Paul G. Schmidt

Analysis and Numerical Approximation of a Stationary MHD Flow Problem with Nonideal Boundary

We are concerned with the steady flow of a conducting fluid, confined to a bounded region of space and driven by a combination of body forces, externally generated magnetic fields, and currents entering and leaving the fluid through electrodes attached to the surface. The flow is governed by the Navier--Stokes equations (in the fluid region) and Maxwells equations (in all of space), coupled via Ohms law and the Lorentz force. By means of the Biot--Savart law, we reduce the problem to a system of integro-differential equations in the fluid region, derive a mixed variational formulation, and prove its well-posedness under a small-data assumption. We then study the finite-element approximation of solutions (in the case of unique solvability) and establish optimal-order error estimates. Finally, an implementation of the method is described and illustrated with the results of some numerical experiments.

A velocity-current formulation for stationary MHD flow

Abstract We derive a velocity-current formulation for the equations of stationary, incompressible magnetohydrodynamics under natural interface conditions for the magnetic field and prove its well-posedness for small data by means of a variational principle.

Numerical Simulation of Steady Liquid-Metal Flow in the Presence of a Static Magnetic Field

We describe a novel approach to the mathematical modeling and computational simulation of fully three-dimensional, electromagnetically and thermally driven, steady liquid-metal flow. The phenomenon is governed by the Navier-Stokes equations, Maxwells equations, Ohms law, and the heat equation, all nonlinearly coupled via Lorentz and electromotive forces, buoyancy forces, and convective and dissipative heat transfer. Employing the electric current density rather than the magnetic field as the primary electromagnetic variable, it is possible to avoid artificial or highly idealized boundary conditions for electric and magnetic fields and to account exactly for the electromagnetic interaction of the fluid with the surrounding media. A finite element method based on this approach was used to simulate the flow of a metallic melt in a cylindrical container, rotating steadily in a uniform magnetic field perpendicular to the cylinder axis. Velocity, pressure, current, and potential distributions were computed and compared to theoretical predictions.

Nonexistence criteria for polyharmonic boundary-value problems

Abstract We discuss some known results and open problems pertaining to the nonexistence of nontrivial solutions for semilinear polyharmonic equations under Dirichlet or Navier boundary conditions. Precise control of the boundary terms in a Pohozaev-type identity allows us to establish a sharp nonexistence criterion for radially symmetric solutions, which closes a gap in the literature.

Experimental Measurements of Velocity, Potential, and Temperature Distributions in Liquid Aluminum During Electromagnetic Stirring

An experimental technique has been developed to measure both axial and transverse velocities and temperature distribution in molten aluminum. Couette flow of liquid aluminum, lead, tin, and low melting alloy in cylindrical container was chosen for calibration of the experimental technique and the magnetic probe. Velocity and temperature profiles for liquid aluminum rotating in cylindrical container at different angular velocities are obtained for two different values of the depth. We determined that the velocity values increase with magnetic induction.

Branches of Stationary Solutions for Parameter-dependent Reaction-Diffusion Systems from Climate Modeling

We are concerned with a parameter-dependent reaction-diffusion system on a two-dimensional compact connected oriented Riemannian manifold M without boundary (e.g., the two-sphere):

A TWO-LEVEL DISCRETIZATION METHOD FOR THE STATIONARY MHD EQUATIONS

{c_j}{\partial _t}{u_j} - div\left( {{k_j}\;grad\;{u_j}} \right) = {f_j}\left( {\mu ;x,{u_1},{u_2},{u_3}} \right){\mkern 1mu} \quad \left( {j = 1,2.3} \right).

A parabolic system involving a quadratic gradient term related to the Boussinesq approximation

(*)