By Radyadour Kh. Zeytounian

This learn monograph bargains with a modeling conception of the approach of Navier-Stokes-Fourier equations for a Newtonian fluid governing a compressible viscous and warmth engaging in flows. the most aim is threefold. First , to 'deconstruct' this Navier-Stokes-Fourier method with the intention to unify the puzzle of a number of the partial simplified approximate versions utilized in Newtonian Classical Fluid Dynamics and this, first aspect, have evidently a demanding method and a vital pedagogic impression at the college schooling.

The moment aspect of the most target is to stipulate a rational constant asymptotic/mathematical idea of the of fluid flows modeling at the foundation of a regular Navier-Stokes-Fourier preliminary and boundary price challenge. The 3rd side is dedicated to a demonstration of our rational asymptotic/mathematical modeling conception for varied technological and geophysical stiff difficulties from: aerodynamics, thermal and thermocapillary convections and in addition meteofluid dynamics.

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24b) which is a closed system of two equations for the velocity components ui and the density r, when we assume that the viscosity coefficients, l and m, are a function only of r. 23b) with p ¼ p(r), or Eqs. 23b with p ¼ p(r), for the unknowns ui, p and r, which are mainly used by applied mathematicians in their rigorous mathematical analyses (see, for instance, P. L. Lions (1998) [51]) does not have any physical reality, mainly because just viscosity always generates entropy (baroclinity). For this, in particular, the various rigorous mathematical results concerning the so called “incompressible limit”, related to the limiting process M # 0, in the framework of above, compressible barotropic (p ¼ p(r)) and viscous, two systems, seems (to me) very questionable.

20b) For a perfect (absence of viscosity) fluid, the pressure has already appeared as a dynamical variable in Euler Eq. 4. Characteristic of the discipline of gas dynamics is the postulate that the thermodynamic pressure, introduced via functional relations among the state variables (see Sect. 3), is equal to this dynamical pressure. When the deformation D(u) ¼ 0, for a perfect fluid, p is the thermodynamic pressure when the fluid is compressible, while p is simply an independent dynamical variable otherwise.

19) The coefficients l and m (of viscosity) being of scalar functions of the thermodynamic state (considered in Sect. 3) and I is the unit tensor, with dij (the so-called Kronecker symbol with dkk ¼ 1 and dij ¼ 0 if i  j) as components. Indeed, the fully general expression is Poisson’s (1831) relation [41] – but the name of Poisson is rarely quoted today. 19), is absent. 19) in the special case when m ¼ m0 ¼ const and mv  l + (2/3)m ¼ 0, which is the so-called (1845) “Stokes relation” [48]. 19) for the viscous NS motion, discovered by Cauchy in 1828.

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