [7]. Peristaltic transport of a Newtonian fluid in an asymmetric channel discussed by Mishra et al [8]. A non-Newtonian fluid flow model for blood flow through a catheterized artery-steady flow studied by Sankar and Hemalatha [9]. Effect of Deborah number and phase difference on perist-altic transport of a third-order fluid in an asymmetric channel discussed by Haroun [10]. Mekheimer [11] studied peristaltic flow of a magneto-micropolar fluid effect of induced magnetic field. There are several investigations to study the effect of the magneto hydrodynamic (MHD) peristaltic flow of a fluid is of interest in connection with certain problems of the movement of conductive physiological fluids e.g. the blood and blood pump machines. Magnetic fluid model induced by peristaltic waves studied by Siddiqui et al.[12]. II. Formulation of the problem

Consider an axisymmetric flow of unsteady incompressible micropolar fluid under the effect of a constant magnetic field 0,0, in an infinite tube of uniform radius d, with sinusoidal waves travelling along the boundary of tube with speed c, small amplitude a and wavelength λ , Fig (1). The wall is assumed to be a flexible membrane.

Fig.(1). Geometry of cylindrical tube with peristaltic wave motion of wall under the effect the magnetic field

Effect of magnetic field and wall properties 1347 The governing equations for the peristaltic motion of an incompressible micropolar fluid in the circular cylindrical coordinates (R, ,Z) are [8]:

0, 1

W , 2

U W∂∂Z

22

1

, 3

U W 21

W∂R , 4

where U, are velocity components in the R and directions respectively. is the component of microrotation in the direction. is pressure, is time and

denotes the density. is dynamic viscosity coefficient. is microinertia constant, and are the viscosity coefficients of micropolar fluid. B0 is constant magnetic field and is the electrical conductivity. At the boundary, the fluid is subjected to the motion of the wall which is of the form:

Z, τ 2 ⁄ , 5

where is the radial displacement from mean position (d) of the wall (Fig.1). It will be assumed that the wall is inextensible and the particles on the wall have no longitudinal displacement and only their lateral motions normal to the under formed positions occur. Further, it is assumed that the microrotation at the wall is zero. Thus we have no slip and no spin conditions on the wall as:

W 0 0 , . 6

Using the theory of stretched membrane with viscous damping force suggested the dynamic boundary condition at the axisymmetric motion of the flexible wall: The dynamic boundary condition at , gives:

∂∂Z

∂L ζ∂Z , 7

where , is the tension in the membrane, is the mass per unit area and is the coefficient of viscous damping force . Using Eq.(3) in Eq.(7) we get at , :

∂L∂Z U W

22

1

, 8


It is assumed that the velocity and the microrotation are finite at 0. Introducing stream function in terms of :

U1

, W1

, 9

and eliminating the pressure from Eqs. (2) - (3), we get differential equations for Ψ and by using d as characteristic length and c as character-ristic velocity:

, , , , , Ψ ,

, ,

, . 10 After ignoring the microinertia effects the governing equations and boundary conditions in non-dimensional form written as:

∂t Ψ1

Ψ Ψ2r Ψ

1Ψ

1Ψ Ψ

22 Ψ

Ψ 1 Ψ 2r

∂g∂r

gr , 11

1r Ψ 2 2 1

2r

∂g∂r

gr 0. 12

The boundary conditions at 1 z, t are : Ψ 0, 0, 13

1Ψ

1Ψ Ψ

1Ψ Ψ

1Ψ Ψ

Ψ

1 22 Ψ

1Ψ

∂g∂r

gr , 14

Where: 1

, z, t cos ,

,

ratio of amplitude to mean radius of the cylindrical wavy wall ,

wave number, Reynolds number,

magnetic parameter , , ,

m , , 2⁄ , 2

⁄.

As observed in the classical peristaltic flow, , and m are the non-dimensional quantities related to the wall motion through the dynamic boundary condition Eq.(14). The parameters and represent the dissipative and rigidity feature of wall

Effect of magnetic field and wall properties 1349

respectively where as m indicates the stiffness property of wall. 0 implies that the wall move up and down with no damping force on it and hence indicates the case of elastic wall. The parameters and are the non-dimensional quantities due to micropolar fluid flow . The number characterizes the coupling of Eqs.(11) and (12). As tends to zero , becomes zero and Eqs. (11) and (12) are uncoupled. Further when and are zero, that is, when becomes zero and tends to infinity, Eqs.(11) and (12) reduce to the classical Navier–Stokes equation.

III. Method of solution

It may be noted that the flow is quite complex because of nonlinearity of the governing equations and the dynamic boundary condition. Thus to solve Eqs.(11) and (12). for the velocity field and the microrotation component, we attempt an approximate solution for Ψ and g as a power series in terms of . Thus, Ψ and g are taken in the following form:

Ψ Ψ Ψ Ψ Ψ , g g g g g , 15 similar expression is assumed for . Substituting Eq.(15) in Eqs.( 11) and (12) and comparing the coefficients of like powers of on both sides of the equations, we obtain sets of the governing equations and boundary conditions for Ψ , Ψ , Ψ , … and g , g , g ,.... .It may be noted that for

0, the zeroth order equations correspond to Hagen–Poiseuille flow of micro polar fluid in a tube [8]. IV. Free pumping

In the following we shall restrict our analysis to the case of pure peristalsis (free pumping) which corresponds to 0 that is the zeroth order pressure gradient is zero. Thus, for this case the zeroth order solution as given by Eringen [5], reduces to: Ψ 0 g 0. 16

IV.I The first order equation:

Governing equations and the corresponding boundary conditions are given by

∂t2

2Ψ 1 Ψ

g2r

∂g∂r

gr , 17

1r 2g 2 1 g

2r

∂g∂r

gr 0, 18


1 0 , g 1 0, 19

12

2 1 1 1 1

1 g 1 g 112 e e , 20

where: and . IV.II The second order equation:

∂t1Ψ Ψ

2r Ψ

1Ψ

1Ψ Ψ

22

1g

2r

∂g∂r

gr 21

1r 2g 2 1 g

2r

∂g∂r

gr 0, 22

1 1 cosα z t 0 , g 1 g 1 cosα z t0, 23

1 1 1 1 1 1 1

2

2 1 1 1 11 1 1

2 1 2 112 3 1 3 1

1 1

g 1 g 1 g 1 2g 1 14 . 24

IV.III First order solution:

the first order flow quantities suggest the following form for and g :

, 25 g , 26

where the asterisk denotes complex conjugate. Substituting Eqs.(25) and (26) in the governing equations of and g and separating the harmonics, we have the following governing equations for and ξ :

1

1

2 1 1

0, 27

Effect of magnetic field and wall properties 1351

2 11 1 1

20, 28

where , . The equations governing and are conjugate. The boundary conditions reduce to:

1 0, 1 0, 29

1 1 2 1 2 μ. 30

Where two more conditions will be taken from the assumption that flow is axisymmetric at 0 . The number within the parentheses indicates the derivative with respect to , ( ). The equations governing and are conjugate to Eqs. (27) - (30). IV.IV Second order solution :

Similarly for Ψ and g , we assume the solution of the form:

, 31 g . 32

Substituting Eqs. (31) and (32) in Eqs. (21) - (24), we have the following equation for and :

1 12

1 1 ξ

22

1

1

22

1

, 33

2 1 2 0, 34

with boundary conditions 1 1 1

0, 35


1 1 10, 36)

1 1 2 12 2 μ

22 μ

1 1 1 112 1 1

1 1 1 1 1 1

1

1 1 1 1 . 37

V. The time mean flow

To calculate the time mean flow for the case of pure peristalsis free pumping The time mean flow is defined as the axial velocity averaged over the period of oscillation of the wave propagation imposed on the flexible wall,

1

1

1Ψ .

For free pumping case 0, and therefore . 38

In the following, we shall find . To obtain and , we shall first solve for and . Since it is not possible to get closed form solution, we solve Eqs. (27) and (28) to get and , numerically using Rung Kutta method. Solution procedure for getting and . VI. Numerical results and discussion

In the absence of magnetic field micropolar fluid parameter this analysis reduces to approximate analytical solution of peristaltic flow of a Newtonian fluid in cylindrical tubes with wall properties . Therefore, the present Rung Kutta technique has been validated. The details of the validation procedure are provided. Fig.(2-3) show the effect of magnetic field and on for 0.5 and various values of , 1.0 and 0.2 . It is seen here that for given M and ,

increases with increasing and decreasing . It can be interpreted that increase in the micropolar viscosity coefficient reduces the mean velocity. When 1.0 and 0.2 , is almost constant. Fig.(2-3) show the distribution of with for α 0.5, 1.0, 1.0, 0, 5.0 and

Effect of magnetic field and wall properties 1353 0.2. For the mean velocity profile sharply decreases as and increases. The influence of are seen by comparing fig 2.0 and fig. 3.0 Fig.(4-5) show the distribution of with for α 1.0, 1.0 and 0.2.. Comparing Fig.(2) and (4), it is observed that decreases with increase in and

. Fig.(5) Show the distribution of with for α 1.0, 1.0, 0.2.. Comparing Fig.(4) and (5), it is observed that decreases with increase in , and . The influence of , and is seen by comparing fig 4.0 and fig. 5.0 In case of dissipative wall ( 0): in Fig.(6), we see the influence of magnetic field

dissipative nature of the wall,( 0) on the flow reversal by taking 1.0 and 0. To see this effect, that is , time mean flow at the boundary is

plotted versus . The analysis is done for 0.2. For, 6.0 and 1.0, it is observed that no flow reversal takes place at the boundary for smaller values of and

, for any value of , 1 is negative for larger values of and , it keeps on decreasing as and increases. Increasing the ratio of viscosity coefficient decreases the values of . Fig. (7) show effect of magnetic field on 1 for case dissipative wall 1.0 and 0. The effect of is to decrease the value of at small values of . It is observed that from comparing Figs(6-7) the effect of is significant on the variation of over .

Fig.(2): Show the distribution of with

for α 0.5, 1.0, 1.0


for α 0.5, 5.0, 1.0


Conclusions

In the present study, an analysis of peristaltic of micropolar motion of fluid in a circular cylindrical tubes with dynamic boundary condition has been presented for the case of free pumping under the effect of magnetic field. The axisymmetric study agrees with peristaltic motion in a two-dimensional channel. It is observed that 1 decreases when increases and as M decreases. For non-zero viscous damping, flow reversal is found at the wall of flexible tube. Numerical results are presented in a graph


for α 1.0, 1.0, 1.0


for α 1.0, 1.0, 1.0

Fig.(6): Show effect of viscous damping

on for case dissipative wall 1.0

Fig.(7): Show effect of viscous damping

on for case dissipative wall 1.0

Effect of magnetic field and wall properties 1355 which exhibits the fact that the effect of magnetic field is to change of time mean velocity profile. Acknowledgement. Special thanks are due to Professor A. M. Abd-Alla [1-3] for his helpful discussions and valuable remarks.

References [1] A. M. Abd-Alla and S. R. Mahmoud "Magneto-thermoelastic problem in rotating non-homogeneous orthotropic hollow cylindrical under the hyperbolic heat conduction model", Meccanica, 45; 4; 451-462, (2010). [2] A.M. Abd-Alla, S. R. Mahmoud and M.I.R.Helmi, "Effect of initial stress and magnetic field on Propagation of shear wave in non homogeneous Anisotropic medium under gravity field", The Open Applied Mathematics Journal,3; 49-56, (2009). [3] A.M. Abd-Alla, S. R. Mahmoud and M.I.R.Helmi, " Influences of Rotation, Magnetic Field, Initial Stress and Gravity on Rayleigh Waves in a Homogeneous Orthotropic Elastic Half-Space.", Applied Mathematical Sciences, vol.4; 2; 91 – 108, ( 2010). [4] N. A.S Afifi,. & Gad,N. S ,'' Interaction of peristaltic flow with pulsatile magneto-fluid through a porous medium' ; Acta Mech (149)229-237, (2001). [5] N.A .S Afifi, , ''Aspects of a magneto-fluid with suspended particles ''; Appl Math , Infor Sci, (1) 103-112, (2007). [6] J.C Burns,. & Parkes ,T.,'' Peristaltic motion '' ;J. Fluid Mech (29) 731, ( 1967). [7] A. C Eringen , ''Theory of micropolar fluids ''. J. Math . Mech (16 ) 1–18, (1966). [8] M.Mishra, & Rao, A.R., "Peristaltic transport of a Newtonian fluid in an asymmetric channel", Z. Angew. Math. Phys. (54) 532–550, (2004). [9] D.S Sankar,. & Hemalatha , K."A non-Newtonian fluid flow model for blood flow through a catheterized artery-steady flow" .Appl Math Model (37)253-218, (2006). [10] M.H Haroun,. "Effect of Deborah number and phase difference on peristaltic transport of a third-order fluid in an asymmetric channel ", Commun . Nonlinear. Sci. Numer. Simul. (12) 1464–1480, (2007).


[11] ,Kh. S Mekheimer.:'' Peristaltic flow of a magneto-micropolar Fluid: Effect of Induced Magnetic Field'' .J. Appl Math, (2008). [12] A.M Siddiqui,. Hayat, T . & Khan M." Magnetic fluid model induced by peristaltic waves" J Phys Soc Jpn .7,32,14–27, (2004). Received: December, 2010

effect of magnetic field and wall properties on

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