MHD peristaltic transport of a micropolar fluid in an asymmetric channel with porous medium

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1 Available online at Advances in Applied Science Research, 06, 7():05-4 ISSN: CODEN (USA): AASRFC MHD peristaltic transport of a micropolar fluid in an asymmetric channel porous medium K. V. V. Satyanarayana, S. Sreenadh, P. Lakshminarayana * and G. Sucharitha Department of Mathematics, Sri Venkateswara University, Tirupati, India Department of Mathematics, Sree Vidyanikethan Engineering College, Tirupati, India ABSTRACT In the present study the peristaltic transport of an incompressible conducting micropolar fluid in an asymmetric channel porous medium has been studied under the assumptions of long wave length and low Reynolds number. Applying wave frame analysis, exact analytical solutions have been obtained for the axial velocity and the microrotation component. Expression for the pressure rise is also obtained. The influence of physical parameters on the velocity, pressure gradient and pressure rise are presented through graphs. The effect of increase in the permeability parameter and the magnetic parameter is to reduce the velocity. Keywords: Peristaltic Transport, micropolar fluid, MHD, Porous medium, Asymmetric channel. INTRODUCTION Peristaltic transport is a mechanism of fluid transport which occurs when a progressive wave of area contraction or expansion propagates along the distensible tube containing fluid. The principle of peristalsis is used for pumping physiological fluids from one place to another in a living body, such as swallowing food through the esophagus, the transport of spermatozoa in the ductus efferentes of the male reproductive tract, movement of chyme in the gastrointestinal tract, urine transport from the kidneys to the urinary bladder through the ureter, movement of ovum in the fallopian tubes, bile flow from the gall bladder into the duodenum and circulation of blood in small blood vessels. Also many practical applications involve in modern biomedical and mechanical systems such as blood pump machine, heart-lung machine, dialysis machine and noxious fluid transport in nuclear industries. Most of the physiological fluids behave like a non-newtonian fluids. Hence the peristaltic transport of non- Newtonian fluids have much attention recently. Among several non-newtonian models proposed for physiological fluids, micropolar fluid has considerable interest due to its importance in engineering and biomedical problems. After the first investigation of Eringen [], this model attracted the attention of many researchers. Lukasazewicz [] gives many important aspects of the theory and applications of micropolar fluids. A few investigators considered the peristaltic flow problems concerning these fluids. Srinivasacharya et al. [3] analyzed the peristaltic transport of a micropolar fluid in a tube. Ali and Hayat [4] discussed the peristaltic flow of a micropolar fluid in an asymmetric channel. Hayat et al. [5] studied the peristaltic flow of a micropolar fluid in a channel different wave forms. Sreenadh et al. [6] investigated the peristaltic flow of micropolar fluid in an asymmetric channel permeable walls. The influence of partial slip on the peristaltic transport of a micropolar fluid in an inclined asymmetric channel is studied by Arun Kumar et al. [7]. Magneto hydrodynamics is the dynamics of electrically conducting fluids. It is now a well accepted fact that the peristaltic flows of magnetohydrodynamic fluids are important in medical sciences and bioengineering. The mutual interaction between the fluid motion and magnetic field is the essential feature of the physical situation in the MHD 05

2 P. Lakshminarayana et al Adv. Appl. Sci. Res., 06, 7():05-4 fluid flow problems. MHD principles are useful in the design of heat exchangers, pumps, radar systems, power generation development of magnetic devices, cancer tumor treatment, hyperthermia and blood reduction during surgeries. It is realized that the principles of magneto hydrodynamics find extensive applications in bioengineering and medical sciences. Hence several scientists having in mind such importance extensively discussed the peristalsis magnetic field effects [8-5]. The study of fluid flows through porous medium has attracted much attention recently. This is primarily because of practical applications of flow through porous media, such as separation process in chemical industries, filtration, transpiration cooling, storage of radioactive nuclear waste material transfer, transport process in aquifers and ground water pollution. Several researchers studied the peristaltic flow porous media [6-3]. In this paper the influence of MHD on peristaltic transport of an incompressible micropolar fluid in an asymmetric channel porous medium has been investigated under the assumptions of long wave length and low Reynolds number. The expression for the velocity, pressure rise are obtained. The effects of different parameters on velocity, pressure gradient and pressure rise are discussed through graphs.. Mathematical formulation Consider the flow of an incompressible micropolar fluid in a two dimensional porous asymmetric channel of width d +, in the presence of a magnetic field. The flow in the channel is governed by micropolar model and is d induced by sinusoidal wave trains propagating constant speed c along the channel wall. The geometry of the walls surfaces is given by ( ) π h ( x, t) d + a cos x ct λ (right hand side wall ) () π (, ) cos ( ) h x t d a x ct + φ λ (left hand side wall ) () where a a and are the amplitudes of the waves, λ is the wave length, c is the wave speed, φ ( φ π ) 0 is the phase difference, X and Y are the rectangular coordinates X measured along the axis of the channel and Y perpendicular to X. Let ( U, V ) be the velocity components in a fixed frame of reference ( X, Y ) should be noted that φ 0 corresponds to a symmetric channel waves out of phase and for φ π the waves are in phase. In the laboratory frame ( X, Y ) the flow is unsteady. However it is observed in a coordinate system ( x, y ) moving at the wave speed c, the flow can be treated as steady. The coordinates and velocities in the wave and lab frames related through the following expressions:. It ( ) ( ) x X ct, y Y, u x, y U c, v u, v V (3) where u and v are the velocity components in the wave frame. Neglecting body force and body couple, the equations governing the steady flow of an incompressible micropolar fluid in the presence of an external magnetic field, on neglecting the induced magnetic field, are given by v + 0, x y p u u w µ ρ u + v + ( µ + k) + + k ( u + c) σ 0 B0 ( u + c), x y x x y y K (4) (5) 06

3 P. Lakshminarayana et al Adv. Appl. Sci. Res., 06, 7():05-4 v v p v v w µ ρ u + v + ( µ + k) + k v, x y y x y y K w w w w v u ρ J u v kw υ k , x y x y x y (6) (7) where u and v are the velocity components in the x and y directions, respectively, w is the microrotation velocity component in the direction normal to both x and yaxes. p is the fluid pressure, σ0 is the electrical B is the strength of magnetic field, K is the permeability, µ coefficient of viscosity,, conductivity, 0 k γ are the viscosity constants of for micropolar fluid, ρ and are the fluid density and microgyration parameter. For the flow under consideration, the velocity field is v ( u, v, 0) We introduce the following non-dimensional quantities: and micro rotation vector is ( 0,0, w ) w. x y u λ v d w c j x y u v w t t j λ,,,,,,, d c d c c λ d h h d ρ c d d K h, h,, Re, p p, K, δ d d λ µ cλ µ d σ a a d σ, M d B, a, b and d K µ 0 0 d d d (8) where Re is the Reynolds number, δ is the dimensionless wave number, σ is the permeability parameter, M is the Hartmann number and p is the pressure. The governing quantities in dimensionless form can be written as + 0 x (9) p w u u Re δ u + v + N + δ + ( σ M ) ( u ) + + x x N x (0) 3 v v p δ w v v Reδ u + v + N + δ + δ σ v x N x x ( ) Re jδ N w w v u N w w u + v w + δ + δ + N x x m x () () where N parameter. k + k, is the coupling number ( 0 N ), m d k (µ + k) ( γ ( µ + k) ) µ is the micropolar Using long wavelength and low Reynolds number approximations, (9) to () reduce to 07

4 P. Lakshminarayana et al Adv. Appl. Sci. Res., 06, 7(): x p w u ( N) N + ( N )( σ + M )( u + ) p 0 x ( N) w w + m 0 (3) (4) (5) (6) The corresponding non-dimensional boundary conditions in the wave frame are u at y h and h w 0 at y h and h (7) The dimensional volume flow rate in the laboratory frame is h( X, t) Q U ( X, Y, t ) dy h ( X, t) (8) In the wave frame the above equation reduces to h( x) q u ( x, y) d y (9) h ( x) From equations (3), (8) and (9), we get ( ) ( ) Q q + ch x ch x (0) The time averaged flow over a period T at a fixed position X is Q T T 0 Q dt () Using (0) in () and then integrating we get Q q + cd + cd () Defining the dimensionless mean flow Θ in the laboratory frame and F in the wave frame as Q q Θ, F cd cd (3) 08

5 P. Lakshminarayana et al Adv. Appl. Sci. Res., 06, 7():05-4 Equation () reduces to Θ F + + d (4) in which F h h u dy (5) where h ( x) + a cos( π x), h ( x) d b cos( πx + ) φ 3. Solution of the problem By differentiating equation (6) respect to y and then using in equation (4) we obtain ( N ) + + ( N )( + M )( u + ) ( N ) m x 3 w ( N ) w p σ 3 On simplification we get 3 w ( N) w p u N N 3 N σ M ( ) + ( ) ( )( + ) m x (6) using equation (6) in (6) we obtain 4 w w 4 A + wb 0 y (7) the general solution of equation (7) is given by w c cosh θ y+ c sinh θ y+ c cosh θ y+ c sinh θ y (8) 3 4 where θ A A 4B A A 4B +, θ A ( N)( σ + M ) + m, B m ( N)( σ + M ) ( N) Substituting (7) in (5) we get p u L ( c sinh θ y + c cosh θ y) + L ( c3 sinh θ y + c4 cosh θ y) ( σ + M ) x (9) The arbitrary constants 3 presented below c, c, c and c4 are obtained by applying the boundary conditions (7) and they are c L dp 9, c L dp 0, c3 L dp, c4 L dp dx dx dx dx From equation (5) we find that dp F + ( h h ) dx L + L L (30) 09

6 P. Lakshminarayana et al Adv. Appl. Sci. Res., 06, 7():05-4 where F Θ d The non-dimensional form of the pressure rise per wavelength P is given by dp p dx dx 0 (3) An important property of the micropolar fluid is that the stress tensor is not symmetric. The non-dimensional shear stresses in the problem under consideration are given by xy N w N N + w N N τ (3) τ (33) yx RESULTS AND DISCUSSION Equation (9) gives the expression for velocity as a function of y. The velocity profiles are plotted in figures from () to (5) to study the effects of different parameters such as coupling number N, microrotation parameter m, permeability parameter σ, phase difference φ, and magnetic parameter M on the velocity distribution in the asymmetric channel. Figures and are plotted for different values of coupling number N and microrotation parameter m. It is observed that velocity profiles are parabolic. Also we noticed that the velocity decreases increasing N where as it increases for increasing microrotation parameter m. Figures 3 and 4 depict that the increase in magnetic parameter M and permeability parameter σ reduces the velocity in the mid way of the channel. Further from figure 5 it can be found that the velocity decreases increasing φ. Fig. Velocity profiles for different N fixed : a 0.5, b 0.5, d, x 0., φ π 4, m, dp / dx 0, σ 0.4, M. Fig. Velocity profiles for different m fixed : a 0. 5, b 0.5, d, x 0., φ π 4, N 0., dp / dx 0, σ 0.4, M. 0

7 P. Lakshminarayana et al Adv. Appl. Sci. Res., 06, 7():05-4 Fig.3 Velocity profiles for different σ fixe d : a 0.5, b 0.5, d, x 0., φ π / 4, m, N 0., dp / dx 0, M Fig.4 Velocity profiles for different M fixed : a 0. 5, b 0.5, d, x 0., φ π / 4, m, N 0., dp / dx 0, σ 0.4 The expression for the pressure gradient d p Fig.5 Velocity profiles for different φ fixed : a 0.5, b 0.5, d, x 0., m, N 0., dp / dx 0, σ 0.4, M. d x is given by equation (30). The variation of pressure gradient the wave length for different values of coupling number N and microrotation parameter m is shown in figures 6 and 7. It is observed that in the wider part of the channel the pressure gradient is relatively small while in a narrow part of the channel a much pressure gradient is required to maintain the flux. On other wards the magnitude of the pressure gradient increases increasing N where as it decreases increasing m. From figures 8 and 9 we observe that d p d x increases increasing Magnetic parameter M and permeability parameter σ. Figure 0 depicts that the amplitude of the pressure gradient decreases increasing φ. Fig.6 Variation of pressur gradient versus wavelength for different N fixed : a 0.5, b 0.5, d, φ π / 4, m 0.5, σ 0.3, M, Θ - Fig.7 Variation of pressur gradient versus wavelength for different m fixed : a 0.5, b 0.5, d, φ π / 4, N 0., σ 0.3, M, Θ -

8 P. Lakshminarayana et al Adv. Appl. Sci. Res., 06, 7():05-4 The equation (3) gives the expression for the pressure rise p in terms of the mean flow Θ. The variation of pressure rise the mean flow for different values of coupling number N is shown in figure. It is observed that the pressure rise increases increasing N and the pumping curves meet at the point (0.6, 0). Further apposite behavior is observed. Figure shows that the pressure rise increases decreasing microrotation parameter m and the pumping curves are intersect at the point (0.6, 0). After this point situation is reversed. From figures 3 and 4 we noticed that the pressure rise increases increasing permeability parameter σ and Magnetic parameter M and the pumping curves are meet at Θ 0.. Further apposite behavior is observed. Figure 5 depicts that the pressure rise decreases increasing phase difference φ. Fig.8 Variation of pressur gradient versus wavelength for different σ fixed : a 0.5, b 0.5, d, φ π / 4, N 0., m 0.5, M, Θ - Fig.9 Variation of pressur gradient versus wavelength for different M fixed : a 0.5, b 0.5,d, φ π / 4, N 0., m 0.5, σ 0. 3, Θ- Fig.0 Variation of pressur gradient versus wavelength for differen t φ fixed : a 0.5, b 0.5,d, N 0., m 0.5, σ 0.3, M, Θ- Fig. Variation of pressurerise versus flow rate for different N fixed : a 0.5, b 0.5, d, φ π / 4, m 0.5, σ 0.4, M.5 Fig. Variation of pressurerise versus flow rate for different m fixed : a 0.5, b 0.5,d, φ π / 4, N 0., σ 0.3, M

9 P. Lakshminarayana et al Adv. Appl. Sci. Res., 06, 7():05-4 Fig.3 Variation of pressurerise versus flow rate for different σ fixed : a 0.5, b 0.5, d, φ π / 4, N 0., m 0.5,M Fig.4 Variation of pressurerise versus flow rate for different M fixed : a 0.5, b 0.5,d, φ π / 8, N 0., m 0.5, σ 0.4 Fig.5 Variation of pressurerise versus flow rate for different φ fixed : a 0.5, b 0.5, d, N 0., m 0.5, σ 0.4, M Appendix θ θ ( N ) θ( ) ( N ) θ ( ) L m, L m, ( N)( σ + M ) ( N)( σ + M ) (cosh θh cosh θh ) L3, L4 L (sinh θh cosh θh sinh θh cosh θh ) ( σ + M ) L L (cosh θ h cosh θ h cosh θ h cosh θ h ), 5 L L (sinh θ h cosh θ h sinh θ h cosh θ h ), 6 L (cosh θ h sinh θ h cosh θ h sinh θ h ), 7 L (sinh θ h sinh θ h sinh θ h sinh θ h ), 8 L (cosh θ h sinh θ h cosh θ h sinh θ h ), 9 L ( L sinh θ h sinh θ h L cosh θ h cosh θ h ), 0 L ( L cosh θ h sinh θ h L sinh θ h cosh θ h), 3

10 P. Lakshminarayana et al Adv. Appl. Sci. Res., 06, 7():05-4 dp sinh θh L L (sinh θh cosh θh ), L3 dx ( σ + M ) L L L L L, L L L L L, L L L L L, L3 L9 L8 L5 L6 L7 L4 L L6 L0, L8 L5 L L6 L, L9 L4 L8 L5 L L0 L L L L, L L L L L L, L5 L L L sinhθ h L L coshθ h L L sinhθ h ( σ + M ) L L coshθh L L3 [ L9 (cosh θh cosh θh ) + L0 (sinh θh sinh θh )], θ L L4 [ L(cosh θh cosh θh ) + L (sinhθ h sinh θh )] θ ( h h ) L5 ( σ + M ) 9 0 REFERENCES [] A.C. Eringen, J. Math. Mech., 966, 6, -6. [] G. Lukasazewicz, Birkhausr, Boston., 999. [3] D. Srinivasacharya, M. Mishra, A.R. Rao, Acta. Mech., 003, 6, [4] N. Ali, T. Hayat, Computers and Mathematics applications, 008, 55, [5] T. Hayat, N. Ali, Z. Abbas, Phys. Lett. A, 008, 370, [6] S. Sreenadh, P. Lakshminarayana, G. Sucharitha, Int. J. of Appl. Math. and Mech., 0, 7(0), [7] M. Arun Kumar, S. Venkataramana, S. Sreenadh, P. Lakshminarayana, 0, 3 (), [8] Kh.S. Mekheimer, Appl. Math. Comput., 004, 53, [9] M. Kothandapani, S. Srinivas, Int. J. Nonlinear Mech., 008, 43, [0] T. Hayat, N. Ali, Commun.Nonlinear Sci. Num. Simu., 008, 3(7), [] Kh.S. Mekheimer, Y. ABD Elmaboud, Phys. Lett., 008, A 37, [] S. Noreen, T. Hayat, A. Alsaedi, Int. J. Phys. Sci., 0; 6: [3] P. Lakshminarayana, S. Sreenadh, G. Sucharitha, Advances in applied science research, 0, 3 (5), [4] P. Lakshminarayana, S. Sreenadh, G. Sucharitha, Procedia Engineering, 05,7, [5] K. Vajravelu, S. Sreenadh, G. Sucharitha, P. Lakshminarayana, International Journal of Biomathematics, 04, 7 (6), (5 pages). [6] S. Sreenadh, P. Lakshminrayana, M. Arun kumar, G. Sucharitha, International Journal of Mathematical Sciences and Engineering Applications (IJMSEA), 04, 8 (V), [7] S. Srinivas, M. Kothandapani, Appl. Math. Comput., 009, 3, [8] G. Sucharitha, S. Sreenadh, P. Lakshminarayana, Int. J. of Eng. Research & Technology, 0, (0), -0. [9] M. Mishra, A. Ramachandra Rao, J. Non-Newtonian Fluid Mech., 004,, [0] K. Vajravelu, S. Sreenadh, P. Lakshminarayana, Commun. Non- linear Sci. Numer. Simulat., 0, 6, [] P. Lakshminarayana, S. Sreenadh, G. Sucharitha, Advances In Applied Science Research, 05,6 (), [] S. Sreenadh, P. Lakshminarayana, V. Jagadesh, G. Sucharitha, K. Nandagopal, Global Journal of Pure and Applied Mathematics, 05. (), [3] K. Vajravelu, S. Sreenadh, P. Lakshminarayana, G. Sucharitha, International Journal of Biomathematics, 06, 9 (),65003(5 pages). 4