On the Oberbeck-Boussinesq approximation for gases

The Oberbeck-Boussinesq (OB) approximation for a compressible fluid in Bénard’s problem geometry is studied. We present a new derivation of the classical and ’extended’ OB models for a fluid obeying the ideal gas state equation by means of an asymptotic limit for vanishing temperature difference. The ‘extended’ OB approximation takes account of the compressibility effects in the energy balance, due to the work of hydrostatic pressure inside the stratified fluid, while the isochoric character of the motion is unchanged. The scaling laws on the material and experimental parameters realizing the asymptotic systems are characterized. Moreover, the ranges of validity of the approximations are described in terms of temperature difference and thickness of the fluid layer and two other characteristic lengths associated to the material parameters. The problem of non-linear stability of the rest state in the extended OB model is traced back to the known stability analysis of the classical OB model by means of a rescaling of the temperature perturbation