A. J. Meir

On the existence, uniqueness, and finite element approximation of solutions of the equations of stationary, incompressible magnetohydrodynamics

The authors consider the equations of stationary, incompressible magneto-hydrodynamics posed in a bounded domain in three dimensions and treat the full, coupled system of equations with inhomogeneous boundary conditions. Under certain conditions on the data, they show that the existence and uniqueness of the solution of a weak formulation of the equations can be guaranteed. They discuss a finite element discretization of the equations and prove an optimal estimate for the error of the approximate solution.

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.

Normal Pulse Voltammetry as Improved Quantitative Detection Mode for Amperometric Solvent Polymeric Membrane Ion Sensors

A normal pulse voltammetric detection mode for amperometric solvent polymeric membrane ion sensors is described. These sensors function on the basis of ion transfer voltammetry into an organic membrane phase of high viscosity. To avoid sensor drift, it is required that sample ions extracted within a measurement period are quantitatively stripped off the sensing membrane before the next measurement step. The time required for complete back extraction of previously extracted ions must be substantially longer than for the uptake process. Indeed, more than 40% of extracted ions are predicted to remain in the membrane phase if the stripping time equals the uptake time. This suggests that cyclic voltammetry is generally an inadequate method for a reliable application/characterization of these sensors. The pulsed method imposes discrete potential pulses onto the membrane that are incrementally changing with time to cover the total desired potential range. Between each uptake pulse a sufficiently long stripping pulse around 0 V is applied. Optimization of uptake and stripping times are performed, and comparative data with cyclic voltammetry are shown. Normal pulse voltammetric detection scans show strictly the current response for the ion uptake process, and are free of superimposed stripping waves. This characteristic aids in elucidating the nature of each observed wave and can therefore also be used for qualitative purposes. The scans also show higher sensitivity than in classical cyclic voltammetry. Experiments are here limited to ionophore-free membranes as model systems.

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.

The equations of stationary, incompressible magnetohydrodynamics with mixed boundary conditions

Abstract This paper deals with the questions of existence, uniqueness, and finite element approximation of solutions to the equations of steady-state magnetohydrodynamics with mixed boundary conditions, posed on a bounded, three-dimensional domain. The boundary conditions for the velocity equations are of Dirichlet, Neumann, and mixed type. These boundary conditions are important when considering free boundary value problems, problems on artificially truncated domains, and control problems which are governed by these equations.

Boundary optimal control of MHD flows

This paper deals with some optimal control problems associated with the equations of steady-state, incompressible magnetohydrodynamics. These problems have direct applications to nuclear reactor technology, magnetic propulsion devices, and design of electromagnetic pumps. These problems are first put into an appropriate mathematical formulation. Then the existence of optimal solutions is proved. The use of Lagrange multiplier techniques is justified and an optimality system of equations is derived. The theory is applied to an example.

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.

How Do Pulsed Amperometric Ion Sensors Work? A Simple PDE Model

The heat equation posed on the half-line may be used as a simple mathematical model describing the operation of an amperometric ion sensor. These sensors represent the next generation of sensors that are in routine use today. Such sensors may be used to measure ion concentrations in the laboratory, for clinical analysis, environmental monitoring, process and quality control, biomedical analysis, and physiological applications. Study of the heat equation and its solutions provides insight into the operation of these ion sensors.

Finite element analysis of magnetohydrodynamic pipe flow

Abstract This paper deals with magnetohydrodynamic flows in pipes with arbitrary cross-section and arbitrary wall conductivities under the influence of a transverse magnetic field. The equations studied are of considerable practical interest since they model induction flow meters, electromagnetic pumps, magnetic propulsion devices and generators and have application in medicine, in power generation and in nuclear reactor technology. We are primarily interested in the questions of existence, uniqueness and finite element approximation of solutions to the equations of steady-state magnetohydrodynamics with arbitrary boundary conditions which describe this phenomenon. We derive error estimates for the approximate solution and show that one obtains optimal accuracy when using the finite element method to construct approximate solutions. We illustrate these results with some numerical experiments.

Thermally coupled magnetohydrodynamics flow

Abstract This paper deals with the questions of existence and uniqueness of solutions to the equations of stationary, incompressible magnetohydrodynamics (MHD) when bouyancy effects due to temperature differences in the flow cannot be neglected. To that effect we couple the MHD equations to the heat equation and employ the well-known Boussinesq approximation. We consider the equations posed on a bounded three-dimensional domain. We point out that these equations model such phenomena as the cooling of nuclear reactors by electrically conducting fluids, continuous metal casting, crystal growth, and semiconductor manufacture.