Skip to main content
    • Aa
    • Aa

The approach to normality of the concentration distribution of a solute in a solvent flowing along a straight pipe

  • P. C. Chatwin (a1)

Taylor (1953, 1954a) showed that, when a cloud of solute is injected into a pipe through which a solvent is flowing, it spreads out, so that the distribution of concentration C is eventually a Gaussian function of distance along the pipe axis. This paper is concerned with the approach to this final form. An asymptotic series is derived for the distribution of concentration based on the assumption that the diffusion of solute obeys Fick's law. The first term is the Gaussian function, and succeeding terms describe the asymmetries and other deviations from normality observed in practice. The theory is applied to Poiseuille flow in a pipe of radius a and it is concluded that three terms of the series describe C satisfactorily if Dt/a2 > 0·2 (where D is the coefficient of molecular diffusion), and that the initial distribution of C has little effect on the approach to normality in most cases of practical importance. The predictions of the theory are compared with numerical work by Sayre (1968) for a simple model of turbulent open channel flow and show excellent agreement. The final section of the paper presents a second series derived from the first which involves only quantities which can be determined directly by integration from the observed values of C without knowledge of the velocity distribution or diffusivity. The latter series can be derived independently of the rest of the paper provided the cumulants of C tend to zero fast enough as t → ∞, and it is suggested, therefore, that the latter series may be valid in flows for which Fick's law does not hold.

Hide All
Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions. New York: Dover.
Aris, R. 1956 Proc. Roy. Soc. A 235, 67.
Aris, R. 1958 Proc. Roy. Soc. A 245, 277.
Batchelor, G. K. 1966 Proc. 2nd Aust. Conference on Hydraulics and Fluid Mechanics.
Elder, J. W. 1959 J. Fluid Mech. 5, 54.
Fischer, H. B. 1966 California Institute of Technology, Rep. KH-R-12.
Gill, W. N. 1967 Proc. Roy. Soc. A 298, 335.
Kendall, M. G. & Stuart, A. 1958 The Advanced Theory of Statistics, Vol. i. London: Griffin.
Lighthill, M. J. 1966 J. Inst. Math. Applic. 2, 9.
Longuet-Higgins, M. S. 1963 J. Fluid Mech. 17, 45.
O'Hara, K. 1969 M.Sc. Thesis, University of Liverpool.
Sayre, W. W. 1968 Colorado State University Hydraulics Paper no. 3, Fort Collins.
Taylor, G. I. 1953 Proc. Roy. Soc. A 219, 186.
Taylor, G. I. 1954a Proc. Roy. Soc. A 223, 446.
Taylor, G. I. 1954b Proc. Roy. Soc. A 225, 473.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 53 *
Loading metrics...

Abstract views

Total abstract views: 157 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 19th October 2017. This data will be updated every 24 hours.