Anisotropic micropolar fluids subject to a uniform microtorque: the unstable case

TL;DR

This paper investigates the nonlinear instability of a three-dimensional anisotropic micropolar fluid under a uniform microtorque, demonstrating that the unique equilibrium state is unstable through a bootstrap argument.

Contribution

It provides a rigorous proof of nonlinear instability for anisotropic micropolar fluids with microtorque, a novel result in fluid dynamics theory.

Findings

Unique nontrivial equilibrium exists under microtorque

The equilibrium state is proven to be nonlinearly unstable

Instability identified at the $L^2$-level using bootstrap methods

Abstract

We study a three-dimensional, incompressible, viscous, micropolar fluid with anisotropic microstructure on a periodic domain. Subject to a uniform microtorque, this system admits a unique nontrivial equilibrium. We prove that this equilibrium is nonlinearly unstable. Our proof relies on a nonlinear bootstrap instability argument which uses control of higher-order norms to identify the instability at the $L^2$-level.