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 [Lt ] D −1/3 ), diffusion dominates advection yielding a spherically symmetric Gaussian dispersion cloud. At large times ( t [Gt ] 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 [Gt ] t [Gt ] D −1 ), the longitudinal diffusion is anomalous with a width proportional to t [Lt ] 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.