Applied Mathematics. 2011; 1(2): 112-121 DOI: 10.5923/j.am.20110102.19 On the Slow Viscous Flow through a Swarm of Solid Spherical Particles Covered by Porous Shells Pramod Kumar Yadav Department of Mathematics, National Institute of Technology Patna, Patna, 800005, Bihar, India Abstract This paper concerns the slow viscous flow through a swarm of concentric clusters of porous spherical parti- cles. An aggregate of clusters of porous spherical particles is considered as a hydro-dynamically equivalent to a porous spherical shell enclosing a solid spherical core. The Brinkman equation inside and the Stokes equation outside the porous spherical shell in their stream function formulations are used. As boundary conditions, continuity of velocity, continuity of normal stress and stress-jump condition at the porous and fluid interface, the continuity of velocity components on the solid spherical core are employed. On the hypothetical surface, uniform velocity and Happel boundary conditions are used. The drag force experienced by each porous spherical shell in a cell is evaluated. As a particular case, the drag force experienced by a porous sphere in a cell with jump is also investigated. The earlier results reported for the drag force by Davis and Stone[5] for the drag force experienced by a porous sphere in a cell without jump, Happel[2] for a solid sphere in a cell and Qin and Kaloni[4] for a porous sphere in an unbounded medium have been then deduced. Representative results are pre- sented in graphical form and discussed. Keywords Spherical Porous Shell, Cell Model, Permeability, Brinkman Equation, Drag Force 1. Introduction The study of viscous flow through a porous medium has gained importance in recent years because of its natural occurrence and of its importance in bio-mechanics, physical sciences and chemical engineering etc. The flow of fluids through a swarm of porous particles has many industrial and engineering applications, such as, in flow through porous beds, in petroleum reservoir rocks, in flow sedimentation, etc. Several researchers have considered the flow of viscous fluid past and through solids or porous bodies with different models. For effective use of a porous medium in the above areas, the structure of porous layer should be viewed from all angles. There are many physical situations arises in which the porous particles moving in the viscous fluid. The flow in most of the above process is creeping because the Reynolds number is smaller than unity. The two terms that play an important role for analytical treatment of the problems re- lated to flow through porous media, are porosity and per- meability. The porosity The porosity represents the ratio of volume of openings (voids or pores) to the volume of the material. It seems that more the number of pores, easier will be the flow through the medium, which is not correct and this can be explained by permeability. Permeability is a measure * Corresponding author: pramod547@gmail.com (Pramod Kumar Yadav) Published online at http://journal.sapub.org/am Copyright © 2011 Scientific & Academic Publishing. All Rights Reserved of interconnectivity of voids (pores) in the medium. Hence, its number of voids along with their interconnectivity both that determine the ease with which fluid will flow through the medium. For the medium of high porosity, the sum suggested by Brinkman[1] is more suitable for describing the flow through the porous medium. Brinkman[1] evaluated the viscous force exerted by a flowing fluid on a dense swarm of particles by introducing modified Darcy’s equation for po- rous medium, which is commonly known as Brinkman equation. In the analysis of flow through swarm of particles, we get cumbersome calculations, if we consider the solution of the flow field over the entire swarm by taking exact positions of the particles. In order to avoid the above complication, it is sufficient to obtain the analytical expression by considering the effects of the neighboring particles on the flow field around a single particle of the swarm, which can be used to develop relatively simple and reliable models for heat and mass transfer. This has lead to the development of parti- cle-in-cell models. Uchida[2] proposed a cell model for a sedimenting swarm of particles, considering spherical particle surrounded by a fluid envelope with cubic outer boundary. This was accu- rately solved by Brenner[3]. Happel[4, 5] proposed cell models in which both particle and outer envelope are spherical. Happel solved the problem when the inner sphere is solid with respective boundary conditions on the cell sur- face. The Happel model assumes uniform velocity condition and no tangential stress at the cell surface. The merit of this