DOI: 10.1017/s0022112088002083. Arrested shear dispersion and other models of anomalous diffusion

Arrested shear dispersion and other models of anomalous diffusion

10.1017/s0022112088002083

Crossref journal-article

Cambridge University Press (CUP)

Journal of Fluid Mechanics (56)

Abstract

The macroscopic dispersion of tracer in microscopically disordered fluid flow can ultimately, at large times, be described by an advection-diffusion equation. But before this asymptotic regime is reached there is an intermediate regime in which first and second spatial moments of the distribution are proportional to tν. Conventional advection-diffusion (which applies at large times) has ν = 1 but in the intermediate regime ν < 1. This phenomenon is referred to as ‘anomalous diffusion’ and this article discusses the special case ν = ½ in detail. This particular value of ν results from tracer dispersion in a central pipe with many stagnant side branches leading away from it. The tracer is “held up” or ‘arrested’ when it wanders into the side branches and so the dispersion in the central duct is more gradual than in conventional advection-diffusion (i.e. ν = ½ < 1). This particular example serves as an entry point into a more general class of models which describe tracer arrest in closed pockets of recirculation, permeable particles, etc. with an integro-differential equation. In this view tracer is arrested and detained at a particular site for a random period. A quantity of fundamental importance in formulating a continuum model of this interrupted random walk is the distribution of stopping times at a site. Distributions with slowly decaying tails (long sojourns) produce anomalous diffusion while the conventional model results from distributions with short tails.

Bibliography

Young, W. R. (1988). Arrested shear dispersion and other models of anomalous diffusion. Journal of Fluid Mechanics, 193, 129â149.

Authors 1

W. R. Young (first)

References 29 Referenced 35

Montroll, E. W. & West, B. J. 1979 On an enriched collection of stochastic processes. In Fluctuation Phenomena (ed. E. W. Montroll & J. L. Lebowitz ). (10.1016/B978-0-444-85248-9.50005-4)
Nadim, A. , Pagitsas, M. & Brenner, H. 1986b Higher-order moments in macrotransport processes.J. Chem. Phys. 85,5238–5245. (10.1063/1.451664)
Sahimi, M. , Hughes, B. D. , Scriven, L. E. & Davis, H. T. 1983 Stochastic transport in disordered systems.J. Chem. Phys. 78,6849–6864. (10.1063/1.444631)
Nadim, A. , Cox, R. G. & Brenner, H. 1986a Taylor dispersion in concentrated suspensions of rotating cylinders.J. Fluid Mech. 164,185–215. (10.1017/S0022112086002525)
Derrida, B. & Pomeau, Y. 1982 Classical diffusion on a random chain.Phys. Rev. Lett. 48,627–630. (10.1103/PhysRevLett.48.627)
Pomeau, Y. , Pumir, A. & Young, W. R. 1987 Transients in the advection and diffusion of impurities.C. R. Acad. Sci. Paris. to be published.
Shen, C. & Floryan, J. M. 1985 Low Reynolds number flow over cavities.Phys. Fluids 28,3191–3202. (10.1063/1.865366)
Smith, R. 1981 A delay-diffusion description for contaminant dispersion.J. Fluid Mech. 105,469–486. (10.1017/S0022112081003297)
Bretherton, F. P. & Haidvogel, D. B. 1976 Two-dimensional turbulence above topography.J. Fluid Mech. 78,129–154. (10.1017/S002211207600236X)
Guyon, E. , Hulin, J. P. , Baudet, C. & Pomeau, Y. 1987 Dispersion in the presence of recirculation zones.Proceedings of Chaos 1987. To be published in Nucl. Phys. (10.1016/0920-5632(87)90023-5)
Rhines, P. B. & Young, W. R. 1983 How rapidly is a passive scalar mixed within closed streamlines? J. Fluid Mech. 133,133–145. (10.1017/S0022112083001822)
Sneddon, I. N. 1972 The Use of Integral Transforms .McGraw Hill,539 pp.
Gill, W. M. & Sankarasubramanian, R. 1972 Dispersion of nonuniformly distributed time variable continuous sources in a time-dependent flow..Proc. R. Soc. Lond. A327,191–208. (10.1098/rspa.1972.0040)
Mandelbrot, B. B. 1983 The Fractal Geometry of Nature .Freeman,468 pp.
Brenner, H. 1980 Dispersion resulting from flow through spatially periodic porous media..Phil. Trans. R. Soc. Lond. A297,81–133. (10.1098/rsta.1980.0205)
Coullet, P. H. & Spiegel, E. A. 1983 Amplitude equations for systems with competing instabilities.SIAM J. Appl. Maths. 43,776–821. (10.1137/0143052)
Koch, D. L. & Brady, J. F. 1985 Dispersion in fixed beds.J. Fluid Mech. 154,399–427. (10.1017/S0022112085001598)
Feller, W. 1971 An Introduction to Probability Theory and its Applications , vol. 2.Wiley 669 pp.
Roberts, A. J. 1988 The application of centre manifold theory to the evolution of systems which vary slowly in space.Bull. Austral. Math. Soc. B (to appear). (10.1017/S0334270000005968)
Taylor, G. I. 1953 Dispersion of soluble matter in solvent flowing slowly through a tube..Proc. R. Soc. Lond. A219,186–203. (10.1098/rspa.1953.0139)
Whittaker, E. T. & Watson, G. N. 1927 A Course of Modern Analysis ,4th edn. Cambridge University Press,608 pp.
Gill, A. E. & Smith, R. K. 1970 On similarity solutions of the differential equation = 0.Proc. Camb. Phil. Soc. 67,163–171. (10.1017/S0305004100057200)
Aris, R. 1956 On the dispersion of solute in a fluid flowing through a tube..Proc. R. Soc. Lond. A235,67–77. (10.1098/rspa.1956.0065)
Stratonovich, R. L. 1967 Topics in the Theory of Random Noise , vol. 2.Gordon and Breach,329 pp.
Saffman, P. N. 1959 A theory of dispersion in porous media.J. Fluid Mech. 6,321–349. (10.1017/S0022112059000672)
Hughes, B. D. , Montroll, E. W. & Shlesinger, M. F. 1982 Fractal random walks.J. Stat. Phys. 28,111–126. (10.1007/BF01011626)
Bleistein, N. & Handelsman, R. A. 1975 Asymptotic Expansions of Integrals .Dover,425 pp.
Smith, R. 1983 Effect of boundary absorption upon longitudinal dispersion in shear flows.J. Fluid Mech. 134,161–177. (10.1017/S0022112083003286)
Gill, W. M. & Sankarasubramanian, R. 1970 Exact analysis of unsteady advection diffusion..Proc. R. Soc. Lond. A316,341–350. (10.1098/rspa.1970.0083)

Dates

Type	When
Created	19 years, 4 months ago (April 21, 2006, 2:16 p.m.)
Deposited	2 months, 1 week ago (June 21, 2025, 3:22 a.m.)
Indexed	2 months, 1 week ago (June 22, 2025, 12:02 a.m.)
Issued	37 years, 1 month ago (Aug. 1, 1988)
Published	37 years, 1 month ago (Aug. 1, 1988)
Published Online	19 years, 4 months ago (April 21, 2006)
Published Print	37 years, 1 month ago (Aug. 1, 1988)

Funders 0

None

BibTeX

@article{Young_1988, title={Arrested shear dispersion and other models of anomalous diffusion}, volume={193}, ISSN={1469-7645}, url={http://dx.doi.org/10.1017/s0022112088002083}, DOI={10.1017/s0022112088002083}, journal={Journal of Fluid Mechanics}, publisher={Cambridge University Press (CUP)}, author={Young, W. R.}, year={1988}, month=aug, pages={129–149} }

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