The initial development of longitudinal dispersion in straight tubes

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Abstract
When a contaminant molecule is released in a laminar flow in a straight tube its motion differs from that of the fluid particle with which it initially coincided because of its random motion, whose intensity is measured by the molecular diffusivity κ. For T = kt/a2 ≤ 0·25, where t is the time after release and a is a length characteristic of the cross-section, the statistics of its motion can be determined in the way described by Taylor (1953), Aris (1956) and Chatwin (1970). However, in many applications, including blood flow, the values of T which are attained are much smaller, and the purpose of this paper is to present first approximations for T [Gt ] 1 to some of the statistics measuring the deviations between the motion of the molecule and that of the fluid particle with which it initially coincided. To obtain these a technique due to Saffman (1960) is used for molecules released well away from the tube wall, and an extension of the same technique is used for molecules released near the tube wall. It is shown how the results can be used to describe the initial stages of dispersion of a cloud of contaminant molecules distributed arbitrarily over the cross-section in any laminar flow, steady or unsteady. Comparisons with some exact results for steady Poiseuille flow in a circular tube confirm that the approximations are formally correct, but show that in certain cases their practical use is limited to very small values of T, because of the high coefficients of the first two terms neglected in deriving the approximation.