The existence of proper weak solutions of the Dirichlet-Cauchy problem constituted by the Navier-Stokes-Fourier system which characterizes the incompressible homogeneous Newtonian fluids under thermal effects is studied. We call proper weak solutions such weak solutions that verify some local energy inequalities in analogy with the suitable weak solutions for the Navier-Stokes equations. Finall...

It is shown that the generalization of the Navier-Stokes equations to a theory with N “internal state” copies of the velocity fields is a step in a wrong direction: the N → ∞ limit has no physical sense and produces wrong results, whereas the treatment of the first order terms in 1/N is even more complicated than the initial problem of description of turbulence in the frame of the Navier-Stokes...

We consider the Navier-Stokes equations of an incompressible fluid in a three dimensional curved domain with permeable walls in the limit of small viscosity. Using a curvilinear coordinate system, adapted to the boundary, we construct a corrector function at order ε , j = 0, 1, where ε is the (small) viscosity parameter. This allows us to obtain an asymptotic expansion of the Navier-Stokes solu...

Navier-Stokes equation. The expression is the nonlinear part of the equation and features in the approximation of the flow’s Reynold’s number (see Remarks 5.2(3), on page 721 of [2]). In other words, we discretize the linearized , non-homogeneous Navier-Stokes problem, representing the 3-D slow flow of a fluid. A typical example of a slow Navier-Stokes fluid flow is ground water through an aqui...

The aim of this work is to justify mathematically the derivation of a viscous free/congested zones two–phase model from the isentropic compressible Navier-Stokes equations with a singular pressure playing the role of a barrier. Titre et résumé en Français. Modèle bi-phasique gérant zones libres/zones congestionnées comme limite singulière d’un système de Navier-Stokes compressible. Le but de ce...

We derive a hierarchy of PDEs for the leading-order evolution of wall-based quantities, such as the skin-friction and the wall-pressure gradient, in two-dimensional fluid flows. The resulting Reduced Navier-Stokes (RNS ) equations are defined on the boundary of the flow, and hence have reduced spatial dimensionality compared to the Navier-Stokes equations. This spatial reduction speeds up numer...

In this paper, we study the solvability of inhomogeneous boundary value problems for the three-dimensional Oseen and Navier–Stokes equations in the following formulation: given function spaces for Dirichlet boundary conditions, initial values, and right-hand side forcing functions, find function spaces for solutions such that the operator generated by the boundary value problem for the Oseen eq...