Stabilized finite element approximation of the incompressible MHD equations

No es frecuente encontrar un campo donde dos ramas principales de la Física estén involucradas. La Magnetohidrodinámica es uno de tales campos debido a que involucra a la Mecánica de Fluidos y al Electromagnetismo. Aun cuando puede parecer que esas dos ramas de la Física tienen poco en común, comparten similitudes en las ecuaciones que gobiernan los fenómenos involucrados en ellas. Las ecuaciones de Navier-Stokes y las ecuaciones de Maxwell, ambas en la raíz de la Magnetohidrodinámica, tienen una condición de divergencia nula y es esta condición de divergencia nula sobre la velocidad del fluido y el campo magnético lo que origina algunos de los problemas numéricos que surgen en la modelación de los fenómenos donde el flujo de fluidos y los campos magnéticos están acoplados.El principal objetivo de este trabajo es desarrollar un algoritmo eficiente para la resolución mediante elementos finitos de las ecuaciones de la Magnetohidrodinámica de fluidos incompresibles.Para lograr esta meta, los conceptos básicos y las características de la Magnetohidrodinámica se presentan en una breve introducción informal.A continuación, se da una revisión completa de las ecuaciones de gobierno de la Magnetohidrodinámica, comenzando con las ecuaciones de Navier-Stokes y las ecuaciones de Maxwell. Se discute la aproximación que da origen a las ecuaciones de la Magnetohidrodinámica y finalmente se presentan las ecuaciones de la Magnetohidrodinámica.Una vez que las ecuaciones de gobierno de la Magnetohidrodinámica han sido definidas, se presentan los esquemas numéricos desarrollados, empezando con la linealización de las ecuaciones originales, la formulación estabilizada y finalmente el esquema numérico propuesto. En esta etapa se presenta una prueba de convergencia.Finalmente, se presentan los ejemplos numéricos desarrollados durante este trabajo.Estos ejemplos pueden dividirse en dos grupos: ejemplos numéricos de comparación y ejemplos de internes tecnológico. Dentro del primer grupo están incluidas simulaciones del flujo de Hartmann y del flujo sobre un escalón. El segundo grupo incluye simulaciones del flujo en una tobera de inyección de colada continua y el proceso Czochralski de crecimiento de cristales.It is not frequent to find a field where two major branches of Physics are involved. Magnetohydrodynamics is one of such fields because it involves Fluid Mechanics and Electromagnetism. Although those two branches of Physics can seem to have little in common, they share similarities in the equations that govern the phenomena involved. The Navier-Stokes equations and the Maxwell equations, both at the root of Magnetohydrodynamics, have a divergence free condition and it is this divergence free condition over the velocity of the fluid and the magnetic field what gives origin to some of the numerical problems that appear when approximating the equations that model the phenomena where fluids flow and magnetic fields are coupled.The main objective of this work is to develop an efficient finite element algorithm for the incompressible Magnetohydrodynamics equations.In order to achieve this goal the basic concepts and characteristics of Magnetohydrodynamics are presented in a brief and informal introduction.Next, a full review of the governing equations of Magnetohydrodynamics is given, staring from the Navier-Stokes equations and the Maxwell equations. The MHD approximation is discussed at this stage and the proper Magnetohydrodynamics equations for incompressible fluid are reviewed.Once the governing equations have been defined, the numerical schemes developed are presented, starting with the linearization of the original equations, the stabilization formulations and finally the numerical scheme proposed. A convergence test is shown at this stage.Finally, the numerical examples performed while this work was developed are presented. These examples can be divided in two groups: numerical benchmarks and numerical examples of technological interest. In the first group, the numerical simulations for the Hartmann flow and the flow over a step are included. The second group includes the simulation of the clogging in a continuous casting nozzle and Czochralski crystal growth process.Postprint (published version

The materials processing research base of the Materials Processing Center

An annual report of the research activities of the Materials Processing Center of the Massachusetts Institute of Technology is given. Research on dielectrophoresis in the microgravity environment, phase separation kinetics in immiscible liquids, transport properties of droplet clusters in gravity-free fields, probes and monitors for the study of solidification of molten semiconductors, fluid mechanics and mass transfer in melt crystal growth, and heat flow control and segregation in directional solidification are discussed

Halevi's extension of the Euler-Drude model for plasmonic systems

The nonlocal response of plasmonic materials and nanostructures is usually described within a hydrodynamic approach which is based on the Euler-Drude equation. In this work, we reconsider this approach within linear response theory and employ Halevi's extension to this standard hydrodynamic model. After discussing the impact of this improved model, which we term the Halevi model, on the propagation of longitudinal volume modes, we accordingly extend the Mie-Ruppin theory. Specifically, we derive the dispersion relation of cylindrical surface plasmons. This reveals a nonlocal, collisional damping term which is related to earlier phenomenological considerations of limited-mean-free-path effects and influences both, peak width and amplitude of corresponding resonances in the extinction spectrum. In addition, we transfer the Halevi model into the time-domain thereby revealing a novel, diffusive contribution to the current which shares certain similarities with Cattaneo-type currents and analyze the resulting hybrid, diffusive-wave-like motion. Further, we discuss the relation of the Halevi model to other approaches commonly used in the literature. Finally, we demonstrate how to implement the Halevi model into the Discontinuous-Galerkin Time-Domain finite-element Maxwell solver and are able to identify an oscillatory contribution to the diffusive current. The Halevi model thus captures a number of relevant features beyond the standard hydrodynamic model. Contrary to other extensions of the standard hydrodynamic model, its use in time-domain Maxwell solvers is straightforward -- especially due its affinity to a class of descriptions that allow for a clear distinction between bulk and surface response. This is of particular importance for applications in nano-plasmonics where nano-gap structures and other nano-scale features have to be modeled efficiently and accurately.Comment: 20 pages, 7 figures, 1 table