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Numerical solution of free-boundary problems in fluid mechanics. Part 2. Buoyancy-driven motion of a gas bubble through a quiescent liquid

Published online by Cambridge University Press:  20 April 2006

G. Ryskin  and
L. G. Leal
G. Ryskin
Affiliation:
Department of Chemical Engineering, California Institute of Technology, Pasadena, California 91125 Present address: Department of Chemical Engineering, Northwestern University, Evanston, Illinois 60201.
L. G. Leal
Affiliation:
Department of Chemical Engineering, California Institute of Technology, Pasadena, California 91125
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Abstract

In this paper numerical results are presented for the buoyancy-driven rise of a deformable bubble through an unbounded quiescent fluid. Complete solutions, including the bubble shape, are obtained for Reynolds numbers in the range 1 ≤ R ≤ 200 and for Weber numbers up to 20. For Reynolds numbers R ≤ 20 the shape of the bubble changes from nearly spherical to oblate-ellipsoidal to spherical-cap depending on Weber number; at higher Reynolds numbers ‘disk-like’ and ‘saucer-like’ shapes appear at W = O(10). The present results show clearly that flow separation may occur at a smooth free surface at intermediate Reynolds numbers; this fact suggests a qualitative explanation of the often-observed irregular (zigzag or helical) paths of rising bubbles.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 148 , November 1984 , pp. 19 - 35
Copyright
© 1984 Cambridge University Press

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References

Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Bhaga, D. & Weber, M. E. 1981 Bubbles in viscous liquids: shapes, wakes and velocities. J. Fluid Mech. 105, 61–85.Google Scholar
Brabston, D. C. & Keller, H. B. 1975 Viscous flows past spherical gas bubbles. J. Fluid Mech. 69, 179–189.Google Scholar
Brignell, A. S. 1973 The deformation of a liquid drop at small Reynolds number. Q. J. Mech. Appl. Maths 26, 99–107.Google Scholar
Collins, R. 1966 A second approximation for the velocity of a large gas bubble rising in an infinite fluid. J. Fluid Mech. 25, 469–480.Google Scholar
Davies, R. M. & Taylor, G. I. 1950 The mechanics of large bubbles rising through extended liquids and through liquids in tubes. Proc. R. Soc. Lond. A 200, 375–390.Google Scholar
Haberman, W. L. & Morton, R. K. 1953 An experimental investigation of the drag and shape of air bubbles rising in various liquids. David Taylor Model Basin Rep. 802.Google Scholar
Hartunian, R. A. & Sears, W. R. 1957 On the instability of small gas bubbles moving uniformly in various liquids. J. Fluid Mech. 3, 27–47.Google Scholar
Hnat, J. G. & Buckmaster, J. D. 1976 Spherical cap bubbles and skirt formation. Phys. Fluids 19, 182–194.Google Scholar
Kojima, M., Hinch, E. J. & Acrivos, A. 1984 The formation and expansion of a toroidal drop moving in a viscous fluid. Phys. Fluids 27, 19–32.Google Scholar
Lane, W. R. & Green, H. L. 1956 The mechanics of drops and bubbles. In Surveys in Mechanics (ed. G. K. Batchelor & R. M. Davies), pp. 162–215. Cambridge University Press.
Leal, L. G. & Acrivos, A. 1969 The effect of base bleed on the steady separated flow past bluff objects. J. Fluid Mech. 39, 735–752.Google Scholar
MacCready, P. B. & Jex, H. R. 1964 Study of sphere motion and balloon wind sensors. NASA TM X-53089.Google Scholar
Miksis, M., Vanden-Broeck, J.-M. & Keller, J. B. 1981 Axisymmetric bubble or drop in a uniform flow. J. Fluid Mech. 108, 89–100.Google Scholar
Miksis, M., Vanden-Broeck, J.-M. & Keller, J. B. 1982 Rising bubbles. J. Fluid Mech. 123, 31–41.Google Scholar
Moore, D. W. 1959 The rise of a gas bubble in a viscous liquid. J. Fluid Mech. 6, 113–130.Google Scholar
Moore, D. W. 1965 The velocity of rise of distorted gas bubbles in a liquid of small viscosity. J. Fluid Mech. 23, 749–766.Google Scholar
Nakamura, I. 1976 Steady wake behind a sphere. Phys. Fluids 19, 5–8.Google Scholar
Preukschat, A. W. 1962 Measurements of drag coefficients for falling and rising spheres in free motion. M.S. thesis, Department of Aeronautical Engineering, California Institute of Technology.
Rivkind, V. Y. & Ryskin, G. 1976 Flow structure in motion of a spherical drop in a fluid medium at intermediate Reynolds numbers. Fluid Dyn. 11, 5–12.Google Scholar
Ryskin, G. & Leal, L. G. 1984 Numerical solution of free-boundary problems in fluid mechanics. Part 1. The finite-difference technique. J. Fluid Mech. 148, 1–17.Google Scholar
Saffman, P. G. 1956 On the rise of small air bubbles in water. J. Fluid Mech. 1, 249–275.Google Scholar
Taylor, T. D. & Acrivos, A. 1964 On the deformation and drag of a falling viscous drop at low Reynolds number. J. Fluid Mech. 18, 466–476.Google Scholar
Tsuge, H. & Hibino, S. I. 1977 Onset conditions of oscillatory motion of single gas-bubbles rising in various liquids. J. Chem. Engng Japan 10, 66–68.Google Scholar
Wegener, P. P. & Parlange, J.-Y. 1973 Spherical-cap bubbles. Ann. Rev. Fluid Mech. 5, 79–100.Google Scholar
Willmarth, W. W., Hawk, N. E. & Harvey, R. L. 1964 Steady and unsteady motions and wakes of freely falling disks. Phys. Fluids 7, 197–208.Google Scholar