Taylor (1953, 1954a) showed that, when a cloud of solute is injected into a pipe through which a solvent is flowing, it spreads out, so that the distribution of concentration C is eventually a Gaussian function of distance along the pipe axis. This paper is concerned with the approach to this final form. An asymptotic series is derived for the distribution of concentration based on the assumption that the diffusion of solute obeys Fick's law. The first term is the Gaussian function, and succeeding terms describe the asymmetries and other deviations from normality observed in practice. The theory is applied to Poiseuille flow in a pipe of radius a and it is concluded that three terms of the series describe C satisfactorily if Dt/a2 > 0·2 (where D is the coefficient of molecular diffusion), and that the initial distribution of C has little effect on the approach to normality in most cases of practical importance. The predictions of the theory are compared with numerical work by Sayre (1968) for a simple model of turbulent open channel flow and show excellent agreement. The final section of the paper presents a second series derived from the first which involves only quantities which can be determined directly by integration from the observed values of C without knowledge of the velocity distribution or diffusivity. The latter series can be derived independently of the rest of the paper provided the cumulants of C tend to zero fast enough as t → ∞, and it is suggested, therefore, that the latter series may be valid in flows for which Fick's law does not hold.

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The paper examines how a passive contaminant disperses along the axis of a tube in which the flow is driven by a longitudinal pressure gradient varying harmonically with time. This problem is of intrinsic interest and is relevant to some important practical problems. Two examples are dispersion in estuaries and in the blood stream. By means both of statistical arguments and an analysis like that used by Taylor (1953) in the case of a steady pressure gradient it is shown that eventually the mean distribution of concentration satisfies a diffusion equation (and is therefore a Gaussian function of distance along the axis) with an effective longitudinal diffusion coefficient K(t) which is a harmonic function of time with a period equal to one half of that of the imposed pressure gradient. Contrary to the supposition made in most previous work on this problem it is shown by examining some special cases that the harmonic terms in K(t) may have a noticeable effect on the dispersion of the contaminant and in particular on the rate at which it is spreading axially. The size of the effect depends on both the frequency and the Schmidt number and is particularly large at low frequencies. The paper concludes with an analysis of a model of dispersion in estuaries which has been used frequently and it is concluded that here too oscillatory effects may often be noticeable.