In this paper, we propose a new paradigm for solving Navier–Stokes equations. The proposed methodology is based on a streamfunction–velocity formulation of the two-dimensional steady-state Navier–Stokes equations representing incompressible fluid flows in two-dimensional domains. Similar formulations are also possible for three-dimensional fluid flows. The main advantage of our formulation is t...

A reconstruction-based discontinuous Galerkin (RDG) method is presented for the solution of the compressible Navier-Stokes equations on arbitrary grids. The RDG method, originally developed for the compressible Euler equations, is extended to discretize viscous and heat fluxes in the Navier-Stokes equations using a so-called inter-cell reconstruction, where a smooth solution is locally reconstr...

In an early approach, we proposed a kinetic model with multiple translational temperature [K. Xu, H. Liu and J. Jiang, Phys. Fluids 19, 016101 (2007)] to simulate non-equilibrium flows. In this paper, instead of using three temperatures in the x−, y−, and z-directions, we further define the translational temperature as a second-order symmetric tensor. Based on a multiple stage BGK-type collisio...

We consider the Navier-Stokes equations with Navier friction boundary conditions and prove two results. First, in the case of a bounded domain we prove that weak Leray solutions converge (locally in time in dimension ≥ 3 and globally in time in dimension 2) as the viscosity goes to 0 to a strong solution of the Euler equations provided that the initial data converges in L2 to a sufficiently smo...

We investigate the stabilizing effect of convection in three-dimensional incompressible Euler and Navier-Stokes equations. The convection term is the main source of nonlinearity for these equations. It is often considered destabilizing although it conserves energy due to the incompressibility condition. In this paper, we show that the convection term together with the incompressibility conditio...

We study the inviscid limit of the free boundary Navier-Stokes equations. We prove the existence of solutions on a uniform time interval by using a suitable functional framework based on Sobolev conormal spaces. This allows us to use a strong compactness argument to justify the inviscid limit. Our approach does not rely on the justification of asymptotic expansions. In particular, we get a new ...

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...