Hydrodynamic interactions of two permeable particles moving slowly along their centerline

Abstract

The hydrodynamic interaction between two permeable spherical particles moving with constant velocity along the line of their centers in a fluid at infinity at rest is analyzed. The motion of the fluid in the domain between and around the two spheres is described by Stokes' equation, whereas the motion of the fluid in the interior of the porous spheres is assumed to obey Darcy's law. At the porous boundaries, continuity of the pressure and normal fluid velocity is used complemented by the simplified Beavers-Joseph slip condition. Analytical solutions for the flow field in the exterior and for the pressure field in the interior of the spheres are obtained using the stream function formulation in bispherical coordinates. Quasistationary vortices are obtained when the two spheres approach each other or when a porous sphere moves towards a porous planar wall over a wide range of permeability values. The hydrodynamic forces exerted by the fluid on the two spheres are determined as functions of the permeabilities of the spheres, the separation distance, the sphere size ratio, and the velocity ratio. Finally, the distribution of the normal and shear stresses exerted by the fluid on the surfaces of two simultaneously moving porous spheres is calculated for various permeability values. The numerical results of this work along with the qualitative conclusions drawn from them are useful in the modeling of cluster—particle or cluster—cluster agglomeration, particle—cluster deposition in crossflow and dep filtration, hydrodynamic interaction of macromolecule coils with each other and container or pore walls, and other similar phenomena.