Narrow Results

The $z$-pinch is a classical steady state for the MHD model, where a confined plasma fluid is separated by vacuum, in the presence of a magnetic field which is generated by a prescribed current along the $z$-direction. We develop a scaled variational framework to study its stability in the presence of viscosity effect, and demonstrate that any such $z$-pinch is always unstable. We also establish the existence of a largest growing mode, which dominates the linear growth of the linear MHD system.

We investigate the existence and properties of traveling wave solutions for the hyperbolic-elliptic system of conservation laws describing the dynamics of van der Waals fluids. The model is based on a constitutive equation of state containing two inflection points and incorporates nonlinear viscosity and capillarity terms. A global description of the traveling wave solutions is provided. We distinguish between classical and non-classical trajectories and, for the latter, the existence and properties of kinetic functions is investigated. An earlier work in this direction (cf. [4]) was restricted to dealing with one inflection point only. Specifically, given any left-hand state and any shock speed (within some admissible range), we prove the existence of a non-classical traveling wave for a sequence of parameter values representing the ratio of viscosity and capillarity. Our analysis exhibits a surprising lack of monotonicity of traveling waves. The behavior of these non-classical trajectories is also investigated numerically.

The aim of this study is the numerical computation of the wave propagation in crack geological solids. The finite difference method was applied to solve the differential equations involved in the problem. Since the problem is symmetric, we prefer to use this technique instead of the finite element method and/or boundary elements technique. A comparison of the numerical results with analytical solutions is provided.

The level set method for compressible flows [13] is simple to implement, especially in the presence of topological changes. However, this method was shown to suffer from large spurious oscillations in [11]. In [4], a new Ghost Fluid Method (GFM) was shown to remove these spurious oscillations by minimizing the numerical smearing in the entropy field with the help of an Isobaric Fix [6] technique. The original GFM was designed for the inviscid Euler equations. In this paper, we extend the formulation of the GFM and apply the extended formulation to the viscous Navier-Stokes equations. The resulting numerical method is robust and easy to implement along the lines of [15].