Simulating the Hele-Shaw flow in the presence of various obstacles and moving particles

A generalization of the approach developed in the recent papers by the authors is presented. It aims to provide a description of the Hele-Shaw cell in the presence of multiple small obstacles/moving particles. We perform an asymptotic analysis of the dynamics of the moving boundary and the moving particles. For this, a modification of Maz’ya–Movchan–Nieves uniform asymptotic formula for the Green’s function of the mixed boundary value problem for the Laplace equation in a multiply connected domain is utilized. The paper contains extensive numerical analysis, accounting for various physical mechanisms of particle movement in the Hele-Shaw flow.

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