Here u, b describe the flow velocity vector and the magnetic field vector respectively, p is a scalar pressure, ν > 0 is the kinematic viscosity, η > 0 is the magnetic diffusivity, while u0 and b0 are the given initial velocity and initial magnetic field with ∇ · u0 = ∇ · b0 = 0. If ν = η = 0, (1.1) is called the ideal MHD equations. As same as the 3D Navier-Stokes equations, the regularity of ...

The problem considered is the Stokes and Navier-Stokes flow through a system of two pipes in the gravity field; inside a vertical pipe there is a free heavy piston. Theoretical analysis, the existence and non-uniqueness of solution, has been completed recently by the authors. Here we present numerical analysis, using finite elements methods, of the stationary state with respect to the angle bet...

In this paper we prove the existence and uniqueness of strong solutions for the stochastic Navier-Stokes equation in bounded and unbounded domains. These solutions are stochastic analogs of the classical Lions-Prodi solutions to the deterministic Navier-Stokes equation. Local monotonicity of the nonlinearity is exploited to obtain the solutions in a given probability space and this significantl...

The direction splitting approach proposed earlier in [7], aiming at the efficient solution of Navier-Stokes equations, is extended and adopted here to solve the Navier-Stokes-Brinkman equations describing incompressible flows in pure fluid and in porous media. The resulting pressure equation is a perturbation of the incompressibility constraint using a direction-wise factorized operator as prop...

We construct the weakly nonlinear-dissipative approximate system for the general compressible Navier-Stokes system in a periodic domain. It was shown in [11] that because the Navier-Stokes system has an entropy structure, its approximate system will have Leray-like global weak solutions. These solutions decompose into an incompressible part governed by an incompressible Navier-Stokes system, an...

We give two results that indicate that the relaxation time for the flow governed by the Navier-Stokes-Voigt (NSV) model is sinificantly larger than for the Navier-Stokes equations. We first show that for the Green-Taylor vortex decay problem, NSV admits an exact solution which has a significantly larger half life than for true fluid flow. Second, we observe in a channel flow test that NSV provi...

Here ν = 0 for the Euler equation and ν > 0 for the Navier–Stokes equation. We assume that the initial vector field, v(x, 0) = u(x), is given by a smooth, bounded, divergence–free function u(x) of finite energy. Then a smooth solution v(x, t) is known to exist in some finite time interval 0 ≤ t < T . It is an open problem, for both the Euler and the Navier–Stokes equations, if blow–up can occur...

In the following paper we will consider Navier-Stokes problem and it's interpretation by hyperbolic waves, focusing on wave propagation. We will begin with solution for linear waves, then present problem for non-linear waves. Later we will derive for numerical solution using PDE's. Also we will design a Matlab program to solve and simulate wave propagation. Keywords1: Navier-Stokes equations, F...

We consider the inviscid limit of the stochastic damped 2D NavierStokes equations. We prove that, when the viscosity vanishes, the stationary solution of the stochastic damped Navier-Stokes equations converges to a stationary solution of the stochastic damped Euler equation and that the rate of dissipation of enstrophy converges to zero. In particular, this limit obeys an enstrophy balance. The...

In this article, we discuss the numerical solution of the Stokes and Navier-Stokes equations completed by nonlinear slip boundary conditions of friction type in two and three dimensions. To solve the Stokes system, we first reduce the related variational inequality into a saddle point-point problem for a well chosen augmented Lagrangian. To solve this saddle point problem we suggest an alternat...