Mathematical Sciences: Viscous Incompressible Magnetohydrodynamics: Analysis and Numerical Approximation

点击翻译按钮获取中文摘要

9404440 Schmidt Under the assumptions of the magnetohydrodynamic (or MHD) approximation, the motion of a viscous, incompressible, electrically conducting fluid is governed by the Navier-Stokes equations coupled to Maxwell's equations for the magnetic field. The latter will in general transcend the region of conducting fluid and, ideally, extend to all of space. Typically, both the interior and exterior magnetic fields must be determined and suitably matched at the interface separating the fluid from the solid material or vacuum outside. Most prior mathematical work has been concerned with special situations where boundary conditions for the magnetic field (or related quantities) are given on the surface of the fluid region and the exterior field can be disregarded. In the general case, little is known beyond basic existence and uniqueness results. Even if the problem can be shown to be analytically well-posed, formidable difficulties occur when trying to numerically approximate the solution. Major drawbacks include the unbounded support of the magnetic field and the fact that it exhibits jump discontinuities across interfaces separating media with different magnetic properties. It is the goal of the proposed research to develop general analytical and numerical methods tailored for the rigorous treatment of increasingly complex problems of this kind, in stationary as well as time-dependent situations. The numerical and computational feasibility of the approach will be investigated and compared to that of alternative procedures. It will then be studied to what extent the approach carries over to more complex situations featuring many magnetically different media and nonsmooth geometries. At later stages, some extended models of MHD will be considered, including non-Newtonian fluids, Hall current plasmas, and thermally coupled flows. Another long-term goal is the analytical and numerical investigation of MHD problems involving free boundaries. Magnet ohydrodynamics (or MHD) is the theory of the macroscopic interaction of electrically conducting fluids with a magnetic field. It is of importance in connection with many engineering problems, such as sustained plasma confinement for controlled thermonuclear fusion, liquid-metal cooling of nuclear reactors, and electromagnetic casting of metals. It also finds applications in geophysics and astronomy, where prominent examples are the so-called dynamo problem, that is, the question of the origin of the Earth's magnetic field in its liquid metal core, and the equilibrium and stability of magnetic stars in their own gravitational fields. The difficulties addressed by this research are most pronounced in models describing typical applications in engineering, such as the magnetic shaping of liquid metals or induction manufacturing of specialized materials, where one encounters fairly complicated geometries with several conducting fluids and current-carrying solid conductors interacting at a distance via the universal magnetic field.