Diffusion Cascades and Mutually Coupled Diffusion Processes

Abstract

In this paper, we define and investigate a system of coupled regular diffusion equations in which each concentration acts as a driving term in the next diffusion equation. Such systems can be understood as a kind of cascade process which appear in different fields of physics like diffusion and reaction processes or turbulence. As a solution, we apply the time-dependent self-similar Ansatz method, the obtained solutions can be expressed as the product of a Gaussian and a Kummer’s function. This model physically means that the first diffusion works as a catalyst in the second diffusion system. The coupling of these diffusion systems is only one way. In the second part of the study we investigate mutually coupled diffusion equations which also have the self-similar trial function. The derived solutions show some similarities to the former one. To make our investigation more complete, different kinds of couplings were examined like the linear, the power-law, and the Lorentzian. Finally, a special coupling was investigated which is capable of describing isomerization with temporal decay.

Keywords

• diffusion
• self-similar solution