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 C̄ , 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 κ t/a² < 0.25(ūa/κ)-2/3 , where ū is the discharge velocity, the effect of longitudinal diffusion dominates over that of the interaction, which is, in turn, dominant for κ t/a² > 2.5(ūa/κ)-2/3 , when C̄ is close to the form described by Lighthill (1966).