Estimates for stress concentration between two adjacent rigid inclusions in Stokes flow

TL;DR

This paper derives optimal estimates for stress concentration and gradient blow-up rates in Stokes flow around two closely spaced rigid inclusions, providing sharp bounds in three dimensions and beyond.

Contribution

It establishes the optimal blow-up rates of the gradient and stress tensor in Stokes flow with two rigid inclusions, resolving previously open questions.

Findings

Optimal blow-up rate of the gradient in 3D

Optimal blow-up rate of the Cauchy stress tensor

Upper bounds for gradient in higher dimensions

Abstract

In this paper, we establish the estimates for the gradient and the second-order partial derivatives for the Stokes flow in the presence of two closely located strictly convex inclusions in dimension three. Moreover, the blow-up rate of the gradient is showed to be optimal by a pointwise upper bound and a lower bound in the narrowest region. We also show the optimal blow-up rate of Cauchy stress tensor. In dimensions greater than three, the upper bounds of the gradient are established. These results answer the questions raised in [25].