I n this paper the streamwise growth of a passive contaminant cloud in the laminar flow within a uniform conduit is investigated. A probabilistic formulation, based on the Lagrangian motion of typical marked fluid molecules, is used to gain insight into the complex dispersion problem that exists at times that are significantly smaller (and often of more practical relevance) than those required for the asymptotic case discussed by Taylor (1953) to be valid.
Previous investigations of the small-time spread of a contaminant cloud in a tube by Lighthill (1966) and Chatwin (1976, 1977) were primarily concerned with a cloud near a tube axis as may be appropriate to an injection into the flow in arteries. When the contaminant is more uniformly spread over the conduit cross-section it is shown that, even at quite small times, the conduit boundary has a very pronounced influence on the streamwise contaminant distribution. Such a situation occurs, for example, when extracting sample fluid from a flow by means of a small-diameter sampling tube.
The streamwise spread of the contaminant cloud that results from an initial sheet of contaminant, spread uniformly over the conduit cross-section, is shown to depend critically on the Lagrangian mean-velocity history of a typical fluid molecule. This mean-velocity history function generally (and necessarily) is distinguished by a ‘hump’ whose location is determined by the proximity of the molecule's release position to the nearest conduit boundary. The ‘hump’ is a more-pronounced feature for release positions near a boundary than i t is for contaminant molecules released near the conduit centre, where the ‘hump’ becomes almost indiscernible.
The specific case of flow between parallel plates is investigated using a random-walk model of the process. A significant difference is found from the results of an analysis that excludes the influence of the conduit boundaries on the streamwise contaminant distribution at times $t = O[d^4/\kappa U^2]^{\frac{1}{3}}$, where U is the mean-flow velocity, κ is the molecular diffusivity and d is the plate separation distance.