Free Boundary Formulation Research Articles

On the stability of steadily rotating waves in the free boundary formulation

We perform a linear stability analysis of the steadily rotating solutions found from the free-boundary problem of wave propagation in single-diffusive excitable media. Two kind of modes can be considered: a tangential mode which moves the spiral tip tangentially to the circle limiting the unexcited region; and a radial mode which moves the spiral tip outwards the same circle in the radial direction. The study of the second kind of modes is complicated by a technical difficulty associated to the divergence of the refractory field perturbations at the spiral tip (square root difficulty). The tangential modes are found to be stable whatever the angular velocity of the steady solution.

Mathematical analysis of the influence of power-law fluid rheology on a capillary diffusion zone

Non-Newtonian fluids are used in current oil recovery processes. These fluids do not satisfy the linear Darcy Law for flow through porous media. A generalization is needed to model the flow processes involved. Furthermore, when two immiscible fluids are present in a porous medium, capillary pressure will cause a transition zone to develop between them. This transition zone may lead to early breakthrough of water into an oil well. In this paper, we study the effect of the non-Newtonian behaviour of fluids on a capillary transition zone. A general framework is set up for modelling processes involving two-phase flow of non-Newtonian, immiscible and incompressible fluids in a porous medium. The equations are applied to the one-dimensional diffusion process of power-law fluids. The model allows for general capillary pressure and relative permeability functions. The mathematical model consists of a degenerate diffusion equation, giving rise to a free boundary formulation. The free boundaries represent the endpoint of the diffusion zone. Qualitative properties, and some analytical solutions, can be obtained for the saturation profile. Two numerical methods are presented. One is applicable if the rheology of both fluids is modelled with equal powers. The other is applicable to the general situation. Both methods make use of the qualitative, analytical properties of the solutions, which clearly improved the results obtained by standard methods. One-dimensional results can be used to interpret the general multi-dimensional flow behaviour of non-Newtonian fluids when capillarity is considered. They can also be used to test numerical algorithms developed for multi-dimensional displacement processes.

Stability of spiral waves in the large-hole-radius limit.

We demonstrate that, when the trajectory of a tip is a circle of large radius, the steadily rotating spirals found from the free-boundary formulation are stable, in agreement with numerical simulations of reaction-diffusion systems.

Universal limit of spiral wave propagation in excitable media.

We demonstrate that for a large class of excitable media the minimum excitability required for spiral wave propagation follows the universal scaling law ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{min}}$=const\ifmmode\times\else\texttimes\fi{}${\mathrm{\ensuremath{\varepsilon}}}^{1/3}$, where \ensuremath{\varepsilon} is the usual small parameter associated with the abruptness of excitation. The prefactor of the scaling law is obtained by solving the free-boundary problem of wave propagation. The quantitative validity of the free-boundary formulation is tested for the first time by direct comparison with the results of numerical simulation of a specific two-variable model of excitable kinetics.

The Princeton spectral equilibrium code: PSEC

A fast computer code has been developed to calculate free-boundary solutions to the plasma equilibrium equation that are consistent with the currents in external coils and conductors. The free-boundary formulation is based on the minimization of a mean-square error ε while the fixed-boundary solution is based on a variational principle and spectral representation of the coordinates x(Ψ, θ) and z(Ψ, θ). Specific calculations using the Columbia University Torus II, the Poloidal Divertor Experiment (PDX), and the Tokamak Fusion Test Reactor (TFTR) geometries are performed.

A finite element free boundary formulation for the problem of multiphase heat conduction

AbstractThe problem of heat conduction in a multiphase medium is a free boundary problem in which the free boundary is the phase transition line. The solution to this problem is highly irregular in that the temperature gradient is discontinuous at the free boundary. Two‐dimensional, discontinuous, space–time finite elements are introduced to obtain a finite element, free boundary formulation for this problem. The jump in heat flux and position of the phase transition line become dependent variables in the finite element model. The resulting models give accurate results while allowing the calculation of free boundary problems with a fixed finite element mesh.

Preparation and properties of inp/cds solar cells

A fast computer code has been developed to calculate free-boundary solutions to the plasma equilibrium equation that are consistent with the currents in external coils and conductors. The free-boundary formulation is based on the minimization of a mean-square error ε while the fixed-boundary solution is based on a variational principle and spectral representation of the coordinates x(Ψ, θ) and z(Ψ, θ). Specific calculations using the Columbia University Torus II, the Poloidal Divertor Experiment (PDX), and the Tokamak Fusion Test Reactor (TFTR) geometries are performed.

Transforming Boundary Conditions to Initial Conditions for Ordinary Differential Equations

One of the interesting and important classes of transformations in the solution of boundary value problems is the class of transformations from a boundary value to an initial value problem. A classical example is given in Blasius' solution of the steady, two-dimensional, incompressible boundary layer equations with uniform mainstream velocity [1]. This technique, when it applies, simplifies the process of obtaining numerical solutions. Otherwise a trial-and-error technique generally has to be used in order to match the boundary condition at the other point. A recent contribution by Klamkin [2] has greatly extended the range of application of the technique, and as a result, a much wider class of problems, including some simultaneous differential equations, can be solved in this manner. In the present paper, the technique is extended to cases where the boundary conditions are specified over finite intervals, i.e., from zero to a finite value of the independent variable, with one or more boundary conditions required to be specified at the initial point. It also shows that the transformation treated in previous works is not the only type of transformation possible. Similar to the class of problems treated by Blasius [1] and Klamkin [2], the boundary conditions at the initial point have to be homogeneous for the method to apply. It can be homogeneous or nonhomogeneous at the other point. The method given in this paper is developed in terms of a oneor two-parameter group of transformations, which is seen to be as simple here as in its application to the similarity analysis [3]. The concept can best be illustrated by an example. Consider the problem of steady heat conduction with linear heat generation and power-law thermal conductivity. The energy equation can be written as

On the Transformation of a Class of Boundary Value Problems into Initial Value Problems for Ordinary Differential Equations

On the Transformation of a Class of Boundary Value Problems into Initial Value Problems for Ordinary Differential Equations