A lubrication theory for permeable particles

Hydrodynamics interactions of permeable particles play a key role in applications including particle filtration processes and enhanced mass transport in fluidized catalytic reactors. This talk is about the effect of particle permeability on the hydrodynamic interactions of spherical particles immersed in viscous liquids. Under weak permeability conditions, K=k/a² << 1 , where k is the permeability and a is the reduced radius of the particles, the near-contact hydrodynamic interactions are qualitatively affected. A lubrication theory will be presented for permeable particles undergoing axisymmetric and transverse motions and the complete resistance matrix is derived. An integral Reynolds lubrication equation arises due to non-local coupling between pressure in the gap and the intraparticle pressure. Competition between liquid draining from the thin gap between two particles and draining into the particles imposes a characteristic lateral length scale L=K^1/5 a. By contrast to the singular resistances that characterize all near-contact relative motions of impermeable hard spheres, axisymmetric resistances and transverse rolling motions of contacting permeable particles are non singular. At contact, the axisymmetric mobility under the action of oppositely-directed forces is U/U₀=d₀ K^2/5, where U is the relative velocity, U₀ is the velocity in the absence of hydrodynamic interactions, and d₀=1.332. Under the action of a constant tangential force, a particle in contact with a permeable half-space rolls without slipping with velocity U/U₀=d₁ (d₂ + log K^-1)^-1, where d₁=3.125 and d₂=6.666; in shear flow, the same expression holds with d₁=7.280.