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 thermodynamical-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.

从长远来看，一种流体占据了一个机械隔离的容器，并且其壁保持在空间不均匀的温度，从长远来看，这种流体会达到空间不均匀的稳态。不管初始条件如何，速度场都将消失，而温度场将由稳态热方程完全确定。但是，使用相应的控制方程很难证明这种简单的观察结果。主要困难是温度演化方程中存在耗散加热项，并且对通过边界的热通量缺乏控制。使用基于热力学的论据表明，可以克服证明中的这些困难，