Asymptotic aspects of the Boussinesq approximation for gases and liquids

摘要

The asymptotic formulation of the Boussinesq approximation relates the pressure of the fluid to a thermodynamical quantity involving the heat capacitycPo. In this paper we examine the implications of such a scaling, in particular: (i) the singular degeneracy of the equation of state #x3C1; = #x3C1;* (1 #x2212;#x3B1;* (T#x2212;;T*)) of a liquid: this equation of state is valid only for small values of the coefficient #x3B1;T*; (ii) in which manner the scaling introduces the Mach number of the flow as a small parameterefor a compressible fluid. The equations at order zero with respect to #x3F5; are the same equations for gases and for liquids only if the thermodynamics of the medium is described by using the Brunt-V#xE4;is#xE4;l#xE4; frequency instead of the temperature.