Collision rates of permeable particles in creeping flows

Binary collision rates are calculated for the permeable particles undergoing (i) Brownian motion, (ii) gravity sedimentation, (iii) uniaxial straining flow, and (iv) shear flow. Darcy's law is used to describe the flow inside the permeable particles, and no-slip boundary conditions are applied at particle surfaces. A leading-order asymptotic solution of the problem is developed for the weak permeability regime K = k / a 2 ≪ 1, where k = 1 2 ( k 1 + k 2 ) is the mean permeability and a = a 1 a 2 / ( a 1 + a 2 ) is the reduced radius; ai, ki (i = 1, 2), respectively, is the radius and permeability of each particle. The resulting collision rates are given by the quadrature of the pair mobility functions for permeable particles in the near-contact lubrication region and size-ratio-dependent parameters obtained from standard hard-sphere pair mobility functions. Collision rates in shear flow vanish below a critical value of the permeability parameter K * that increases with diminishing size ratio. The analogous problem of pair collision rates of particles with small-amplitude surface roughness δa is also analyzed. The formulas for the collision rates of rough particles provide accurate analytical approximations for the collision rates of permeable particles for all four aggregation mechanisms and a wide range of size ratios using an equivalent roughness δ = 0.72 K 2 / 5.