Near-contact approach of two permeable spheres

Abstract

An analysis is presented for the axisymmetric lubrication resistance between permeable spherical particles. Darcy's law is used to describe the flow in the permeable medium and a slip boundary condition is applied at the interface. The pressure in the near-contact region is governed by a non-local integral equation. The asymptotic limit $K=k/a^{2} \ll 1$ is considered, where $k$ is the arithmetic mean permeability, and $a^{-1}=a^{-1}_{1}+a^{-1}_{2}$ is the reduced radius, and $a_1$ and $a_2$ are the particle radii. The formulation allows for particles with distinct particle radii, permeabilities and slip coefficients, including permeable and impermeable particles and spherical drops. Non-zero particle permeability qualitatively affects the axisymmetric near-contact motion, removing the classical lubrication singularity for impermeable particles, resulting in finite contact times under the action of a constant force. The lubrication resistance becomes independent of gap and attains a maximum value at contact $F=6{\rm \pi} \mu a W K^{-2/5}\tilde {f}_c$, where $\mu$ is the fluid viscosity, $W$ is the relative velocity and $\tilde {f}_c$ depends on slip coefficients and weakly on permeabilities; for two permeable particles with no-slip boundary conditions, $\tilde {f}_c=0.7507$; for a permeable particle in near contact with a spherical drop, $\tilde {f}_c$ is reduced by a factor of $2^{-6/5}$.