Peristaltic Transport of Micropolar Fluid in a Tubes Under Influence of Magnetic Field and Rotation A.M.Abd-Alla a, G.A.Yahya b,c, H.S.

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1 International Journal of Engineering & Technology IJET-IJENS Vol: 11 No: 1 17 Peristaltic Transport of Micropolar Fluid in a Tubes Under Influence of Magnetic Field and Rotation A.M.Abd-Alla a, G.A.Yahya b,c, H.S.Al :Osaimi a a Mathematics Department Faculty of Science Taif University-K.S.A. b Physics Department Faculty of Science Taif University-K.S.A. c Physics Department Faculty of Science in Aswan Sauth Valley University- Egypt Abstract-- In this paper, The peristaltic flow of micropolar fluid in a flexible tube with viscoelastic is studied under effect of magnetic field and rotation. The Runge-Kutta method is used to solve the governing equations of motion resulting from a perturbation technique for small values of amplitude ratio. The time mean axial velocity profiles are presented for the case of free pumping and analyzed to observe the influence of rotation and magnetic field for various values of radial r and wave numbe. The numerical results of the time mean velocity profile are discussed in detail for homogeneous fluid under effect of magnetic field and rotation for different cases by figures. The results indicate that the effect of rotation and magnetic field are very pronounced. Numerical results are given and illustrated graphically. Index Term-- Peristaltic motion, micropolar fluid, magnetic field, rotation, Runge-Kutta method, velocity profiles. I. INTRODUCTION Peristaltic transport is a form of material transport induced by a progressive wave of area contraction or expansion along the length of a distensible tube, mixing and transporting the fluid in the direction of the wave propagation. This phenomenon is known as peristalsis. It plays an indispensable role in transporting many physiological fluids in the body such as urine transport from kidney to bladder, the movement of chime in the gastrointestinal tracts, the transport of spermatozoa in the ductus efferentes of the male reproductive tract, the movement of ovum in the fallopian tubes, the swallowing of food through esophagus and the vasomotion of small blood vessels. Many modern mechanical devices have been designed on the principle of peristaltic pumping for transporting fluids without internal moving parts, for example, the blood pump in the heart-lung machine and the peristaltic transport of noxious fluid in nuclear industry. The transportation of fluid by a wave of contraction and expansion from a region of lower pressure to higher pressure is called peristaltic pumping. The study of peristaltic motion gained considerable interest because of its extensive applications. Thermal convection in a microplar fluid which occupie arotating horizontal layer has been studied by Qin and Kaloni [1] and by Sharma and Kumar [2]. Elshehawey and Sobh [3] introduced peristaltic viscoelastic fluid motion in a tube. Peristaltic motion in circular cylindrical tubes: Effect of wall properties is studied by Muthu, et al [4]. Abd El Naby, et al [5] studied the separation in the flow through peristaltic motion of a carreau fluid in uniform tube. Hayat, et al [6] studied Couette and Poiseuille flows of fluid with magnetic field. Siddiqui, et al [7] discussed magnetic fluid model induced by peristaltic waves. Mekheimer [8] described the peristaltic flow of blood under effect of a magnetic field in a non-uniform channels. Hariharan [9] studied peristaltic pressure-flow relationship of non-newtonian fluids in distensible tubes with limiting wave forms. In another attempt, Haroun [1] described the effect of relaxation and retardation time on peristaltic transport of the Oldroydian viscoelastic fluid. Srinivas and Pushparaj [11] studied the nonlinear peristaltic transport in an inclined asymmetric channel.a mathematical model for blood flow in magnetic field is studied by Tzirtzilakis [12]. Radhakrishnamacharya and Srinivasulu [13] studied the influence of wall properties on peristaltic transport with heat transfer. Recently, Srinivasacharya, et al [14] Influence of wall properties on peristalsis in the presence of magnetic field.wang, et al [15] studied MHD peristaltic motion of a Sisko fluid in a symmetric channel..ali,et al [16] investigated peristaltic mechanism of a Maxwell fluid in an asymmetric channel. Peristaltic flow of viscoelastic fluid with fractional Maxwell model through a channel is studied by Tripathi, et al.[17]. In this paper, we discuss the effect of magnetic field and rotation on peristaltic motion of micropolar fluid in a flexible tube with uniform magnetic field. Here the governing equations are nonlinear in nature, we used regular perturbation method to obtain linearized system of coupled differential equations which are then solved numerically using Runge-Kutta method. Results have been discussed for time mean velocity profile to observe the effect of rotation and magnetic field in the presence of micropolarity effects. The numerical results of the time mean velocity profile are discussed in detail for homogeneous fluid under effect of magnetic field and rotation for different cases by figures. II. FORMULATION OF THE PROBLEM Consider an axisymmetric flow of unsteady incompressible micropolar fluid in an infinite rotating tube of uniform radius d, with sinusoidal waves travelling along the boundary of tube with speed c, small amplitude a,the constant angular velocity'ω' and wavelength λ, (Fig. 1) The wall is assumed to be a flexible membrane.

2 International Journal of Engineering & Technology IJET-IJENS Vol: 11 No: 1 18 assumed that the microrotation at the wall is zero. Thus we have no slip and no spin conditions on the wall as: 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: Where is the tension in the membrane, is the mass per unit area and is the coefficient of viscous damping force. Using equation (3) in equation (7) we get at : Fig. 1. Geometry of cylindrical tube with peristaltic wave motion of wall The governing equations for the peristaltic motion of an incompressible micro polar fluid in the circular cylindrical coordinates (R,,Z) are It is assumed that the velocity and the micro rotation are finite at Introducing stream function in terms of : and eliminating the pressure from equations (2) and (3), we get differential equations for and by using d as characteristic length and c as character-ristic velocity: After ignoring the microinertia effects the governing equations and boundary conditions, in non-dimensional form written as: where are the velocity components in the and directions respectively. is the component of microrotation in the direction. is the pressure, is the time and is the density. is the dynamic viscosity coefficient. the is microinertia constant, and are the viscosity coefficients of micropolar fluid, is defined as Lorenttz s force, which may be written as, is the magnetic permeability and is the intensity of the uniform magnetic field, parallel to Z-axes. At the boundary, the fluid is subjected to the motion of the wall which is of the form: where is the radial displacement from mean position (d) of the wall as in 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 undeformed positions occur. Further, it is The boundary conditions at are : Where:

3 International Journal of Engineering & Technology IJET-IJENS Vol: 11 No: 1 19 As observed in the classical peristaltic flow. and are the non-dimensional quantities related to the wall motion through the dynamic boundary condition equation (14). The parameters and represent the dissipative and rigidity feature of wall respectively, where as indicates the stiffness property of wall []. 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 equations (11) and (12). As tends to zero, becomes zero and equations (11) and (12) are uncoupled. Further when and are zero, that is, when becomes zero and tends to infinity, equations (11) and (12) reduce to the classical Navier Stokes equation. b- The Second order equation Where: 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 equations (11) and (12). for the velocity field and the micro rotation component, we attempt an approximate solution for and g as a power series in terms of. Thus, and g are taken in the following form: Similar expression is assumed for. Substituting equation (15) in equations (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 and.... It may be noted that for, the zeroth order equation correspond to Hagen Poiseuille flow of micro polar fluid in a tube. IV. FREE PUMPING In the following we shall restrict our analysis to the case of pure peristalsis (free pumping) which corresponds to that is the zeroth order pressure gradient is zero. Thus, for this case the zeroth order solution reduces to a- The first order equation Governing equations and the corresponding boundary conditions are given by c- First order solution the first order flow quantities suggest the following form for and : where the asterisk denotes complex conjugate. Substituting equations (25) and (26) in the governing equations of and and separating the harmonics, we have the following governing equation for and Where.The equation governing and are conjugate. The boundary conditions reduce to: Where two more conditions will be taken, from the assumption that flow is axisymmetric at. The number within the parentheses indicates the derivative with respect to (for example, ). The equations governing and are conjugate to equations (27) and (3).

4 International Journal of Engineering & Technology IJET-IJENS Vol: 11 No: 1 2 d- Second order solution Similarly for and, we assume the solution of the form: Substituting equations (31) and (32) in equations (21) and (24), we have the following equation for and : with boundary conditions 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, The boundary conditions are: for free pumping case, and therefore, 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 equations (27) and (28) to get and, numerically, using Rung Kutta method. Solution procedure for getting and. Using equations (27) and (28), we have the governing equation for as: For the solution of we substitute the solution of in equation (27) and solve that with the following boundary conditions: From equations (33) and (34) we get equation for and boundary conditions as:

5 International Journal of Engineering & Technology IJET-IJENS Vol: 11 No: 1 21 Where: Then we will find from the known in equation (33), with the following boundary conditions m And the other two boundary condition given in equations (35) and (37). VI. NUMERICAL RESULTS AND DISCUSSION It may be noted that in the absence of micropolar fluid parameters the present analysis reduces to approximate analytical solution of peristaltic flow of a Newtonian fluid in cylindrical tubes with wall properties [11]. Therefore, the present Rung-Kutta method has been validated against the analytical results of Muthu et al.[11]. In order to illustrate theoretical results obtained in the preceding section, we now present some numerical results. The physical data for which is given below:, k2,.1,.5, Re 1., k3 1., M 1., 1 4. effect of rotation and magnetic field are to increase the value of w(r) at small values of k 2. 1 Fig. 2: shows the relation between the time mean velocity profile w and the radial r in the circular cylindrical tubes for different values of. It is clear from the Figure that w(r) increases continuously as r decrease. Fig. 3: shows the relation between the time mean velocity profile w and the intensity of the uniform magnetic field in the circular cylindrical tubes for different values of Ω. It is clear from the Figure that w(r) increases continuously as increase. Fig. 4: shows the relation between the time mean velocity profile w and the radial r in the circular cylindrical tubes for different values of Ω at =.5. It is clear from the Figure that w(r) increases continuously as r decrease. Fig. 5: shows the relation between the time mean velocity profile w and the wave number in the circular cylindrical tubes for different values of at Ω=.5. It is clear from the Figures that w(r) increases continuously as α increase. Fig. 6 : shows the relation between Reynolds number R e and the wave number in the circular cylindrical tubes for different values of at Ω =.5. It is clear from the Figures that α increases continuously as R e increase. Also, the relation between and R e is direct proportional. Fig.7: shows the relation between Reynolds number R e and the wave number in the circular cylindrical tubes for different values of Ω at =.5. It is clear from the Figures that α increases continuously as R e increase. Finally, the VII. CONCLUSIONS In the present study an analysis of peristaltic motion of micropolar fluid in a circular tubes with dynamic boundary condition has been presented for the case of free pumping. The axisymmetric study agrees with peristaltic motion in a two-dimensional channel [11]. It is observed that w(r) increases when the rotation and magnetic field increase. The effect of the rotation and magnetic field on the w(r) are studied. Finally, the results are discussed and illustrated graphically.

6 w (r) w (r) w (r) w (r) International Journal of Engineering & Technology IJET-IJENS Vol: 11 No: =. =.1 =.3 = =.1 =.3 =.5 = Fig. 2. Variation of the time mean velocity profile w with the radial r at =.,.1,.3,.5 and = r Fig. 5. Variation of the time mean velocity profile w with the wave number α at =.1,.3,.5,.7 and = =.1 =.3 =.5 = R e Fig. 3. Variation of the time mean velocity profile w with the at =.,.5,1., Fig. 6. Variation of the Reynolds number R e with the wave number α at =.1,.3,.5,.7 and = R e Fig. 4. Variation of the time mean velocity profile w with the radial r at =.,.5,1.,1.5 and =.5 r Fig. 7. Variation of the Reynolds number R e with the wave number α at =.5, 1., 1.5, 2. and =.3.

7 International Journal of Engineering & Technology IJET-IJENS Vol: 11 No: 1 23 REFERENCES [1] Y. Qin and P.N.Kaloni, A thermal instability problem in arotating microplar fluid, International Journal of Engineering Science, Vol.3, pp ,1992. [2] R.C. Sharma and P.Kumar, Effect of rotation on thermal convection in microplar fluids, International Journal of Engineering Sciences, Vol.32,pp ,1994. [3] E.F. Elshehawey, A.M.F. Sobh, Peristaltic viscoelastic fluid motion in a tube, IJMMS, Vol. 26, pp , 21. [4] P. Muthu, B.V. Rathish Kumar,and P. Chandra, Peristaltic motion of micropolar fluid in circular cylindrical tubes: Effect of wall properties, Appl. Math. Model, Vol. 32, pp , 28. [5] A. Abd El Naby, A. E. M. El Misery and M. F. Abd El Kareem, Separation in the flow through peristaltic motion of a carreau fluid in uniform tube, Phys. A, Vol. 343, No. 1 4, pp. 1 14, 24. [6] T.Hayat, M.Khan andm.ayub,couette and Poiseuille flows of an Oldroyd 6-constant fluid with magnetic field,j. Math. Anal. Appl.,Vol.298, pp ,24. [7] A.M. Siddiqui, T. Hayat and M. Khan, Magnetic fluid model induced by peristaltic waves, J. Phy. Soc. Japan, Vol. 73, pp ,24. [8] Kh.S. Mekheimer, Peristaltic flow of blood under effect of a magnetic field in a non-uniform channels, Appl. Math. Comput., Vol. 153, pp , 24. [9] P. Hariharan, Peristaltic pressure-flow relationship of non- Newtonian fluids in distensible tubes with limiting wave forms, M.S. Thesis, Univ. Cincinnati, 25. [1] M.H. Haroun, Effect of relaxation and retardation time on peristaltic transport of the Oldroydian viscoelastic fluid, J. Appl. Tech. Phys., Vol. 46, pp , 25. [11] S.Srinivas and V.Pushparaj, Nonlinear peristaltic transport in an inclined asymmetric channel, Comm. Nonlinear Sci. Numer. Simul, Vol. 13, pp , 28. [12] E.E. Tzirtzilakis, A mathematical model for blood flow in magnetic field, Phys. Fluids, Vol.17, pp ,25. [13] G. Radhakrishnamacharya and Ch. Srinivasulu, Influence of wall properties on peristaltic transport with heat transfer, C. R. Mecanique, Vol.335, pp ,27. [14] D. Srinivasacharya, G. Radhakrishnamarya and Ch. Srinivasulu, Influence of wall properties on peristalsis in the presence of magnetic field, Int. J. Fluid Mech. Res., Vol.34, pp ,27. [15] Y. Wang, T. Hayat, N. Ali and M. Oberlack, Magnetohydrodynamic peristaltic motion of a Sisko fluid in a symmetric or asymmetric channel, Physica A, Vol. 387, pp ,28. [16] N. Ali, T. Hayat and S. Asghar, Peristaltic flow of a Maxwell fluid in a channel with compliant walls, Chaos Solitons Fract, Vol. 39, pp ,29. [17] D. Tripathi, S.K. Pandey and S. Das,Peristaltic flow of viscoelastic fluid with fractional Maxwell model through a channel, Appl. Math. Comput., Vol. 215, pp ,21.