Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-22T01:51:26.405Z Has data issue: false hasContentIssue false
  • English
  • Français

Axisymmertic Stokes flow images in spherical free surfaces with applications to rising bubbles

Published online by Cambridge University Press:  17 February 2009

J. F. Harper
J. F. Harper
Affiliation:
Department of mathematics, Victoria University of Wellington, Private Bag, Wellington, New Zealand.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A theorem is derived for the hydrodynanuc image of an axially symmetric slow viscous (Stokes) flow in a sphere which is impermeable and free of shear stress. A second theorem establishes a sense in which such a flow past an arbitrary rigid surface or shear-free sphere becomes, on inversion in an arbitrary sphere with its centre on the axis of symmetry, a flow past the rigid or shear-free inverse of that surface or sphere.

The theorems are used to simplify the proofs of a number of known results for images of point singularities in plane and spherical rigid and free boundaries, and for a pair of bubbles rising steadily in line in a viscous fluid. They also give for the first time accurate numerical solutions for the velocities of each of a larger number of spherical bubbles rising quasi-steadily in line. These enable one to assess the accuracy of simple approximations to those velocities.

Type
Research Article
Information
The ANZIAM Journal , Volume 25 , Issue 2 , October 1983 , pp. 217 - 231
Copyright
Copyright © Australian Mathematical Society 1983

References

[1]Aderogba, K. and Blake, J. R., ‘Action of a force near the planar surface between two semi-infinite immiscible liquids at very low Reynolds numbers”, Bull. Austral. Math. Soc. 18 (1978), 345–356, and 19 (1979), 309–318.CrossRefGoogle Scholar
[2]Bailey, W. N., Generalised hypergeometric series (Cambridge University Press, 1932).Google Scholar
[3]Berdan, C. and Leal, L. G., “Motion of a sphere in the presence of a deformable interface, I, perturbanation of the interface from flat: the effects of drag and torque. II, A numerical study of the translation of a sphere normal to an interface”, J. Colloid Interface Sci. 87 (1982), 62–80 and 81–106.CrossRefGoogle Scholar
[4]Blake, J. R. and Chwang, A. T., “Fundamental singularities of viscous flow. Part I: The image systems in the vicinity of a stationary no-slip boundary”, J. Engrg. Math. 8 (1974), 23–29.CrossRefGoogle Scholar
[5]Burns, J. C., “A generalization of Milne-Thomson's circle theorem”, J. Math. Phys. Sci. 7 (1973), 373–382.Google Scholar
[6]Butler, S. F. J., “A note Stokes ’ Stream function for motion with a spherical boundary”, Proc. Cambridge Philos. Soc. 49 (1953), 169–174.CrossRefGoogle Scholar
[7]Bromwich, T. J. I'A., An intriduction to the theory of infinite series, 2nd ed. (Macmillan, 1926), p. 387.Google Scholar
[8]Chwang, A. T. and Wu, T. Y. T., ‘Hydromechanics of low-Reynolds number flow. Part 2. Singularity method for Stokes flows”, J. Fluid Mech. 67 (1975), 787–815.CrossRefGoogle Scholar
[9]Collins, W. D.. “A note on Stokes' stream-function for the slow steady motion of viscous fluid before plane and spherical boundaries”, Mathematika 1 (1954), 125–130.CrossRefGoogle Scholar
[10]Collins, W. D., ‘Note on a sphere theorem for the axisymmetric Stokes flow of a viscous fluid”, Mathematika 5 (1958), 118–121.CrossRefGoogle Scholar
[11]Gluckman, M. J., Pfeffer, R. and Weinbaum, S., “A new technique for treating multiparticle slow viscous flow: axisymmetric flow past spheres and spheroids”, J. Fluid Mech. 50 (1971), 705–740.CrossRefGoogle Scholar
[12]Hansen, E. R.. A table of series and products (Prentice-Hall, Englewood Cliffs, 1975).Google Scholar
[13]Happel, J. and Brenner, H., Low Reynolds number hydrodynamics (Prentice-Hall, Englewood Cliffs, 1965).Google Scholar
[14]Harper, J. F.. ‘The motion of bubbles and drops through liquids”, Adv. Appl. Mech. 12 (1972), 59–129.CrossRefGoogle Scholar
[15]Hartland, S., “The effect of circulation patterns on the drainage of the film between a liquid drop and a deformable liquid-liquid interface”, Chem. Engrg. Sci. 24 (1969), 611–613.CrossRefGoogle Scholar
[16]Hetsroni, G. and Haber, S., “The flow in and around a droplet or bubble submerged in an unbounded velocity field”. Rheol. Acta 9 (1970), 488–496.CrossRefGoogle Scholar
[17[Lee, S. H., Chadwick, R. S. and Leal, L. G., “Motion of a sphere in the presence of a plane interface. Part I. An approximate solution by generalization of the method of Lorentz”, J. Fluid Mech. 93 (1979), 705–725.CrossRefGoogle Scholar
[18[Lee, S. H. and Leal, L. G., “Motion of a sphere in the presence of a plane interface. Part II. An exact solution in bipolar coordinates”, J. Fluid Mech. 98 (1980), 193–224.CrossRefGoogle Scholar
[19]Morrison, F. A. Jr, “Breakup of a bubble chain”, Chem. Engrg. Sci. 28 (1973), 1115–1116.CrossRefGoogle Scholar
[20]Nigam, S. D. and Srinivasan, V., [No-slip images in a sphere], J. Math. Phys. Sci. 9 (1975), 389–398.Google Scholar
[21]Pedoe, D., A course of geometry for colleges and universities (Cambridge University Press, 1970).Google Scholar
[22]Pruppacher, H. R. and Klett, J. D., Microphysics and clouds and precipitation (Reidel, Dordrecht, 1978), p. 636.CrossRefGoogle Scholar
[23]Usha, R., “Flow at small Reynolds numbers”, Ph. D. Thesis, Indian Institute of Technology, Madras, 1980.Google Scholar
[24]Wacholder, E. and Weihs, D., “Slow motion of a fluid sphere in the vicinity of another sphere or a plane boundary”, Chem. Engrg. Sci. 27 (1972), 1817–1828.CrossRefGoogle Scholar