We consider the Navier-Stokes equation in a domain with rough boundaries. The small irregularity is modeled by a small amplitude and small wavelength boundary, with typical lengthscale ε ¿ 1. For periodic roughness, it is well-known that the best homogenized (that is regular in ε) boundary condition is of Navier type. Such result still holds for random stationary irregularities, as shown recent...

We consider the convergence in the L norm, uniformly in time, of the Navier-Stokes equations with Dirichlet boundary conditions to the Euler equations with slip boundary conditions. We prove that if the Oleinik conditions of no back-flow in the trace of the Euler flow, and of a lower bound for the Navier-Stokes vorticity is assumed in a Kato-like boundary layer, then the inviscid limit holds. M...

We present a numerical simulation of two coupled Navier-Stokes flows, using operator-splitting and optimization-based nonoverlapping domain decomposition methods. The model problem consists of two Navier-Stokes fluids coupled, through a common interface, by a nonlinear transmission condition. Numerical experiments are carried out with two coupled fluids; one with an initial linear profile and t...

A class of sufficient conditions of local regularity for suitable weak solutions to the nonstationary three-dimensional Navier-Stokes equations are discussed. The corresponding results are formulated in terms of functionals which are invariant with respect to the Navier-Stokes equations scaling. The famous Caffarelli-Kohn-Nirenberg condition is contained in that class as a particular case. 1991...

We consider the zero viscosity limit of long time averages of solutions of damped and driven Navier-Stokes equations in R. We prove that the rate of dissipation of enstrophy vanishes. Stationary statistical solutions of the damped and driven Navier-Stokes equations converge to renormalized stationary statistical solutions of the damped and driven Euler equations. These solutions obey the enstro...

We consider the dyadic model, which is a toy model to test issues of wellposedness and blow-up for the Navier–Stokes and Euler equations. We prove well-posedness of positive solutions of the viscous problem in the relevant scaling range which corresponds to Navier–Stokes. Likewise we prove wellposedness for the inviscid problem (in a suitable regularity class) when the parameter corresponds to ...

4 Phenomenologi al Equation Closure. 7 4 1 Examples 8 Example: River Flow 8 Quasi-equilibrium approximation 8 Example: TraÆ Flow 9 Example: Heat Condu tion 10 Fi k's Law 10 Thermal ondu tivity, difusivity, heat equation 10 Example: Granular Flow 11 Example: Invis id Fluid Flow 12 In ompressible Euler Equations 12 In ompressible Navier-Stokes Equations 12 Gas Dynami s 13 Equation of State 13 Ise...

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...