CREEPING FLOW AROUND AND THROUGH A PERMEABLE SPHERE MOVING WITH CONSTANT VELOCITY TOWARDS A SOLID WALL

Abstract

The problem of creeping flow of a newtonian fluid around and through a permeable sphere that is moving towards an impermeable wall with constant velocity is solved in terms of the streamfunction and the pressure. The permeability of the sphere is assumed to be continuous, uniform and isotropic, The flow in the sphere is modelled with Darcy's law, and a Beavers-Joseph-Saffman slip-flow boundary condition is assumed at the boundary. Sample streamlines and isobars are calculated. The hydrodynamic correction factor to Stoke's law, ƒ is calculated as a function of the dimensionless permeability, κ, the dimensionless slip factor, β, and the dimensionless gap length, δ. As expected, for typical κ and β values the values of ƒ are substantially smaller than those for an impermeable sphere. An important result is that as δ decreases the value of ƒincreases much more slowly than it does for an impermeable sphere; furthermore, ƒ is finite as δ→0. For moderate and large κ values the ƒ vs δ curve has a maximum.

KEYWORDS:

• Creeping flow
• Permeable
• Sphere
• Constant velocity