Squeezing of a periodic emulsion through a cubic lattice of spheres

Abstract

Squeezing of a periodic, highly concentrated emulsion of deformable drops through a dense, simple cubic array of solid spherical particles at zero Reynolds number is simulated by considering one drop in a periodic cell. The particles are rigidly held in space. The drops with nondeformed diameter comparable with the particle size (and considerably larger than the interparticle constrictions) squeeze under a specified average pressure gradient. This three dimensional problem serves as a useful prototype model of drop-solid interaction for emulsion flow through granular materials. The solution allows us to study permeabilities for both phases in detail and determine the critical conditions when the drop phase flow stops due to blockage in the pores by capillary forces. The algorithm employs a boundary-integral formulation with periodic Green’s function, Hebeker representation for solid-particle contributions, and recent desingularization tools [A. Z. Zinchenko and R. H. Davis, J. Fluid Mech. 564, 227 (2006)] to alleviate difficulties with lubrication. Calculations are challenging in that tens of thousands of boundary elements per surface and 10 000–20 000 time steps are required for near-critical squeezing conditions, and the use of multipole acceleration is crucial to make such simulations feasible. The results are presented for 36% and 50% concentrated emulsions flowing through an array of almost packed particles, at drop-to-medium viscosity ratios of 1 and 4. Scaling for the squeezing time of the drop phase at near-criticial capillary numbers is extracted from the calculations. For all the simulated cases, the drops move, on average, faster than the continuous phase.