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Integral Equation Method for Creeping Flow around a Solid Body Near a Porous Slab
( 1 Laboratoire d'Ingénierie Mathématique (LIM), École Polytechnique de Tunisie, rue El Khawarezmi, BP 743, La Marsa, Tunisia 2 Laboratoire PMMH, École Supérieure de Physique et Chimie Industrielles, 10 rue Vauquelin, 75005 Paris, France )
The boundary integral equation method is extended to calculate the creeping flow around a solid body near a porous slab. The boundary condition on the slab is taken automatically into account by an appropriate Green's function calculated by Elasmi and Feuillebois (Z.A.M.M. 2001). The boundary condition on the moving solid body is expressed in terms of a distribution of the Green's function on its surface. The density of this distribution represents the stress on this body. It is obtained as the solution of an integral equation on its surface. We prove that this equation has a unique solution.
The integral equation is then integrated numerically. As a test problem, we consider the motion of a solid sphere perpendicular to a thin porous slab in a viscous fluid. A similar problem was solved analytically by Goren (J. Coll. Int. Sci. 1979). For large values of the porosity, the two problems are identical. Our numerical results for the force on the sphere are then in excellent agreement with Goren's analytical results, even for a small non-dimensional gap (10–4) between the sphere and the slab. In the particular case of a solid wall, our numerical results are in excellent agreement with the exact results of Brenner (Chem. Eng. Sci. 1961) and Maude (Br. J. Appl. Phys. 1961). They are also compared with the numerical results of Hsu and Ganatos (J. Fluid Mech. 1989).
Received 6 August 2001. Revised 14 February 2002.