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

Article type: Research Article

Authors: Novotný, Antonín | Růžička, Michael | Thäter, Gudrun^;

Affiliations: Université du Sud Toulon-Var, BP 132, F-83957 La Garde Cedex, France | Angewandte Mathematik, Universität Freiburg, Eckerstr. 1, D-79104 Freiburg, Germany | Angewandte Mathematik IV, Universität Karlsruhe, D-76128 Karlsruhe, Germany

Note: [] Corresponding author: Gudrun Thäter, Angewandte Mathematik IV, Universität Karlsruhe, D-76128 Karlsruhe, Germany. Tel.: +49 721 608 4 5840; Fax: +49 721 608 4 4178; E-mail: [email protected].

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.

Keywords: anelastic approximation

DOI: 10.3233/ASY-2011-1056

Journal: Asymptotic Analysis, vol. 75, no. 1-2, pp. 93-123, 2011