The ultimate purpose of this paper is to present a new analytical formulation to simulate diffusion with retention in a reactive medium under stable thermodynamic conditions. The analysis of diffusion with retention in a continuum medium is developed after the solution of an equivalent problem using a discrete approach. The new law may be interpreted as the reduction of all diffusion processes with retention to a unifying phenomenon that can adequately simulate the retention effect namely a circulatory motion. It is remarkable that the governing equation requires a fourth order differential term as suggested by the discrete approach. The relative fraction of diffusion particles β is introduced as a control parameter in the diffusion-retention law as suggested by the discrete approach. This control parameter is essential to avoid retention isolated from the diffusion process. Two matrices referring to material properties are introduced and related to the real phenomenon through the circulation hypothesis. The governing equation may be highly non-linear even if the material properties are constant, but the retention effect is a function of the concentration level, that is, β is a function of the concentration.

This paper deals with the analysis of diffusion coupled with temporary retention motivated by the challenge to solve the problem of population spreading. Retention may be associated to colonization of the occupied territory in this case. The discrete approach was selected to deal with this problem due to its relative simplicity and straightforward mathematical treatment. Two types of problems are analyzed namely: symmetric spreading with temporary retention, and propagation with temporary retention. It is clearly shown that higher order differential terms must be included in the governing equations of diffusion and propagation to represent the temporary retention effect. Specifically third and fourth order terms are associated to the retention effect in propagation and diffusion processes respectively. Control parameters regulating the relative influence of the diffusion and the retention terms in the governing equations come up naturally from the analysis. After the appropriated operations the finite difference equations reduce to partial differential equations. The control parameters are kept in the partial differential equations. These parameters are essential in the governing equations to avoid uncontrolled accumulation of particles due to the retention effect. The diffusion-retention problem appearing in several physicochemical problems are governed by the same equations derived here. The current literature refers to several types of diffusion-retention problems, but all solutions assume the classical second order equation as the basic reference. A short analysis of the equilibrium conditions for diffusion-retention problems with a source helps to show the coherence of the theory. In order to explore the potentialities of the discrete approach the problem of asymmetric distribution is also analyzed