COLLISION RATES OF SPHERICAL DROPS OR PARTICLES IN A SHEAR-FLOW AT ARBITRARY PECLET NUMBERS

Collision rates of two nondeformable, freely suspended drops (or parti cles) subject to Brownian motion in a simple shear at low Reynolds num ber are calculated from the solution of the full Fokker-Plank equation for the pair distribution function. Unlike previous studies on shear- induced collisions, the solution is presented for arbitrary Peclet num ber (Pe), thus covering a broad range of drop sizes. An efficient nume rical technique includes a mixed Galerkin/finite- difference approxima tion and the ideas of analytical continuation, to represent the soluti on of the discrete problem as a convergent series for all real Pe. The mobility functions are provided from exact two-drop hydrodynamics and near-contact asymptotics. Extensive calculations are presented for th e collision efficiency as a function of the size ratio, drop-to-medium viscosity ratio (($) over cap mu), and Pe less than or equal to 0(10( 2)), for the case of no interdroplet forces. For ($) over cap mu>0, th e correction to the collision efficiency for Pe much greater than 1 is 0(Pe(-1/2)). For bubbles (($) over cap mu=0), there is also an 0(Pe(- 2/3)) correction of opposite sign, resulting in a local minimum for th e collision efficiency. The asymptotic analysis for the opposite limit of Pe much less than 1 is in excellent agreement with the numerical c alculations. For intermediate Pe, the exact numerical solution is comp ared with different ''additive approximations.'' The simple two-term a dditivity approximation is generally unsuccessful, whereas a modified, three-term approximation provides reasonable results except at small size ratios and large viscosity ratios. The effect of the van der Waal s attractions on the collision efficiency for typical emulsion drops o f 1-10 micron size with ($) over cap mu=0(1) is relatively small, of t he order 10% in the Brownian regime. As a limiting case of drops, the collision efficiency for equal-sized solid spheres with van der Waals attractions is calculated for Pe less than or equal to 200; this limit shows a stronger dependence on the Hamaker constant and the retardati on parameter. The solution for solid spheres is in excellent agreement with reported experimental data on flocculation dynamics for suspensi ons with moderate Peclet numbers. (C) 1995 American Institute of Physi cs.