On the Oberbeck–Boussinesq Approximation on Unbounded Domains

Abstract

AbstractWe study the Oberbeck–Boussinesq approximation describing the motion of an incompressible, heat-conducting fluid occupying a general unbounded domain in R 3. We provide a rigorous justification of the model by means of scale analysis of the full Navier–Stokes–Fourier system in the low Mach and Froude number regime on large domains, the diameter of which is proportional to the speed of sound. Finally, we show that the total energy of any solution of the resulting Oberbeck–Boussinesq system tends to zero with growing time.KeywordsOberbeck–Boussinesq systemSingular limitUnbounded domain