On the equations of free convection in an absolutely heat-conducting fluid
We study the Oberbeck-Boussinesq asymptotic model of free convection in the limit case when the thermal conductivity coefficient \(\delta\) tends to \(\infty\) at a constant Rayleigh number. We discuss questions of solvability for initial-boundary value problems. The situation here is the same as that for the system of Navier-Stokes equations: in the two- dimensional case global solvability is provable for both viscous and inviscid fluids, and in the three-dimensional case the existence of smooth solutions can be proved only for a small time, and global existence theorems can be established only for weak solutions and only in the case of a viscous fluid. Finally, we deal with the growth of solutions in time for an inviscid fluid. The kinetic energy (as well as enstrophy -- that is, the integral of the squared vorticity -- in the case of a horizontal strip) is monotone increasing, which results in a slow explosion (passing to infinity at infinite time) or in transition to some state in which the temperature is balanced and the fluid particles move only horizontally. The same happens as the time tends to \(-\infty\).
scientific article; zbMATH DE number 652209
scientific article