Stokes Flow Past Porous Bodies of Arbitrary Shape
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DOI: 10.1007/s13226-020-0462-0
STOKES FLOW PAST POROUS BODIES OF ARBITRARY SHAPE R. Radha and B. Sri Padmavati School of Mathematics and Statistics, University of Hyderabad, Hyderabad 500 046 India e-mails: [email protected]; [email protected] (Received 22 May 2018; after final revision 23 April 2019; accepted 8 July 2019) In this paper we discuss a new approach to discuss Stokes flow past porous bodies of arbitrary shape using the Darcy [1] model and Saffman [2] boundary conditions. Key words : Stokes flow; porous bodies; arbitrary shape; Darcy model; Saffman conditions; approximate solution. 2010 Mathematics Subject Classification : 76D07, 76S05.
1. I NTRODUCTION The study of Stokes flow past porous bodies has been studied extensively owing to important applications which have left an indelible mark in many areas of science and engineering. Many models using different boundary conditions have been employed in these studies. Among them the Darcy [1] model has been largely employed in irrigation and oil industry as it is found to be favourable for low porosity systems. Many boundary conditions have been suggested for boundary value problems using Darcy model. But the most favoured ones have been the boundary conditions suggested by Saffman [2]. Most often, the subject of investigations has been the flows of viscous, incompressible fluids past porous spheres as they have been commonly used to model hydrodynamic interactions of polymer molecules and to model particles in emulsions and biological particles. But these are idealized shapes and it would be more practical to consider non-spherical shapes in these problems. Several researchers have studied such problems using the techniques like boundary element analysis and boundary integral method [3-6] which are suitable for such geometries. In this paper, we present a new method that is amenable to discuss boundary value problems involving non-spherical geometries
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which is different from many other previously known methods like finite difference methods as the solution can be computed at every point in the domain using a simple technique like series expansion in spherical harmonics. We illustrate the technique by applying it to the problem of Stokes flow past a porous body of arbitrary shape and finding its solution analytically. The formulation of the appropriate boundary conditions at the surface of a porous medium was always carried out as an important aspect of many investigations. Initially Joseph and Tao [7] employed the conditions of continuity of pressure, normal velocity and the condition of no-slip for the exterior tangential velocity when they discussed the problem of creeping flow of a viscous fluid past a porous spherical shell when Darcy law was assumed in the porous region. However the experiments of Beavers and Joseph [8] indicated that there is some slip at the boundary and they suggested that the appropriate boundary condition for plane boundaries is
α du = (u − Q), dy k where u is the velocity parallel t
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