Upstream Diffusion in Hagen-Poiseuille Flow

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Abstract
A theoretical study is described for steady state diffusion of a contaminant from a large reservoir of static fluid into a cylindrical tube discharging fluid into the reservoir. Contaminant decay is considered. Numerical results are presented for the rate of decrease along the tube of the concentration of contaminant for Reynolds × Schmidt numbers > 10 and either small or zero decay. The method used explicitly exhibits quantities of physical interest including a simple form for the asymptotic solution. These results are of practical use in estimating the contamination in the tube at points that are relatively close to the mouth of the tube. An estimate is made of the distance from the tube mouth at which the asymptotic solution given becomes valid. Used for the prediction of heat flow, the results are valid for Péclet numbers > 10.

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