Mathematical Sciences: Viscous Incompressible Magnetohydrodynamics: Analysis and Numerical Approximation

数学科学：粘性不可压缩磁流体动力学：分析和数值逼近

• 批准号: 9404440
• 负责人: Paul Schmidt
• 金额: $ 4.35万
• 依托单位: Auburn University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-07-15 至 1996-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9404440&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Viscous; Incompressible; Magnetohydrodynamics

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.

9404440施密特在磁流体力学近似的假设下，粘性、不可压缩、导电流体的运动由耦合到麦克斯韦磁场方程的纳维斯托克斯方程所支配。后者一般会超越导电流体的区域，理想情况下，会延伸到整个空间。通常，内部和外部磁场都必须在将流体与固体材料或外部真空分开的界面上确定并适当匹配。大多数以前的数学工作都是关于特殊情况的，其中磁场(或相关量)的边界条件是在流体区域的表面上给出的，而外部场可以忽略。在一般情况下，除了基本的存在性和唯一性结果外，鲜有人知道。即使可以证明问题在分析上是适定的，当试图用数字近似解决方案时，也会出现巨大的困难。主要缺点包括磁场的无限支撑以及它在分离具有不同磁性的介质的界面上表现出跳跃不连续性的事实。拟议研究的目标是开发通用的分析和数值方法，以便在静止和依赖时间的情况下严格处理这类日益复杂的问题。将研究该方法的数值和计算可行性，并将其与其他方法的可行性进行比较。然后将研究这种方法在多大程度上适用于具有许多磁性不同的介质和非光滑几何形状的更复杂的情况。在以后的阶段，将考虑一些扩展的MHD模型，包括非牛顿流体、霍尔流等离子体和热耦合流动。另一个长期目标是对涉及自由边界的MHD问题进行分析和数值研究。磁流体力学(MHD)是关于导电流体与磁场宏观相互作用的理论。它在许多工程问题上都具有重要意义，例如用于受控热核聚变的持续等离子体约束、核反应堆的液态金属冷却以及金属的电磁铸造。它在地球物理和天文学中也有应用，其中突出的例子是所谓的发电机问题，也就是地球液态金属核心中磁场的起源问题，以及磁星在其自身引力场中的平衡和稳定性问题。这项研究解决的困难在描述工程中典型应用的模型中最为明显，例如液态金属的磁性成形或特殊材料的感应制造，在这些应用中，人们遇到了相当复杂的几何形状，几个导电流体和载流固体导体通过宇宙磁场在一定距离上相互作用。