Abstract
Although the process of hydrodynamic dispersion has been studied for many years, the description of solute spreading at early times has proved to be challenging. In particular, for some kinds of initial conditions, the solute evolution may exhibit a second moment that decreases (rather than increases, as is typically observed) in time. Most classical approaches would predict a negative effective hydrodynamic dispersion coefficient for such a situation. This creates some difficulties: not only does a negative dispersion coefficient lead to a violation of the second law of thermodynamics, but it also creates a mathematically ill-posed problem. We outline a set of four desirable qualities in a well-structured theory of unsteady dispersion as follows: (i) positivity of the dispersion coefficient, (ii) non-dependence upon initial conditions, (iii) superposability of solutions and (iv) convergence of solutions to classical asymptotic results. We use averaging to develop an upscaled result that adheres to these qualities. We find that the upscaled equation contains a source term that accounts for the relaxation of the initial configuration. This term decreases exponentially fast in time, leading to correct asymptotic behaviour while also accounting for the early-time solute dynamics. Analytical solutions are presented for both the effective dispersion coefficient and the source term, and we compare our upscaled results with averaged solutions obtained from numerical simulations; both averaged concentrations and spatial moments are compared. Error estimates are quantified, and we find good correspondence between the upscaled theory and the numerical results for all times.
Highlights
Taylor dispersion, the process of solute spreading in a capillary tube, is the archetype for hydrodynamic dispersive processes
The results were able to capture some features of the early-time behaviour, but were not able to capture exponentially decaying-in-time modes; most likely, this was because the initial condition was not represented as a source term in the averaged equation. (vi) Formulations that include an initial condition source
In the remainder of the background section, we elaborate a little more on two specific issues that arise in the previous work summarized above: (i) The use of infinite-order (Kramers–Moyal-type) expansions, and (ii) the neglect of source terms arising from the initial condition
Summary
The process of solute spreading in a capillary tube, is the archetype for hydrodynamic dispersive processes. C is the concentration of the solute, the angled brackets indicate cross-section average, D∗ is the effective dispersion coefficient, U is the cross-sectional-averaged longitudinal velocity and s∗ is a non-conventional source term that is exponentially decaying in time; z and t are the independent variables representing space and time, respectively While this form for a macroscale balance equation may seem unusual, the source term s∗ is an essential component that arises directly from the upscaling analysis. We give an overview of the literature on Taylor dispersion, with a specific focus on methods that have been developed to handle early time evolution from the initial conditions
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