Stress concentration factors for the Stokes flow with two nearly touching rigid particles

TL;DR

This paper develops a mathematical framework to analyze stress concentration in Stokes flow around two nearly touching rigid particles, revealing conditions for stress blow-up and providing optimal estimates of stress behavior.

Contribution

It introduces a unified approach to quantify stress concentration factors in all dimensions for nearly touching particles in viscous flow, advancing understanding of stress singularities.

Findings

Stress tensor blows up as particles approach

Pressure contributes most to stress singularity

Optimal gradient estimates established for the flow

Abstract

In this paper, a mathematical model of two adjacent rigid particles immersed into a viscous incompressible fluid is considered. The main feature of the flow is that the Cauchy stress tensor consisting of the strain tensor and the pressure will appear blow-up as the distance between these two particles tends to zero. For the purpose of making clear this high concentration, a family of unified stress concentration factors are precisely captured in all dimensions, which determine whether the Cauchy stress tensor will blow up or not. As a direct application, we establish optimal gradient estimates and asymptotics of the Cauchy stress tensor for Stokes flow, which indicate that its maximal singularity comes from the pressure.