Hostname: page-component-848d4c4894-mwx4w Total loading time: 0 Render date: 2024-06-29T00:42:26.358Z Has data issue: false hasContentIssue false
  • English
  • Français

A uniformly asymptotic approximation for the development of shear dispersion

Published online by Cambridge University Press:  26 April 2006

C. G. Phillips  and
S. R. Kaye
C. G. Phillips
Affiliation:
Physiological Flow Studies Group, Centre for Biological and Medical Systems, Imperial College of Science, Technology and Medicine, Prince Consort Road, London SW7 2BY, UK Department of Mathematics, Imperial College of Science, Technology and Medicine, Prince Consort Road, London SW7 2BY, UK
S. R. Kaye
Affiliation:
Physiological Flow Studies Group, Centre for Biological and Medical Systems, Imperial College of Science, Technology and Medicine, Prince Consort Road, London SW7 2BY, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we consider the development of shear dispersion following the introduction of a diffusing tracer substance into a tube or duct containing flowing fluid, with emphasis on the characterization of the temporal variation of concentration at a fixed axial position. Asymptotic results are derived by assuming that the distance downstream of the point of tracer introduction, appropriately non-dimensionalized, is large. First, we consider the central moments of the temporal concentration variation, including their dependence on transverse position and on the initial transverse distribution of tracer. The moments for finite Péclet number are expressed in terms of their infinite-Péclet-number counterparts, and the latter are given explicitly for Poiseuille flow. Then, assuming the Péclet number is infinite, we derive an approximate solution for the Green's function expressing tracer concentration following its introduction at an arbitrary point within the tube. The solution is expressed in terms of three numerically evaluated functions of a dimensionless time variable, with parametric dependence on the distance downstream of the point of tracer release. The method is illustrated by calculation of the approximate solution for dispersion in Poiseuille flow. Unlike previous approximations, the present solution is uniformly asymptotic and represents the tails of the concentration distribution as well as the approximately Gaussian central part; in these three regions, simpler analytic forms of the approximation are given. Comparison with previous computational solutions suggests the present approximation remains reasonably accurate even at quite short distances from the point where tracer is released.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 329 , 25 December 1996 , pp. 413 - 443
Copyright
© 1996 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aris, R. 1956 On the dispersion of a solute in a fluid flowing through a tube. Proc. R. Soc. Lond. A 235, 67–77.Google Scholar
Camacho, J. 1993 Purely global model for Taylor dispersion. Phys. Rev. E 48, 310–321.Google Scholar
Chatwin, P. C. 1970 The approach to normality of the concentration distribution of a solute in a solvent flowing along a straight pipe. J. Fluid Mech. 43, 321–352.Google Scholar
Chatwin, P. C. 1980 Presentation of longitudinal dispersion data. J. Hydraul. Div. ASCE 106, 71–83.Google Scholar
Gill, W. N. & Sankarasubramanian, R. 1970 Exact analysis of unsteady convective diffusion. Proc. R. Soc. Lond. A 316, 341–350.Google Scholar
Gill, W. N. & Sankarasubramanian, R. 1971 Dispersion of a non-uniform slug in time-dependent flow. Proc. R. Soc. Lond. A 322, 101–117.Google Scholar
Levenspiel, O. & Smith, W. K. 1957 Notes on the diffusion-type model for the longitudinal mixing of fluids in flow. Chem. Engng Sci. 6, 227–233.Google Scholar
Lighthill, M. J. 1966 Initial development of diffusion in Poiseuille flow. J. Inst. Maths Applics. 2, 97–108.Google Scholar
Shankar, A. & Lenhoff, A. M. 1989 Dispersion in laminar flow in short tubes. AIChE J. 35, 2048–2052.Google Scholar
Shankar, A. & Lenhoff, A. M. 1991 Dispersion in round tubes and its implications for extracolumn dispersion. J. Chromatogr. 556, 235–248.Google Scholar
Smith, R. 1981a A delay-diffusion description for contaminant dispersion. J. Fluid Mech. 105, 469–486.Google Scholar
Smith, R. 1981b The importance of discharge siting upon contaminant dispersion in narrow rivers and estuaries. J. Fluid Mech. 108, 43–53.Google Scholar
Smith, R. 1982a Gaussian approximation for contaminant dispersion. Q. J. Mech. Appl. Maths 35, 345–366.Google Scholar
Smith, R. 1982b Non-uniform discharges of contaminants in shear flows. J. Fluid Mech. 120, 71–89.Google Scholar
Smith, R. 1984 Temporal moments at large distances downstream of contaminant releases in rivers. J. Fluid Mech. 140, 153–174.Google Scholar
Smith, R. 1985 Contaminant dispersion as viewed from a fixed position. J. Fluid Mech. 152, 217–233.Google Scholar
Smith, R. 1987a Diffusion in shear flows made easy: the Taylor limit. J. Fluid Mech. 175, 201–214.Google Scholar
Smith, R. 1987b Shear dispersion looked at from a new angle. J. Fluid Mech. 182, 447–466.Google Scholar
Smith, R. 1990 An easy-to-use formula for contaminant dispersion. J. Fluid Mech. 215, 195–207.Google Scholar
Smith, R. 1995 Multi-mode models of flow and of solute dispersion in shallow water. Part 1. General derivation. J. Fluid Mech. 283, 231–248.Google Scholar
Stokes, A. N. & Barton, N. G. 1990 The concentration distribution produced by shear dispersion of solute in Poiseuille flow. J. Fluid Mech. 210, 201–221.Google Scholar
Taylor, G. I. 1953 Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Soc. Lond. A 219, 186–203.Google Scholar
Taylor, G. I. 1954 Conditions under which dispersion of a solute in a stream of solvent can be used to measure molecular diffusion. Proc. R. Soc. Land. A 225, 473–477.Google Scholar
Tsai, Y. H. & Holley, E. R. 1978 Temporal moments for longitudinal dispersion. J. Hydraul. Div. ASCE 104, 1617–1634.Google Scholar
Tsai, Y. H. & Holley, E. R. 1980 Temporal moments for longitudinal dispersion. J. Hydraul. Div. ASCE 106, 2063–2066.Google Scholar
Ultman, J. S. & Weaver, D. W. 1979 Concentration sampling methods in relation to the computation of a dispersion coefficient. Chem. Engng Sci. 34, 1172–1174.Google Scholar
Young, W. R. & Jones, S. 1991 Shear dispersion. Phys. Fluids A 3, 1087–1101.Google Scholar