Creeping axisymmetric flow around a solid particle near a permeable obstacle

Abstract

AbstractThe hydrodynamic interaction between a solid particle and a porous obstacle, both of spherical shape, the former moving slowly along the line of their centers and the latter held stationary in an external axisymmetrical flow field, is analyzed. Owing to the linearity of the creeping motion equations and the boundary conditions, this general problem can be decomposed into two simpler problems: I. the motion of the solid sphere relative to the porous one in a fluid at rest; II. an axisymmetrical streaming flow past the two spheres held stationary. The solution to problem II requires further decomposition into the problem of undisturbed flow in the absence of the two spheres and that of the two spheres following each other in a fluid at rest (problem III). The above component flow problems are solved analytically using the stream function formulation in bispherical coordinates. The flow and pressure fields, and the drag forces exerted on both spheres are determined as functions of the permeability, the slip factor, the gap length, and the relative size of the two spheres. In problem I it is found that the drag force exerted on the solid particle increases with decreasing permeability for any value of the gap length. The opposite behavior is observed in problems III (and II). In all cases, however, the drag force exerted on the porous sphere increases as the permeability decreases for any separation distance. In the region of very small separation distances the drag forces on the two spheres in problem I attain a weak maximum at a critical gap length which is a function of the obstacle permeability and the sphere size ratio. The behavior of the drag forces in problems II (and III) is more complicated and depends strongly on the sphere size ratio. The effects of the slip velocity and the particle to streaming velocity ratio (in the composite problem) on the values of the drag forces are also examined.