A boundary-integral study of a drop squeezing through interparticle constrictions

Abstract

A three-dimensional boundary-integral algorithm is developed to study the squeezing of a deformable drop through a tight constriction formed by several solid particles rigidly held in space. The drop is freely suspended and driven by a flow that is uniform away from the solid obstacles. Particular emphasis is on the trapping mechanism and flow conditions close to critical, when the drop squeezes with high resistance. The problem is a close prototype of drop–solid interactions for emulsion flow through a granular material; such interactions are much more lubrication-sensitive than drop–drop interactions and require advanced numerical tools to succeed. The algorithm is based on the Hebeker representation for the solid–particle contribution, leading to a well-behaved system of second-kind integral equations, combined with novel regularization techniques for singular and near-singular boundary integrals; high-order near-singularity subtraction for the solid-to-drop double-layer contribution is the most crucial element. Simulations are performed for drop squeezing between (i) two close spheres, (ii) two parallel spheroidal disks, and (iii) three close spheres forming an equilateral triangle (including the case of close solid–solid contact). The drop non-deformed diameter is from two to several times larger than the inner constriction diameter and, in some simulations, the drop decelerates $10^3$ – $10^4$ times in the throat before being able to pass through. The effects of the constriction type, capillary number, and viscosity ratio on the drop velocity in the throat, exit time, and drop–solid spacing (of the order of 1% of the particle size) are explored in detail; critical capillary numbers (below which trapping occurs) are accurately determined. Even for a substantially supercritical capillary number, the drop has to nearly coat solid particles to be able to pass through a tight constriction. The ability of the algorithm to simulate both supercritical and subcritical conditions (when the drop is trapped, with a small but non-zero drop–solid spacing) is vital for future applications to large-scale simulations of emulsion flow through granular media.