GRAVITY-INDUCED COALESCENCE OF DROPS AT ARBITRARY PECLET NUMBERS

The collision efficiency in a dilute suspension of sedimenting drops i s considered, with allowance for particle Brownian motion and van der Waals attractive force. The drops are assumed to be of the same densit y, but they differ in size. Drop deformation and fluid inertia are neg lected. Owing to small particle volume fraction, the analysis is restr icted to binary interactions and includes the solution of the full qua si-steady Fokker-Planck equation for the pair-distribution function. U nlike previous studies on drop or solid particle collisions, a numeric al solution is presented for arbitrary Peclet numbers, Pe, thus coveri ng the whole range of particle size in typical hydrosols. Our techniqu e is mainly based on an analytical continuation into the plane of comp lex Peclet number and a special conformal mapping, to represent the so lution as a convergent power series for all real Peclet numbers. This efficient algorithm is shown to apply to a variety of convection-diffu sion problems. The pair-distribution function is expanded into Legendr e polynomials, and a finite-difference scheme with respect to particle separation is used. Two-drop mobility functions for hydrodynamic inte ractions are provided from exact bispherical coordinate solutions and near-held asymptotics. The collision efficiency is calculated for wide ranges of the size ratio, the drop-to-medium viscosity ratio, and the Peclet number, both with and without interdroplet forces. Solid spher es are considered as a limiting case; attractive van der Waals forces are required for non-zero collision rates in this case. For Pe much gr eater than 1, the correction to the asymptotic limit Pe --> infinity i s O(Pe(-1/2)). For Pe much less than 1, the first two terms in an asym ptotic expansion for the collision efficiency are C/Pe+1/2C(2), where the constant C is determined from the Brownian solution in the limit P e --> 0. The numerical results are in excellent agreement with these l imits. For intermediate Pe, the numerical results show that Brownian m otion is important for Pe less than or equal to O(10(2)). For Pe = 10, the trajectory analysis for Pe --> infinity may underestimate the col lision rate by a factor of two. A simpler, approximate solution based on neglecting the transversal diffusion is also considered and compare d to the exact solution. The agreement is within 2-3% for all conditio ns investigated. The effect of van der Waals attractions on the collis ion efficiency is studied for a wide range of droplet sizes. Except fo r very high drop-to-medium viscosity ratios, the effect is relatively small, especially when electromagnetic retardation is accounted for.