Abstract:
We study the Oberbeck–Boussinesq asymptotic model of free convection in the limit case of infinitely large heat conduction coefficient at a fixed Rayleigh number. The equation of heat conduction thus acquires the character of a connection: the temperature is governed merely by the boundary conditions and the instant velocity field. We discuss the questions of unique solvability for an initial-boundary value problem: the situation is the same as that for the system of Navier–Stokes equations. The first passage runs in the same way as in the complete Oberbeck–Boussinesq model. Furthermore, we deal with questions of the growth of solutions in time in the case of a inviscid fluid. Here the kinetic energy, and the enstrophy (i.e., the integral of the squared vortex) in the case of a horizontal strip, are monotone increasing, which results either in a slow explosion (passing to infinity at infinite time) or in transition to a state in which the temperature is in equilibrium and the fluid particles move only in the horizontal direction. The same happens with the time tending to $-\infty$.
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