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On the longitudinal dispersion of passive contaminant in oscillatory flows in tubes

Published online by Cambridge University Press:  29 March 2006

P. C. Chatwin
P. C. Chatwin
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Liverpool
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Abstract

The paper examines how a passive contaminant disperses along the axis of a tube in which the flow is driven by a longitudinal pressure gradient varying harmonically with time. This problem is of intrinsic interest and is relevant to some important practical problems. Two examples are dispersion in estuaries and in the blood stream. By means both of statistical arguments and an analysis like that used by Taylor (1953) in the case of a steady pressure gradient it is shown that eventually the mean distribution of concentration satisfies a diffusion equation (and is therefore a Gaussian function of distance along the axis) with an effective longitudinal diffusion coefficient K(t) which is a harmonic function of time with a period equal to one half of that of the imposed pressure gradient. Contrary to the supposition made in most previous work on this problem it is shown by examining some special cases that the harmonic terms in K(t) may have a noticeable effect on the dispersion of the contaminant and in particular on the rate at which it is spreading axially. The size of the effect depends on both the frequency and the Schmidt number and is particularly large at low frequencies. The paper concludes with an analysis of a model of dispersion in estuaries which has been used frequently and it is concluded that here too oscillatory effects may often be noticeable.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 71 , Issue 3 , 14 October 1975 , pp. 513 - 527
Copyright
© 1975 Cambridge University Press

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References

Aris, R. 1956 On the dispersion of a solute in a fluid flowing through a tube. Proc. Roy. Soc. A 235, 67.Google Scholar
Batchelor, G. K., Binnie, A. M. & Phillips, O. M. 1955 The mean velocity of discrete particles in turbulent flow in a pipe. Proc. Phys. Soc. B 68, 1095.Google Scholar
Batchelor, G. K. & Townsend, A. A. 1956 ‘Turbulent diffusion’. In Surveys in Mechanics (ed. Batchelor & Davies), p. 352. Cambridge University Press.
Bowden, K. F. 1965 Horizontal mixing in the sea due to a shearing current. J. Fluid Mech. 21, 84.Google Scholar
Caro, C. G. 1966 The dispersion of indicator flowing through simplified models of the circulation and its relevance to velocity profile in blood vessels. J. Physiol. 185, 501.Google Scholar
Chatwin, P. C. 1970 The approach to normality of the concentration distribution of solute in a solvent flowing along a straight pipe. J. Fluid Mech. 43, 321.Google Scholar
Chatwin, P. C. 1972 The cumulants of the distribution of concentration of a solute dispersing in solvent flowing through a tube. J. Fluid Mech. 51, 63.Google Scholar
Erdogan, M. E. & Chatwin, P. C. 1967 The effects of curvature and buoyancy on the longitudinal dispersion of solute in a horizontal tube. J. Fluid Mech. 29, 465.Google Scholar
Fischer, H. B. 1966 Longitudinal dispersion in laboratory and natural streams. California Inst. Tech. Rep. KH-R-12.Google Scholar
Fischer, H. B. 1972 Mass transport mechanisms in partially stratified estuaries. J. Fluid Mech. 53, 671.Google Scholar
Fukuoka, A. 1973 Longitudinal dispersion of matter in alternating shear flows. Dept. Engng. James Cook University of North Queensland, Res. Bull. C9.
Holley, E. R. & Harleman, D. R. F. 1965 Dispersion of pollutants in estuary type flows. Hydrodyn. Lab. M.I.T., Rep. no. 74.Google Scholar
Holley, E. R., Harleman, D. R. F. & Fischer, H. B. 1970 Dispersion in homogeneous estuary flow. J. Hyd. Div. A.S.C.E. 96 (HY8), 1691.Google Scholar
Lighthill, M. J. 1966 Initial development of diffusion in Poiseuille flow. J. Inst. Math. Appl. 2, 97.Google Scholar
Mcdonald, D. A. 1960 Blood Flow in Arteries. London: Edward Arnold.
Okubo, A. 1967 The effect of shear in an oscillatory current on horizontal diffusion from an instantaneous source. Int. J. Oceanol. Limnol. 1, 194.Google Scholar
Taylor, G. I. 1921 Diffusion by continuous movements. Proc. Lond. Math. Soc. 20, 196.Google Scholar
Taylor, G. I. 1953 Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. Roy. Soc. A 219, 186.Google Scholar
Taylor, G. I. 1954 The dispersion of matter in turbulent flow through a pipe. Proc. Roy. Soc. A 223, 446.Google Scholar
Watson, E. J. 1975 To be published.