On the sedimentation of a droplet in Stokes flow

This paper is dedicated to the analysis of a mesoscopic model which describes sedimentation of inertia-less suspensions in a viscous flow at mesoscopic scaling. The paper is divided into two parts, the first part concerns the analysis of the transport-Stokes model including a global existence and uniqueness result for $L^1 \cap L^\infty$ initial densities with finite first moment. We investigate in particular the case where the initial condition is the characteristic function of the unit ball and show that we recover Hadamard–Rybczynski result, that is, the spherical shape of the droplet is preserved in time. In the second part of this paper, we derive a surface evolution model in the case where the initial shape of the droplet is axisymmetric. We obtain a 1D hyperbolic equation including nonlocal operators that are linked to the convolution formula with respect to the singular Green function of the Stokes equation. We present a local existence and uniqueness result and show that we recover the Hadamard–Rybczynski result as long as the modelling is well defined and finish with numerical simulations in the spherical case.

Keywords

Stokes flow, transport equation, nonlocal velocity field, hyperbolic equation, local and global existence and uniqueness results for PDEs, numerical simulation

2010 Mathematics Subject Classification

35A01, 35L02, 76D07, 76T20

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Received 12 July 2020

Accepted 21 February 2021

Published 2 August 2021