Hydromagnetic flow between rotating eccentric cylinders with suction at the porous walls

Abstract

The flow of a viscous, incompressible fluid between two eccentric rotating porous cylinders with suction/injection at both the cylinders, for very small clearance ratio is studied. The expressions for various flow characteristics are obtained using per- turbation analysis. Streamlines and pressure plots are shown graphically for various values of flow parameters and discussed. 2000 Mathematics Subject Classification. Primary 76D08. 1. Introduction. In the present investigation we have considered the viscous flow between eccentric porous rotating cylinders. The effect of uniform suction and in- jection is studied on the walls of the cylinders. Several authors have analysed flow through rotating eccentric cylinders for various flow situations. In particular the com- plex hydrodynamical lubrication problem as flow between two rotating eccentric cylin- ders was analysed by Reynolds (10). Wannier (13) presented an exact solution of the Stokes equation for the flow between rotating eccentric cylinders. Here, the pressure distribution reduces to the classical Sommerfeld solution of the Reynolds equation in the limit of small clearance ratio, and provides a correction for curvature effects. Wood (14) has studied the asymptotic expansion at large Reynolds number for steady motion between non-coaxial eccentric rotating cylinders using modified bipolar coor- dinate system. Kamal (6) has considered the effect of the inertial terms on the flow be- tween rotating eccentric cylinders. As no restriction was laid on the clearance ratio, the analysis is somewhat complicated when applied to the lubrication problem. Kulinski and Ostrach (7) also have considered the effect of inertia on the flow in a journal bearing by using a perturbation procedure for small eccentricity. Sood and Elrod (12) have used numerical techniques to solve the full Navier-Stokes equations for the flow between eccentric rotating cylinders but for a clearance ratio of 1.0 only. DiPrima and Stuart (3) have studied the hydrodynamic flow between rotating eccentric cylinders using the modified bipolar coordinate system. They derived the Sommerfeld pressure distribution and associated flow from the Navier-Stokes equation by a straightforward and systematic expansion in the clearance ratio (assumed small). Dintenfass (2) considered the role of cartilage elasticity coupled with the non- newtonian behaviour of synovial fluid to explain the joint lubrication. Dawson (4) suggested the existence of more than one mode of lubrication namely boundary lu- brication, and this idea was supported by McCutchen (9) who concluded that weeping lubrication was not the sum total of joint lubrication but was supplemented with