On the foundations of generalized Taylor dispersion theory

Abstract

Generalized Taylor dispersion theory extends the basic long-time, asymptotic scheme of Taylor and Aris greatly beyond the class of rectilinear duct and channel flow dispersion problems originally addressed by them. This feature has rendered it indispensable for studying flow and dispersion phenomena in porous media, chromatographic separation processes, heat transfer in cellular media, sedimentation of non-spherical Brownian particles, and transport of flexible clusters of interacting Brownian particles, to mention just a few examples of the broad class of non-unidirectional transport phenomena encompassed by this scheme. Moreover, generalized Taylor dispersion theory enjoys the attractive feature of conferring a unified paradigmatic structure upon the analysis of such apparently disparate physical problems. For each of the problems thus treated it provides an asymptotic, macroscale description of the original microscale transport process, being based upon a convective-diffusive ‘model’ problem characterized by a set of constant (position- and time-independent) phenomenological coefficients.The present contribution formally substantiates the scheme. This is accomplished by demonstrating that the coarse-grained (macroscale) transport ‘model’ equation leads to a solution which accords asymptotically with the leading-order behaviour of the comparable solution of the exact (microscale) convective–diffusive problem underlying the transport process. It is also shown, contrary to current belief, that no systematic improvement in the asymptotic order of approximation is possible through the incorporation of higher-order gradient terms into the model constitutive equation for the coarse-grained flux. Moreover, the inherent difference between the present rigorous asymptotic scheme and the dispersion models resulting from Gill–Subramanian moment-gradient expansions is illuminated, thereby conclusively resolving a long-standing puzzle in longitudinal dispersion theory.