CREEPING FLOW OF COMPOSITE SPHERES

Abstract

Hydrodynamic behavior of a composite sphere under creeping flow condition is studied theoretically. The composite particle consists of a solid core and a porous shell. No restriction is placed on the shell thickness relative to the core size. The fluid flow inside the porous shell is governed by the Brinkman equation. Firstly, the boundary effect on the perpendicular motion of a composite sphere towards a plane wall or between two parallel plates is investigated. A boundary collocation method is used to analyze the general case where the shell thickness and separation distance between the particle and the wall can be arbitrary. A lubrication theory is also employed to examine the special case of a particle with a thin permeable layer in near contact with a single plate. A good agreement between the results from both methods is attained. It is found that the hydrodynamic effect of the boundaries on the drag force experienced by a composite sphere or a porous one is weaker than by a solid particle. While the drag force of a porous particle having a low to moderate permeability is a monotonic, decreasing function of the separation distance, a weak maximum drag may occur for a sphere with a very high permeability at a certain distance from the wall. This behavior agrees qualitatively with what Payatakes and Dassios (1987) and Burganos et al. (1992) discovered using Darcy's law to describe the flew inside the porous medium. Secondly, we derived Faxen's laws for a composite sphere. The derivations are carried out by applying reciprocal theorem in combination with fluid velocity and pressure distributions in certain simple flow as a comparison field. In this regard, the fluid velocity disturbance caused by a composite sphere subject to a simple shear flow and a rotational flow are solved individually. The resulting Faxen expressions for the drag force, torque and stress let in the limiting case where the solid core vanishes compare very well with the existing Faxen's laws for a porous sphere. It is found that when the porous layer is thick enough and its permeability is sufficiently low, the hydrodynamic behavior of a composite sphere can be approximated by that of a porous particle with equal permeability. This can be explained by the fact that the relative motion of fluid near the core surface is very weak since the fluid cannot penetrate deeply into a porous layer of low permeability, and thereby the fluid hardly experiences the resistance from the core surface.