Multiple time scale analysis of macrotransport processes

Abstract

Convective and diffusive transport of a Brownian tracer corpuscle is analyzed within a multidimensional space decomposed into local (internal) and global (external) subspaces. Multiple time scale methods are employed to successively eliminate from the kinetic equation governing the multidimensional microtransport process its dependence upon each of the internal coordinates. The resulting long-time equation describing the residual global-space macrotransport process derived by this systematic perturbation procedure is shown to accord with the well-established results of generalized Taylor dispersion theory, heretofore obtained by ad hoc arguments. By way of example, this macrotransport description is used to analyze the Taylor dispersion of a solute in a Poiseuille-type solvent flow occuring within a rectangular duct of small, but non-zero, aspect ratio. Elementary dispersion results obtained by ignoring the side walls apply only for relatively short times; for longer times the perturbing presence of the side walls acts to substantially increase the axial dispersivity, even in the limit of zero aspect ratio-where intuition would strongly suggest otherwise.