Journal of Fluid Mechanics

Transient anomalous diffusion in Poiseuille flow

a1 Harvey Mudd College, Claremont, CA 91711, USA


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 [double less-than sign] D−1/3), diffusion dominates advection yielding a spherically symmetric Gaussian dispersion cloud. At large times (t [dbl greater-than sign] 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 [dbl greater-than sign] t [dbl greater-than sign] D−1), the longitudinal diffusion is anomalous with a width proportional to t [double less-than sign] Dt2 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 D1/2t3/2 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.

(Received January 29 2001)
(Revised March 13 2001)