Stress concentration between two adjacent rigid particles in Navier-Stokes flow

TL;DR

This paper analyzes how stress concentrates around two closely spaced rigid particles in a Navier-Stokes fluid, providing precise bounds on velocity gradients and stress blow-up rates in 2D and 3D.

Contribution

It establishes optimal upper and lower bounds for velocity gradient blow-up rates near closely spaced particles in Navier-Stokes flow, addressing nonlinear challenges.

Findings

Derived sharp upper bounds for velocity gradients as particles approach

Established lower bounds confirming the optimality of blow-up rates

Analyzed stress concentration behavior in both two and three dimensions

Abstract

In this paper we investigate the stress concentration problem that occurs when two convex rigid particles are closely immersed in a fluid flow. The governing equations for the fluid flow are the stationary incompressible Navier-Stokes equations. We establish precise upper bounds for the gradients and second-order derivatives of the fluid velocity as the distance between particles approaches zero, in dimensions two and three. The optimality of these blow-up rates of the gradients is demonstrated by deriving corresponding lower bounds. New difficulties arising from the nonlinear term in the Navier-Stokes equations is overcome. Consequently, the blow up rates of the Cauchy stress are studied as well.