The main approaches of discretising the viscous operator of fluid flow on hybrid meshes are analysed for accuracy, consistence, monotonicity and sensitivity to mesh quality. As none of these approaches is fully satisfactory, a novel method using an approximated finite element approach is presented and analysed. The methods are compared for the linear heat equation and the Navier-Stokes equation...

We study the low Mach low Freude numbers limit in the compressible Navier-Stokes equations and the transport equation for evolution of an entropy variable – the potential temperature Θ. We consider the case of well-prepared initial data on ”flat” tours and Reynolds number tending to infinity, and the case of ill-prepared data on an infinite slab. In both cases, we show that the weak solutions t...

This paper is dedicated to the construction of global weak solutions to the quantum Navier-Stokes equation, for any initial value with bounded energy and entropy. The construction is uniform with respect to the Planck constant. This allows to perform the semi-classical limit to the associated compressible Navier-Stokes equation. One of the difficulty of the problem is to deal with the degenerat...

In this paper, we consider a model equation for the Navier--Stokes strain equation. This has same identity enstrophy growth and number of regularity criteria as full Navier-Stokes equation, is also an evolution on constraint space. We prove finite-time blowup which shows that space are not sufficient their own to guarantee global Navier-Stokes. The mechanism self-amplification strain, consisten...

We prove that any weak space-time L vanishing viscosity limit of a sequence of strong solutions of Navier-Stokes equations in a bounded domain of R satisfies the Euler equation if the solutions’ local enstrophies are uniformly bounded. We also prove that t − a.e. weak L inviscid limits of solutions of 3D Navier-Stokes equations in bounded domains are weak solutions of the Euler equation if they...

We show by nonequilibrium molecular dynamics simulations that the Navier-Stokes equation does not correctly describe water flow in a nanoscale geometry. It is argued that this failure reflects the fact that the coupling between the intrinsic rotational and translational degrees of freedom becomes important for nanoflows. The coupling is correctly accounted for by the extended Navier-Stokes equa...

We consider the attractors for the two-dimensional nonautonomous Navier-Stokes equations in some unbounded domain Ω with nonhomogeneous boundary conditions. We apply the so-called uniformly ω-limit compact approach to nonhomogeneous Navier-Stokes equation as well as a method to verify it. Assuming f ∈ Lloc 0, T ;L2 Ω , which is translation compact and φ ∈ C1 b R ;H2 R1 × {±L} asymptotically alm...

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...

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...