On the hydrodynamic interaction between a particle and a permeable surface

The motion and deposition of a particle translating perpendicular to a rigid, permeable surface is considered. The lubrication approximation is used to derive an equation for the pressure in the gap between the particle and the permeable surface, with a symmetric shape prescribed for the particle. The hydrodynamic force on a particle is, in general, a function of the particle size and shape, the distance from the surface and the surface permeability, and its sign depends on the relative motion of the particle and the background fluid permeating through the surface. As the particle becomes flatter, this force generally increases and is more sensitive to the surface permeability. In the case of a spherical particle, closed-form, approximate solutions are obtained using perturbation methods, in the limits of small permeability and close approach to contact. It is also shown that a sedimenting particle attains a finite velocity on close approach, which scales as k1/2 and k for a sphere and a disc, respectively, where k is the permeability per unit thickness of the surface. In the case of a particle advected toward the surface, as is common in membrane filtration, a balance of electrostatic repulsion and viscous drag is used to calculate a possible equilibrium position of the particle, at some finite distance from the surface. The dependence of the equilibrium and its stability is shown in terms of the ratio of electrostatic and lubrication forces at contact, as well as the ratio of characteristic lengths over which the two forces decay away from the boundary. The latter is found to be a significant factor in determining the conditions under which a stable equilibrium exists. These results are useful for estimating deposition propensity in membrane filtration processes, as affected by operational conditions.