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...

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...

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...

This is the second of two papers on the zero-viscosity limit for the incompressible Navier-Stokes equations in a half-space in either 2D or 3D. Under the assumption of analytic initial data, we construct solutions of Navier-Stokes for a short time which is independent of the viscosity. The Navier-Stokes solution is constructed through a composite asymptotic expansion involving the solutions of ...

Droplet impact onto a solid surface is modeled and numerically simulated using an ALE formulation of the Navier-Stokes free surface equations. The triple line dynamics aspect is modeled implicitely through the Shikhmurzaev theory. Those equations involve a generalized Navier slip boundary conditions with the variation of the surface tension near the triple line (thus a local Marangoni effect). ...

In this paper, we study the 3D axisymmetric Navier-Stokes Equations with swirl. We prove the global regularity of the 3D Navier-Stokes equations for a family of large anisotropic initial data. Moreover, we obtain a global bound of the solution in terms of its initial data in some Lp norm. Our results also reveal some interesting dynamic growth behavior of the solution due to the interaction bet...

This paper focuses on the notion of suitable weak solutions for the three-dimensional incompressible Navier–Stokes equations and discusses the relevance of this notion to Computational Fluid Dynamics. The purpose of the paper is twofold (i) to recall basic mathematical properties of the three-dimensional incompressible Navier-Stokes equations and to show how they relate to LES (ii) to introduce...

This paper gives an example of a periodic, smooth, divergencefree initial vector field and a periodic and bounded external force such that there exist a smooth solution to the 3-dimensional Navier-Stokes equations for incompressible fluid with those initial conditions but the solution cannot be continued to the whole space. The example also shows that the solutions to the Navier-Stokes equation...