Skip to main content
    • Aa
    • Aa

Shear dispersion and residence time for laminar flow in capillary tubes

  • J. E. Houseworth (a1)

The behaviour of passive tracer particles in capillary Poiseuille flow is investigated with regard to the residence time in short axial sections of length z, in which z/a < Va/D, where a is the capillary radius, V is the mean velocity and D the coefficient of molecular diffusion. While methods exist for calculating moments of the cross-sectionally averaged axial concentration distribution as a function of time (e.g. Smith 1982b), much less is known about the distribution of residence time as a function of axial distance. An approximate theoretical solution for point sources in high-Péclet-number flows reveals that the mean residence time 〈t(z)〉, which is asymptotic to z/V0 near the source, will then rise faster than z/V0 before converging to z/V for large z, provided the source is not at the capillary wall. V0 is the advective velocity at the point of release. The variance 〈t2(z)〉 is found to increase initially in proportion to z3 provided the source is not at the capillary wall or on the axis. A Monte Carlo method based on the solution to the diffusion equation in the capillary-tube cross-section is developed to compute particle trajectories which are used to analyse both axial and residence-time distributions. The residence-time distribution is found to display significant changes in character as a function of axial position, for both point sources and a uniform flux of particles along the tube.

Hide All
Aris, R. 1956 On the dispersion of a solute in a fluid flowing through a tube. Proc. R. Soc. Lond. A 235, 67.
Barton, N. G. 1983 On the method of moments for solute dispersion. J. Fluid Mech. 126, 205.
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.
Chatwin, P. C. 1976 The initial dispersion of contaminant in Poiseuille flow and the smoothing of the snout. J. Fluid Mech. 77, 593.
Chatwin, P. C. 1977 The initial development of longitudinal dispersion in straight tubes. J. Fluid Mech. 80, 33.
Crank, J. 1956 The Mathematics of Diffusion. Oxford University Press.
Dewey, R. J. & Sullivan, P. J. 1982 Longitudinal-dispersion calculations in laminar flows by statistical analysis of molecular motions. J. Fluid Mech. 125, 203.
Elder, J. W. 1959 The dispersion of marked fluid in turbulent shear flow. J. Fluid Mech. 5, 554.
Fischer, H. B. 1967 The mechanics of dispersion in natural waters. J. Hydraul. Div. ASCE 93, 187.
Gill, W. N. & Ananthakrishnan, V. 1967 Laminar dispersion in capillaries: part 4. The slug stimulus. AIChE J. 13, 801.
Hammersley, J. M. & Handscomb, D. C. 1979 Monte Carlo Methods. Chapman & Hall.
Jayaraj, K. & Subramanian, R., Shankar 1978 On relaxation phenomena in field-flow fractionation. Sep. Sci. Tech. 13, 791.
Knuth, D. E. 1969 Seminumerical Algorithms - The Art of Computer Programming, vol. 2. Addison-Wesley.
Kraichnan, R. H. 1970 Diffusion by a random velocity field. Phys. Fluids 13, 22.
Saffman, P. G. 1959 A theory of dispersion in a porous medium. J. Fluid Mech. 6, 321.
Saffman, P. G. 1960 On the effect of the molecular diffusivity in turbulent diffusion. J. Fluid Mech. 8, 273.
Saffman, P. G. 1962 The effect of wind shear on horizontal spread from an instantaneous ground source. Q. J. R. Met. Soc. 88, 382.
Smith, R. 1982a Non-uniform discharges of contaminants in shear flows. J. Fluid Mech. 120, 71.
Smith, R. 1982b Gaussian approximation for contaminant dispersion. Q. J. Mech. Appl. Math. 35, 345.
Sullivan, P. J. 1971 Longitudinal dispersion within a two-dimensional turbulent shear flow. J. Fluid Mech. 49, 551.
Taylor, G. I. 1953 Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Soc. Lond. A 219, 186.
Taylor, G. I. 1954 The dispersion of matter in turbulent flow through a pipe. Proc. R. Soc. Lond. A 223, 446.
Tsai, Y. H. & Holley, E. R. 1978 Temporal moments for longitudinal dispersion. J. Hydraul. Div. ASCE 12, 1617.
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: 7 *
Loading metrics...

Abstract views

Total abstract views: 77 *
Loading metrics...

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