Optimal higher derivative estimates for Stokes equations with closely spaced rigid inclusions

TL;DR

This paper derives optimal high-order derivative estimates for Stokes flow around two closely spaced rigid inclusions, revealing precise blow-up rates of stress and derivatives in narrow regions, enhancing understanding of fluid-structure interactions.

Contribution

It introduces a novel high-order derivative estimation method for Stokes equations with rigid inclusions, achieving optimal estimates and detailed blow-up rate analysis.

Findings

Stress amplifies significantly in narrow regions as inclusions approach.

Derived optimal high-order derivative estimates for the Stokes flow.

Established precise blow-up rates of stress and derivatives in the narrow gap.

Abstract

In this paper, we study the interaction between two closely spaced rigid inclusions suspended in a Stokes flow. It is well known that the stress significantly amplifies in the narrow region between the inclusions as the distance between them approaches zero. To gain deeper insight into these interactions, we derive high-order derivative estimates for the Stokes equation in the presence of two rigid inclusions in two dimensions. Our approach resonates with the method used to handle the incompressibility constraint in the standard convex integration scheme. Under certain symmetric assumptions on the domain, these estimates are shown to be optimal. As a result, we establish the precise blow-up rates of the Cauchy stress and its higher-order derivatives in the narrow region.