Doctor of Philosophy
Department of Mathematics
Lee, Alexander Iowa, Mathematics of non-classical diffusion, Doctor of Philosophy thesis, Department of Mathematics, University of Wollongong, 1982. https://ro.uow.edu.au/theses/1554
The theory of double diffusion describes a number of physical situations which are not adequately explained by Fick's laws of diffusion. Some of these applications occur in dislocation-pipe diffusion, diffusion in composite materials and the simultaneous diffusion of two distinct types of point defects. The theory is formulated from a continuum model in which the existence of continuously distributed families of high diffusivity paths is postulated.
Previous authors considered the case in which two families of diffusion paths are present. A number of mathematical results were obtained, including the solution of a coupled system of linear parabolic partial differential equations. This system of equations did not include convection or cross-diffusion terms. In this thesis both of these types of terms are studied. In addition, a study is made of systems in which more than two families of diffusion paths are present.
The inclusion of cross-effects in the coupled equations of existing double diffusion theory yields the general linear system of coupled diffusion equations. This system can be analysed as a pair of simultaneous equations or it can be uncoupled to form a fourth order partial differential equation. For both of these treatments a set of uniqueness results are established. Uniqueness of solution imposes a number of constraints on the various constants appearing in the equations. Furthermore, uniqueness results yield reasonable physical boundary conditions for the concentrations.
The general linear system of coupled diffusion equations with cross-effects is formally analysed by the use of Fourier and Laplace transforms. The analysis leads to a closed form of the source solutions of the system. The formal method requires further constraints on the various constants. From expressions for the Fourier transforms of the solutions, the asymptotic source solutions are obtained for both large and small times. Solutions of the coupled system are also deduced in terms of solutions of the classical diffusion equation. These formulae can be used for the solution of boundary value problems with zero boundary data. For more complicated boundary value problems the latter formulae are 'inverted' to yield solutions of the classical diffusion equation in terms of solutions of the general linear coupled system. This allows the transfer of boundary data from the coupled system to the simpler classical diffusion equation.
The fourth order equation is solved in two ways. Firstly the source solution is obtained by considering Fourier and Laplace transforms. This solution is somewhat restricted and a more general solution is obtained by using the method of separation of variables. In the latter method, use is made of the boundary conditions obtained from the uniqueness results.
A reaction-diffusion system with convection and involving two families of diffusion paths is also considered. Source solutions are found by means of solutions of an equivalent random walk problem. From these formulae, solutions of initial value problems are found. These solutions can be expressed in terms of solutions of the classical diffusion equation involving convection. One limitation for the convection system is that the formulae obtained are not as useful for non-trivial boundary value problems. Asymptotic source solutions are also obtained.
Systems of n diffusion paths (without convection or crossterms) are also discussed. In particular a number of maximum principles are obtained for systems with three diffusion paths. Maximum principles for systems with four or more diffusion paths are briefly discussed. A number of difficulties are outlined for the formal solution of the system with three diffusion paths. The thesis is concluded with some numerical examples.