Journal of Fluid Mechanics



Transient anomalous diffusion in Poiseuille flow


MARCO LATINI a1 and ANDREW J. BERNOFF a1
a1 Harvey Mudd College, Claremont, CA 91711, USA

Abstract

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)



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