A cloud of solute injected into a pipe or channel is known to spread out by a dispersion process based on cross-sectional diffusion across a velocity shear. The original description of the process is due to Taylor (1953, 1954), and an important subsequent contribution was by Aris (1956), who framed and partially solved equations for the integral moments of the cloud of contaminant. The present work resolves some technical difficulties that occur when Aris’ solution method (separation of variables) is pursued in depth. In particular, it is shown that Aris’ technique has to be modified to give the moments at short and moderate times after the injection of solute into the flow. The paper is concerned with dispersion in those parallel flows for which an associated eigenvalue problem has a discrete spectrum of eigenvalues; fortunately, this case appears to be the rule rather than the exception. Expressions are obtained for the second and third moment about the mean, and the theory is applied to three cases of interest.

The classic problem, first treated by Taylor [18], of the dispersion of inert soluble matter in fluid flow continues to attract attention from researchers describing the approach to the asymptotic state [5,17]. The present article considers some of the complications caused when the solute is chemically active. Dispersing chemically active solutes occur in diverse fields such as chromatography, chemical engineering and environmental fluid mechanics.The asymptotic large-time analysis of Chatwin [5] is re-worked to handle the case of reactive solutes dispersing in parallel flow. Matching between moderate and large-time solutions requires consideration of the integral moments of the reactive contaminant cloud, and the Aris method of moments is therefore invoked and modified for reaction effects. The results are applied in detail to the outstanding practical example-the chemical flow reactor (a device used to measure reaction rates for chemical reactions taking place between fluids). For this case, the paper provides a practical alternative to recent variable diffusion coefficient studies [6,7,15], and presents further results for the concentration distribution and for the limiting behaviour under weak and vigorous reactions at the boundary of the flow.

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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.

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