On the significance of higher order differential terms in diffusion processes

Abstract

This paper deals with the analysis of diffusion coupled with temporary retentionThe advance of technological and scientific knowledge introduced new and sophisticated physicochemical processes to deal with new materials and new design concepts. Phenomena that were of little importance for the solution of the usual engineering problems cannot be disregarded anymore when dealing with modern engineering challenges.Some phenomena that would be satisfactorily dealt with the continuum mechanics approach need now to be analyzed at nanoscales. This new trend fostered the search for the correspondence between the responses in terms of macro-variables and continuum mechanics on one hand and micro-variables and micro-mechanics on the other hand. Multiscale analysis, for instance, is a relatively new modeling methodology intended to make the bridge between the state variables at microscales and the corresponding ones at macroscales.The new technological achievements require quick solutions to questions that are not yet completely understood. Pushed to solve a new problem, which is not seldom, the first approach is to apply the closest classical theory with some modification that hopefully would introduce the appropriate corrections. Experimental tests are then used to estimate the values of the critical parameters. This procedure may fail to provide a precise interpretation of the real phenomenon. The experimental results turn to be very restricted to specific problems and the results cannot be extrapolated to other similar cases.The retention effect associated to particle diffusion is an example of such a case where the classical theory is not adequate. To the best of our knowledge, theories appearing in the current literature addressing this question assume the well-known second order parabolic equation as the basic governing equation of the Paper accepted February, 2011. Technical Editor: Fernando A. Rochinha dispersion process with retention. To solve the problem posed by the retention effect either some extra terms are added to the fundamental diffusion equation or the diffusion coefficient is expanded to introduce higher order terms. It is important to remark that when we talk about retention throughout this paper we are referring to temporary retention in contrast with permanent retention which may be simulated by the introduction of a sink in the governing equations.This paper shows that a simple discrete approach may provide fundamental clues to define a consistent constitutive law adequate to take into account the retention effects in the diffusion process. Indeed, the finite difference equation modeling the retentiondiffusion process, after taking the appropriate limits when the time interval and cell size tend to zero, is reduced to a linear fourth order partial differential equation provided that the problem is restricted to processes in thermodynamic equilibrium, the medium is homogeneous, the material coefficients are constant, and the dependent variable is smooth enough with respect to space and time. The ne...

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