Rigorous derivation of the anelastic approximation to the Oberbeck–Boussinesq equations

Abstract

It is shown that the anelastic Oberbeck–Boussinesq system is a small Mach, small Péclet and small Froude number limit of the complete Navier–Stokes–Fourier system for gases with large specific heat at constant volume. This result is obtained on an arbitrary large time interval. The proof allows an intrinsic view into the process of separation of fast oscillating acoustic waves, governed by a Lighthill-type equation, from the equations describing the slow fluid flows. This is a very useful information for numerical analysts.