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On a diphasic low Mach number system. (English) Zbl 1075.35038
The author proposes a diphasic low Mach number (DLMN) system for the modelling of diphasic flows at low Mach number. This system filters out the acoustic waves but keeps all the informations coming from the thermodynamic as the equations of state and the entropy contrary to a standard incompressible diphasic Navier-Stokes system. Thus, the DLMN system is “between” the incompressible diphasic Navier-Stokes system and the compressible diphasic Navier-Stokes system. This DLMN system has good properties. For example, it predicts the dilation and the compression of a bubble under minimal thermodynamic hypothesis which are verified by a large class of generalized van der Waals equations of state. Moreover, the DLMN system is equivalent to a nonlinear heat equation when the two fluids are perfect gases and when the geometry is monodimensional. Using this property, it is possible to build a monodimensional entropic numerical scheme when the two fluids are perfect gases. Moreover, with appropriate modelling hypothesis – again satisfied by a large class of generalized van der Waals equations of state – the DLMN system degenerates (formally) toward the incompressible Navier-Stokes system for one of the two fluids.
The plan of this paper is the following: the second section is devoted to a formal derivation of the DLMN system, in the third section, some basic properties of the DLMN system are described. In the fourth section, the Lagrangian formulation of the DLMN system when each fluid is a perfect gas and when the geometry is monodimensional is considered. In the fifth section, an entropic scheme in monodimensional geometry is proposed. In the sixth section, numerical results are presented.

##### MSC:
 35Q30 Navier-Stokes equations 76T10 Liquid-gas two-phase flows, bubbly flows 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 80A10 Classical and relativistic thermodynamics
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