The concentration distribution produced by shear dispersion of solute in Poiseuille flow

Abstract

One of G. I. Taylor's most famous papers concerns the large-time behaviour of a cloud of soluble matter which has been injected into a solvent in laminar flow in a pipe. In the past thirty years, a number of successful attempts have been made to derive differently or extend Taylor's result, which is that the cloud of solute eventually takes a Gaussian profile in the flow direction. The present paper is another examination of this well-worked problem, but this time from the viewpoint of a formal integral transform representation of the solution. This approach leads to a better understanding of the solution; it also enables efficient numerical computations, and leads to extended and new asymptotic expansions. A Laplace transform in time and a Fourier transform in the flow direction leaves a complicated eigenvalue problem to be solved to give the cross-sectional behaviour. This eigenvalue problem is examined in detail, and the transforms are then inverted to give the concentration distribution. Both numerical and asymptotic methods are used. The numerical procedures lead to an accurate description of the concentration distribution, and the method could be generalized to compute dispersion in general parallel flows. The asymptotic procedures use two different classes of eigenvalues to give leading- and trailing-edge approximations for the solute cloud at small times. Meanwhile, at larger times, one eigenvalue branch dominates the solution and Taylor's result is recovered and extended using'the computer to generate extra terms in the approximation. Sixteen terms in the approximation are calculated, and a continued fraction expansion is deduced to enhance the accuracy.