New Boundary Conditions For Objects In A Continuum Ionized Medium With An External Electric Field
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Mathematical models for conducting or non-conducting spherical objects immersed in a continuum-ionized medium are well known and have wide applications. In the solution of a system of three coupled non-linear equations the classical fluid model consists of: (1) the continuity equation, (2) the transport equations, and (3) the Poisson equation. One important and reasonable approximation consists of replacing the Poisson equation by a Laplace equation. The numerical algorithm becomes very complex if we superpose an external electric field or convection effects. Even if numerical analysts have used upwind methods it is clear that if the wind or the external electric field becomes dominant, the structure of the partial differential equations is modified, and the problem will not be elliptic. In particular, the transition regime between diffusion effects and convection effects is extremely difficult to simulate and numerically unstable. In the classical fluid model, the classical boundary conditions are fixed for the potential and also for the density of species, i.e. ( , ) 0 N a θ ± = where a is, for example, the radius of a dust particle. However, these fixed boundary conditions do not satisfy the convection problem. Using the linearity properties of a Laplace operator, we have introduced new surface boundary conditions. We superpose both boundary conditions for a diffusion problem and a convection problem. Boundary conditions are floating, according to the strength of the external electric field or the convection effects. The numerical scheme becomes stable. Simulations have been carried out for an axisymmetrical problem, using an adaptable computational grid. Keywords: diffusion-convection problems, electric field charging process, floating boundary conditions, upwind schemes, finite element methods.
diffusion-convection problems, electric field charging process, floating boundary conditions, upwind schemes, finite element methods