  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.
