The Benard problem for nonlinear heat conduction: Unconditional stability

An infinite, horizontal layer of viscous fluid is considered, the fluid being everywhere homogeneous and incompressible, except that (i) as usual, the Boussinesq approximation leads to temperature-induced changes in density solely with a view to simulating buoyancy forces, (ii) the diffusivity depends in an arbitrary manner on the temperature, but is bounded below by a given positive constant. The upper and lower faces of the layer are each subject to constant temperatures, that on the bottom being higher than that on the top. The steady, motionless equilibrium state is identified, and its nonlinear stability to spatially periodic perturbations in the plane of the layer is investigated. A Liapunov functional, or measure, is identified, and the perturbations are shown to be asymptotically, exponentially stable in this measure, provided that a certain parameter (called the Rayleigh number), descriptive of the thermal gradient, is sufficiently small. Both fixed-fixed and free-free boundary conditions for velocity are considered.