Exact and numerical solutions to the moving boundary problem resulting from reversible heterogeneous reactions and aqueous diffusion in a porous medium

1986 ◽  
Vol 91 (B7) ◽  
pp. 7531-7544 ◽  
Author(s):  
Peter C. Lichtner ◽  
Eric H. Oelkers ◽  
Harold C. Helgeson
Keyword(s):  
Porous Medium ◽  
Boundary Problem ◽  
Moving Boundary ◽  
Numerical Solutions ◽  
Heterogeneous Reactions ◽  
Moving Boundary Problem

The Method of Fundamental Solutions for the Moving Boundary Problem of the One-Dimension Heat Conduction Equation

2014 ◽  
Vol 1039 ◽  
pp. 59-64 ◽  
Author(s):  
Xin Jiang ◽  
Xiao Gang Wang ◽  
Yue Wei Bai ◽  
Chang Tao Pang
Keyword(s):  
Heat Conduction ◽  
Boundary Problem ◽  
Heat Conduction Equation ◽  
Moving Boundary ◽  
Numerical Solutions ◽  
Fundamental Solutions ◽  
Metal Wire ◽  
Moving Boundary Problem ◽  
Conduction Equation ◽  
Temperature Function

The melting of the material is regarded as the moving boundary problem of the heat conduction equation. In this paper, the method of fundamental solution is extended into this kind of problem. The temperature function was expressed as a linear combination of fundamental solutions which satisfied the governing equation and the initial condition. The coefficients were gained by use of boundary condition. When the metal wire was melting, process of the moving boundary was gained through the conversation of energy and the Prediction-Correlation Method. A example was given. The numerical solutions agree well with the exact solutions. In another example, numerical solutions of the temperature distribution of the metal wire were obtained while one end was heated and melting.

scholarly journals Fluid flow in a medium distorted by a quasiconformal map can produce fractal boundaries

1996 ◽  
Vol 7 (1) ◽  
pp. 1-10 ◽  
Author(s):  
Olli Martio ◽  
Bernt Øksendal
Keyword(s):  
Porous Medium ◽  
Fluid Flow ◽  
Boundary Problem ◽  
Moving Boundary ◽  
Elliptic Boundary ◽  
Homogeneous Medium ◽  
Moving Boundary Problem ◽  
Fractal Boundary ◽  
Elliptic Boundary Value ◽  
Quasiconformal Map

Physical experiments indicate that when an expanding fluid flows through a porous rock then the boundary between the wet and the dry region can be very irregular (e.g. see [OMBAFJ] and the references therein). In fact, it has been conjectured that this boundary is a fractal with Hausdorff dimension about 2.5. The (one-phase) fluid flow in a porous medium can be modelled mathematically by a system of partial differential equations, which, under some simplifying assumptions, can be reduced to a family of semi-elliptic boundary value problems involving the (unknown) pressure p(x) of the fluid (at the point x and at t) and the (unknown) wet region Ut at time t. (See equations (1.5)–(1.7) below). This set of equations, called the moving boundary problem involves the permeability matrix K(x) of the medium at x. A question which has been debated is whether this relatively simple mathematical model can explain such a complicated fractal nature of ∂Ut. More precisely, does there exist a symmetric non-negative definite matrix K(x) such that the solution Ut of the corresponding (expanding) moving boundary problem has a fractal boundary? The purpose of this paper is to prove that this is indeed the case. More precisely, we show that a porous medium which produce fractal wet boundaries can be obtained by distorting a completely homogeneous medium by means of a quasiconformal map.

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