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A fluid occupying a mechanically isolated vessel with walls kept at spatiallynon-uniform temperature is in the long run expected to reach the spatiallyinhomogeneous steady state. Irrespective of the initial conditions the velocityfield is expected to vanish, and the temperature field is expected to be fullydetermined by the steady heat equation. This simple observation is howeverdifficult to prove using the corresponding governing equations. The maindifficulties are the presence of the dissipative heating term in the evolutionequation for temperature and the lack of control on the heat fluxes through theboundary. Using thermodynamically based arguments, it is shown that thesedifficulties in the proof can be overcome, and it is proved that the velocityand temperature perturbations to the steady state actually vanish as the timegoes to infinity.

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