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

TL;DR

This paper derives optimal bounds for the stress concentration in two-dimensional Stokes flow around two closely spaced rigid inclusions of arbitrary shape, providing insights into flow behavior in narrow regions.

Contribution

It establishes pointwise upper bounds for the gradient and second derivatives of Stokes flow near two rigid inclusions, including optimal blow-up rates, for arbitrary shapes.

Findings

Derived upper bounds for flow derivatives in narrow regions

Proved optimality of blow-up rates at the narrowest point

Results applicable to inclusions of arbitrary shape

Abstract

It is vital important in material sciences and fluid mechanics to study the field enhancements in the narrow region between two inclusions. Complex fluids including particle suspensions usually result in complicated flow behavior. In this paper we establish the pointwise upper bounds of the gradient and the second-order partial derivatives for the Stokes flow when two rigid particles are closely spaced suspending in an open bounded domain and away from the boundary in dimension two. Moreover, the lower bounds of the gradient estimates at the narrowest place of the neck region show the optimality of the blow-up rate. These results are valid for inclusions with arbitrary shape.