A uniformly asymptotic approximation for the development of shear dispersion

Abstract

In this paper we consider the development of shear dispersion following the introduction of a diffusing tracer substance into a tube or duct containing flowing fluid, with emphasis on the characterization of the temporal variation of concentration at a fixed axial position. Asymptotic results are derived by assuming that the distance downstream of the point of tracer introduction, appropriately non-dimensionalized, is large. First, we consider the central moments of the temporal concentration variation, including their dependence on transverse position and on the initial transverse distribution of tracer. The moments for finite Péclet number are expressed in terms of their infinite-Péclet-number counterparts, and the latter are given explicitly for Poiseuille flow. Then, assuming the Péclet number is infinite, we derive an approximate solution for the Green's function expressing tracer concentration following its introduction at an arbitrary point within the tube. The solution is expressed in terms of three numerically evaluated functions of a dimensionless time variable, with parametric dependence on the distance downstream of the point of tracer release. The method is illustrated by calculation of the approximate solution for dispersion in Poiseuille flow. Unlike previous approximations, the present solution is uniformly asymptotic and represents the tails of the concentration distribution as well as the approximately Gaussian central part; in these three regions, simpler analytic forms of the approximation are given. Comparison with previous computational solutions suggests the present approximation remains reasonably accurate even at quite short distances from the point where tracer is released.