Transient anomalous diffusion in Poiseuille flow

We revisit the classical problem of dispersion of a point discharge of tracer in laminar pipe Poiseuille flow. For a discharge at the centre of the pipe we show that in the limit of small non-dimensional diffusion, D, tracer dispersion can be divided into three regimes. For small times (t much less than D-1/3), diffusion dominates advection yielding a spherically symmetric Gaussian dispersion cloud. At large times (t much greater than D-1), the flow is in the classical Taylor regime, for which the tracer is homogenized transversely across the pipe and diffuses with a Gaussian distribution longitudinally. However, in an intermediate regime (D-1/3 much less than t much less than D-1), the longitudinal diffusion is anomalous with a width proportional to Dt(2) and a distinctly asymmetric longitudinal distribution. We present a new solution valid in this regime and verify our results numerically. Analogous results are presented for an off-centre release; here the distribution width scales as D(1/2)t(3/2) in the anomalous regime. These results suggest that anomalous diffusion is a hallmark of the shear dispersion of point discharges at times earlier than the Taylor regime.