In this paper, we investigate the system of compressible Navier-Stokes equations with hyperbolic heat conduction, i.e., replacing the Fourier’s law by Cattaneo’s law. First, by using Kawashima’s condition on general hyperbolic parabolic systems, we show that for small relaxation time τ , global smooth solution exists for small initial data. Moreover, as τ goes to zero, we obtain the uniform con...

The thermal expansion of a fluid combined with a temperature-dependent viscosity introduces nonlinearities in the Navier-Stokes equations unrelated to the convective momentum current. The couplings generate the possibility for net fluid flow at the microscale controlled by external heating. This novel thermomechanical effect is investigated for a thin fluid chamber by a numerical solution of th...

We recover the Navier-Stokes equation as the incompressible limit of a stochastic lattice gas in which particles are allowed to jump over a meso-scopic scale. The result holds in any dimension assuming the existence of a smooth solution of the Navier-Stokes equation in a fixed time interval. The proof does not use non-gradient methods or the multi-scale analysis due to the long range jumps.

The performance of multigrid methods for the standard Poisson problem and for the consistent Poisson problem arising in spectral element discretizations of the Navier-Stokes equations is investigated. It is demonstrated that overlapping additive Schwarz methods are effective smoothers, provided that the solution in the overlap region is weighted by the inverse counting matrix. It is also shown ...

Abstract: We study a nonlocal modification of the compressible Navier-Stokes equations in mono dimensional case with a boundary condition characteristic for the free boundaries problem. From the formal point of view our system is an intermediate between the Euler and Navier-Stokes equations. Under certain assumptions, imposed on initial data and viscosity coefficient, we obtain the local and gl...

We propose an original scheme for the time discretization of a triphasic CahnHilliard/Navier-Stokes model. This scheme allows an uncoupled resolution of the discrete CahnHilliard and Navier-Stokes system, is unconditionally stable and preserves, at the discrete level, the main properties of the continuous model. The existence of discrete solutions is proved and a convergence study is performed ...

This is a rather comprehensive study on the dynamics of NavierStokes and Euler equations via a combination of analysis and numerics. We focus upon two main aspects: (a). zero viscosity limit of the spectra of linear Navier-Stokes operator, (b). heteroclinics conjecture for Euler equation, its numerical verification, Melnikov integral, and simulation and control of chaos. Besides Navier-Stokes a...

In this lecture, we will particularly analyze the effect of the shallowness on the Navier Stokes equations, together with anisotropic eddy viscosities. We will derive as an asymptotic model the hydrostatic approximation of the Navier Stokes equations. We will present a stable mixed 3D-FEM discretization, which allows for the computation of the 3D-velocity as a whole. Let us emphasize that verti...

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...