In this paper, we look at the influence of the choice of the Reynolds tensor on the derivation of some multiphasic incompressible fluid models, called Kazhikhov-Smagulov type models. We show that a compatibility condition between the viscous tensor and the diffusive term allows us to obtain similar models without assuming a small diffusive term as it was done for instance by A. Kazhikhov and Sh. Smagulov. We begin with two examples: The first one concerning pollution and the last one concerning a model of combustion at low Mach number. We give the compatibility condition that provides a class of models of the Kazhikhov-Smagulov type. We prove that these models are globally well posed without assumptions between the density and the diffusion terms.

We present a formal derivation of a simplified version of Compressible Primitive Equations (CPEs) for atmosphere modeling. They are obtained from 3-D compressible Navier-Stokes equations with an anisotropic viscous stress tensor where viscosity depends on the density. We then study the stability of the weak solutions of this model by using an intermediate model, called model problem, which is more simple and practical, to achieve the main result.

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