Multiple-timescale analysis of Taylor dispersion in converging and diverging flows

Abstract

A multiple-timescale analysis is employed to analyse Taylor-dispersion-like convective-diffusive processes in converging and diverging flows. A long-time asymptotic equation governing the cross-sectionally averaged solute probability density is derived. The form of this equation is shown to be dependent upon the number of spatial dimensions characterizing the duct or ‘cone’. The two-dimensional case (non-parallel plates) is shown to be fundamentally different from that for three dimensions (circular cone) in that, in two dimensions, a Taylor dispersion description of the process is possible only for small Peclet numbers or angles of divergence. In contrast, in three dimensions, a Taylor dispersion description is always possible provided sufficient time has passed since the initial introduction of solute into the system. The convective Taylor dispersion coefficients versus the half-vertex angle θ0, the respective dispersivity results for the two cases hardly differ from one another, increasing monotonically from 1.0 for θ0 = 0 to approximately 2.6 for a fully flared duct, θ0 = θ/2. Lastly, the techniques developed above for the case of rectilinear channel and duct boundaries are extended to the case of curvilinear boundaries, and an illustrative calculation performed for the case of axisymmetric flow in a flared Venturi tube.