Nonlinear Stability of Taylor-Couette Flows with Heat Buoyancy

TL;DR

This paper analyzes the nonlinear stability of Taylor-Couette flows with thermal buoyancy effects, demonstrating conditions under which solutions remain close to the flow despite destabilizing temperature gradients.

Contribution

It provides a new stability criterion accounting for thermal buoyancy and viscous damping, extending understanding of flow stability in thermally influenced Taylor-Couette systems.

Findings

Stability depends on initial perturbations being bounded by viscosity.

Solutions remain close to Taylor-Couette flow under specified conditions.

Thermal buoyancy introduces destabilizing effects countered by viscous damping.

Abstract

This paper investigates the nonlinear stability of Taylor-Couette (TC) flows incorporating the thermal buoyancy within an annular domain characterized by small viscosity $\nu$ and thermal diffusivity $\mu$. It is well established that the buoyancy induced convection significantly impacts practical industrial applications of Taylor-Couette flow \cite{Chen2006}. In contrast to \cite{An.2024}, we specifically examines the influence of the temperature gradients and the gravity on the stability of Taylor-Couette flows in this article. The thermal buoyancy term introduces a destabilizing radial derivative $\partial_r$ into the rotating TC system. To mitigate this destabilizing effect, we employ estimates involving the negative derivatives. Consequently, the additional viscous damping becomes necessary to counterbalance the buoyancy induced instability. Our stability criterion requires that the initial perturbations from the Taylor-Couette flow are bounded by a suitable power of the viscosity. Under this condition, we prove that solutions to the 2D Boussinesq system on $[1, R] \times \mathbb{S}^1$ remain close to the Taylor-Couette flow at the same order.