Hydrodynamic pair diffusion in isotropic random velocity fields with application to turbulent coagulation

Abstract

Diffusion and coagulation are investigated in a random, isotropic flow in the presence of hydrodynamic interactions, interparticle forces, and Brownian diffusion. Different strain and rotation rate time scales characterize the velocity field and the particles are assumed small compared with the characteristic length of the flow, so that the velocity field is linear in the vicinity of the particles. The pair probability equation for the relative motion of two particles is written in terms of a diffusion tensor and a drift velocity. This technique is valid in the limit of small strain, i.e., when the product of the characteristic velocity gradient and time scale of the fluctuating velocity gradient is small. A consequence of the drift velocity is that, at steady state in a noncoagulating system, the pair probability distribution is nonuniform when hydrodynamic interactions are included, and there is a higher probability of particle pairs at close proximity. The pair probability conservation equation is used to determine the coagulation rate both without and with consideration of interparticle interactions. The stability factor, W, is the ratio of the coagulation rate in the absence and presence of interparticle forces, and W is calculated numerically for different size particles influenced by van der Waals attraction, electrostatic repulsion, hydrodynamic interactions, and Brownian motion. A semi-analytical expression is derived that is valid for large particles that are not influenced by Brownian motion and that experience weak van der Waals attraction. The analysis shows that colloidal stability increases with increasing particle size and shear rate as a result of the hydrodynamic resistance to particle–particle collision. Double layer repulsion can lead to stable colloidal suspensions, but increasing the fluid shear can reduce this effect. Colloid stability for the randomly varying flow considered here is comparable to that obtained for steady linear flows, such as simple shear when only van der Waals attraction is considered. Compared with the steady linear flows, double layer repulsion imparts additional resistance to aggregation in the randomly varying flow. The relevance of applying this analysis to coagulation in isotropic turbulent flows is discussed.