Stokes流中液滴的沉降
On the sedimentation of a droplet in Stokes flow
摘要：
本文致力于分析一个介观模型，该模型描述了无惯性悬浮液在粘性流中的沉降在介观尺度。本文分为两个部分，第一部分是关于迁移-Stokes模型的分析，包括$L^1\cap L^\infty$初始密度在有限阶矩下的整体存在唯一性结果。我们特别调查的情况下，初始条件是单位球的特征函数，并表明，我们恢复Hadamard-Rybczynski结果，即，球形的液滴的形状被保存在时间。在本文的第二部分中，我们推导了液滴初始形状为轴对称时的表面演化模型。我们得到了一个一维双曲型方程，该方程包含与斯托克斯方程的奇异格林函数的卷积公式相联系的非局部算子.我们提出了一个局部的存在性和唯一性的结果，并表明，我们恢复Hadamard-Rybczynski结果，只要模型是定义良好的，并完成数值模拟在球形的情况下。
英文摘要：
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.
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