# Unconditional finite amplitude stability of a fluid in a mechanically isolated vessel with spatially non-uniform wall temperature

## TL;DR

This paper proves that in a thermally isolated vessel with non-uniform wall temperature, the fluid's velocity and temperature perturbations diminish over time, ensuring stability of the steady state despite complex heat flux dynamics.

## Contribution

It provides a thermodynamically grounded proof of the unconditional finite amplitude stability of the fluid's steady state in such a system.

## Key findings

- Velocity field vanishes over time.
- Temperature perturbations decay to steady state.
- Stability holds regardless of initial conditions.

## Abstract

A fluid occupying a mechanically isolated vessel with walls kept at spatially non-uniform temperature is in the long run expected to reach the spatially inhomogeneous steady state. Irrespective of the initial conditions the velocity field is expected to vanish, and the temperature field is expected to be fully determined by the steady heat equation. This simple observation is however difficult to prove using the corresponding governing equations. The main difficulties are the presence of the dissipative heating term in the evolution equation for temperature and the lack of control on the heat fluxes through the boundary. Using thermodynamically based arguments, it is shown that these difficulties in the proof can be overcome, and it is proved that the velocity and temperature perturbations to the steady state actually vanish as the time goes to infinity.