1 Introduction

Rheological fluids have wide coverage in medicine,engineering,and industry. For example,they are important in polymeric and food processes. Further,non-Newtonian electrically conducting fluids in a rotating system are significant in geophysical,cosmical,and astrophysical applications. The Couette flow configuration for rotating frame and porous space has a growing interest among recent modelers and computer scientists. This is in fact because of the occurrence of such flows in petroleum,magnetohydrodynamic (MHD) generators,geothermal processes,underground energy,and metallurgical processes. It is a well-established argument that stretching,bearing,and fluid-driven shear play a key role in synovial joints. Nalim et al.^[1] examined the oscillatory Couette flow mechanical shear loader to simulate in vitro fluiddriven shear. This flow configuration generates shear since the loader enhances understanding of mechanotransduction in the joint tissue. Prasad and Kumar^[2] implemented a boundary layer assumption for the analysis of MHD oscillatory Couette flow with a porous space. The Laplace transform method was used for the velocity field in the flow of a viscous fluid. The hydromagnetic Couette flow of viscous fluid in a rotating channel was investigated by Beg et al.^{[3, 4]} . The lower plate of the channel exhibited non-torsional oscillation. Seth et al.^{[5, 6]} and Guria et al. ^[7] addressed MHD Couette flows in a porous channel and rotating frame.

It is now known that non-Newtonian fluids are more suited in industrial and physiological applications than Newtonian fluids. The expressions for these fluids in a rotating frame are very complicated. Hence,numerical and analytical techniques are implemented to describe the flow of rheological materials by means of various aspects ^{[8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40]} . The purpose of the current work is to advance the knowledge in this area. Thus,a nonlinear problem for the MHD Couette flow of a Sisko fluid between two infinite non-conducting parallel plates in a rotating frame is formulated. The whole system is in a rotating frame of reference about an axis normal to the plates. While the lower plate is kept stationary,the oscillatory flow induced by the rotation of the upper plate is examined. This paper is structured in four sections. The fundamental equations and the mathematical model are given in Section 2. Section 3 comprises the numerical study and discussion,while Section 4 synthesizes conclusions. 2 Problem formulation

An incompressible MHD Sisko fluid is considered between two infinite non-conducting parallel plates at z = 0 and z = d. A constant magnetic field B₀ acts in the z -direction,with the z -axis perpendicular to the plates. The whole system is in a state of rigid body rotation with a uniform angular velocity (0,0,Ω). Figure 1 shows a cross section of the geometry of the problem.

The equations relevant to the present flow configuration can be written as

Here,V denotes the velocity,ρ is the fluid density,t is the time,σ is the electrical conductivity, and the modified pressure including the centrifugal force term is defined as follows: where p is the pressure. S is Oscillatory Couette flow of rotating Sisko fluid 1303 in which a and b are material constants,and tr is the trace of the matrix. The expression for A₁is

Note that Eq. (4) is immediately recognizable for a power law fluid when a = 0. Equation (4) holds for a viscous fluid. We denote the velocity components by (u,v,w). Since we are dealing with infinitely long parallel plates,all physical quantities except the pressure are functions of z and t. The incompressibility condition gives w = 0. The velocity field V is expressed by the following expression:

This definition of velocity satisfies Eq. (1) identically. Utilizing Eqs. (2)-(6) results in the following scalar equations:
and is independent of z . The subjected conditions are Here,ω denotes the oscillating frequency of the upper plate and fluid,and ǫ is a constant. It may be easily verified that E_qs. (7) and (8),after eliminating the modified pressure gradient, satisfy the following equation: where and Introduce 1304 T. HAYAT,S. ABELMAN,and M. HAMESE Then,the dimensionless problem after omitting the asterisks becomes

and the rotation parameter K,the oscillatory Reynolds number β,the Hartman number M , and the nonlinear viscosity parameter L are defined by Note that a and b are nonzero in E_q. (17). 3 Numerical results and discussion

MATLAB pdepe is used for the numerical solution to Eqs. (14)-(16).

In Fig. 2,we plot the fluid velocity profiles q for increasing the Hartman number M . The primary velocity u increases rapidly towards the upper plate for increasing the Hartman number and small β. The secondary velocity v increases only marginally between the two plates,going to zero at the upper plate. For large β,we see that as M increases,the primary velocity u initially increases relatively slow between the plates and then increases rapidly towards the upper plate,while the secondary velocity v remains almost zero.

Figure 3 depicts the velocity q through the increased rotation parameter K. The primary velocity u increases rapidly through large K and small β. However,v increases from the lower plate η = 0 and reaches zero at η = 1. In the vicinity of the upper plate,there exists an incipient flow reversal in the secondary flow direction. For large β,we see that as K increases, the primary velocity u initially increases relatively slow between the plates and then increases rapidly towards the upper plate. The results of v for small and large β are qualitatively similar. Oscillatory Couette flow of rotating Sisko fluid 1305

In Fig. 4,we plot the fluid velocity with increasing values of the nonlinear viscosity parameter L. For small β,as L increases,the value of u increases rapidly,whereas v increases marginally between the plates and vanishes at η = 1.

For large β,we see that as L increases,u initially increases relatively slow between the plates and then increases rapidly towards the upper plate,while v remains almost zero.

Figure 5 displays q through the index n. For small β,u is the largest for the shear thickening case and the smallest for the Newtonian case,whereas v remains almost zero. For large β,the same phenomenon is observed,with u initially increasing relatively slow between the plates and then increasing rapidly towards the upper plate,while v remains almost zero.

In Fig. 6,u increases rapidly for β = 500,while for larger values of β,it initially increases relatively slow between the plates and then increases rapidly near the upper plate,whereas v remains almost zero for increasing values of β. The presence of a porous medium increases the resistance to flow.

In Fig. 7(a),the fluid velocity q for varying values of is plotted for β = 100 and K = 500,100,and 50. We observe that u and v show almost linear behaviours near η = 0 for all values of . Initially,near η = 0,both the fluid velocities u and v increase as increases. Then,when =5,at approximately η = 0.2,the primary velocity u decreases and becomes smaller than the primary velocities for the other two cases as the fluid moves 1306 T. HAYAT,S. ABELMAN,and M. HAMESE towards the upper oscillating plate. A similar occurence is visible for v when = 5 at approximately η = 0.1. For v,there is backflow as the fluid moves towards the upper plate. In Fig. 7(b),a similar behaviour is observed for β = 500 and K = 2 500,500,and 250. We keep the ratio the same in both Figs. 7(a) and 7(b). In Figs 7(a) and 7(b),M = 5,n = 2, t = 5,L = 1,and ε = 0.001.

In Fig. 8,we depict the surface plots for u and v when t ∈ [0,1] and η ∈ [0,5]. Other values of the parameters are β = 500,M = 5,K = 5,n = 2,L = 1,and ε = 0.001. For u,the fluid moves towards the upper oscillating plate,while v remains relatively small between the plates. The periodicity of u is only slightly visible here as we have chosen ε= 0.001.

In Fig. 9,we depict the surface plots for u and v when t ∈ [0,5] and η ∈ [0,1]. Other values for the parameters are β = 500,M = 5,K = 5,n = 2,L = 1,and ε = 1. For u,the fluid moves towards the upper oscillating plate,while v remains relatively small between the plates. The periodicity of u is clearly visible here as we have chosen ε = 1.

4 Conclusions

The oscillatory Couette flow of MHD Sisko fluid between two infinite non-conducting parallel plates in a rotating frame is explored. The key findings of the presented analysis are listed below. 1308 T. HAYAT,S. ABELMAN,and M. HAMESE

(i) The conducted analysis of hydromagnetic Couette flow is significant in MHD power

generators,pumps,flow meters,nuclear reactors,biomagnetic fluid flows,and metallurgy.

(ii) Smaller Reynolds number β corresponds to the creeping flow.

(iii) The behaviour of the secondary fluid velocity v is oscillatory when the rotation parameter K increases.

(iv) The primary fluid velocity u is an increasing function of K.

(v) The secondary fluid velocity v exhibits a marginal change through K,and it remains virtually the same for the increase in M ,L,n,and β.
Acknowledgements S. ABELMAN thanks the University of the Witwatersrand and the NRF,Pretoria,South Africa,for research funding. M. HAMESE was a Bachelor of Science Honours student in the School of Computational and Applied Mathematics,University of the Witwatersrand,Johannesburg in 2011. She contributed to this work through her Bachelor of Science Honours research project. The anonymous reviewers are thanked for their suggested corrections which lead to an improved manuscript