The initial dispersion of contaminant in Poiseuille flow and the smoothing of the snout

In Poiseuille flow in a circular tube passive contaminant initially spread uniformly over the cross-section would be pulled out in a paraboloidal snout in the absence of any diffusive mechanism, and there would be a discontinuity in $\overline{C}$, the mean concentration over the cross-section, associated with the contaminant at the front of the snout. In reality molecular diffusion smooths out this snout in two ways: direct longitudinal diffusion and the interaction between lateral diffusion and advection. The effect of these two mechanisms is discussed, and determined for small values of κt/a², where t is the time since injection, κ is the molecular diffusivity and a is the tube radius. For such values, important in many applications, the tube walls play no part in the smoothing process. It is shown that for $\kappa t/a^2 < 0.25(\overline{u}a/\kappa)^{-\frac{2}{3}}$, where $\overline{u}$ is the discharge velocity, the effect of longitudinal diffusion dominates over that of the interaction, which is, in turn, dominant for $\kappa t/a^2 > 2.5(\overline{u}a/\kappa)^{-\frac{2}{3}}$, when $\overline{C}$ is close to the form described by Lighthill (1966).