On the sedimentation of a droplet in Stokes flow

TL;DR

This paper analyzes a mesoscopic model for sedimentation of inertialess suspensions in viscous flow, proving existence and uniqueness results, and deriving a surface evolution model that preserves spherical shape, supported by numerical simulations.

Contribution

It provides the first rigorous analysis of the transport-Stokes model with global existence, uniqueness, and shape preservation, along with a derived surface evolution equation for axisymmetric droplets.

Findings

Global existence and uniqueness for the transport-Stokes model.

Shape preservation of spherical droplets over time.

Derivation of a 1D hyperbolic surface evolution equation.

Abstract

This paper is dedicated to the analysis of a mesoscopic model which describes sedimentation of inertialess 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 non local 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.