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On the motion of bubbles in a periodic box

Published online by Cambridge University Press:  26 April 2006

Peter Smereka
Peter Smereka
Affiliation:
Department of Mathematics, University of California, Los Angeles, CA 90024-1555, USA
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Abstract

The motion of spherical bubbles in a box containing an incompressible and irrotational liquid with periodic boundary conditions is studied. Equations of motion are deduced using a variational principle. When the bubbles have approximately the same velocity their configuration is Lyapunov stable, provided they are arranged in such a way so as to minimize the effective conductivity of a composite material where the bubbles are treated as insulators. The minimizing configurations become asymptotically stable with the addition of gravity and liquid viscosity. This suggests that a randomly arranged configuration of bubbles, all with approximately the same velocity, cannot be stable. Explicit equations of motion for two bubbles are deduced using Rayleigh's method for solving Laplace's equation in a periodic domain. The results are extended for more than two bubbles by considering pairwise interactions. Numerical simulations with gravity and liquid viscosity ignored show the bubbles form clusters if initially they are randomly arranged with identical velocities. The clustering is inhibited, however, if the initial velocities are sufficiently different. The variance of the bubbles’ velocities is observed to act like a temperature; a lesser amount of clustering occurs with increasing temperature. Finally, in considering the effects of gravity and liquid viscosity, the long time behaviour is found to result in the formation of bubble clusters aggregated in a plane perpendicular to the direction of gravity.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 254 , September 1993 , pp. 79 - 112
Copyright
© 1993 Cambridge University Press

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References

Banerjee, S. & Chan, A. M. C. 1980 Separated flow model I. Analysis of the averaged and local instantaneous formulations. Intl J. Multiphase Flow 6, 1.Google Scholar
Basset, A. B. 1887 On the motion of two spheres in a liquid. Proc. Lond. Math. Soc. 18, 369.Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Batchelor, G. K. 1988 A new theory of the instability of a uniform fluidized bed. J. Fluid Mech. 193, 75.Google Scholar
Beek, P. van 1982 O(α)-accurate modeling of the virtual mass effect in a liquid–bubble dispersion. Appl. Sci. Res. 38, 323.Google Scholar
Bender, C. M. & Orszag, S. A. 1978 Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill.
Benjamin, T. B. 1987 Hamiltonian theory for motions of bubbles in an infinite liquid. J. Fluid Mech. 181, 349.Google Scholar
Biesheuvel, A. & Gorissen, W. C. M. 1990 Void fraction disturbances in a uniform bubbly fluid. Intl J. Multiphase Flow 16, 211.Google Scholar
Biesheuvel, A. & Wijngaarden, L. van 1984 Two-phase flow equations for a dilute dispersion of gas bubbles in liquid. J. Fluid Mech. 148, 301.Google Scholar
Delhaye, J. M. & Achard, J. L. 1976 On the averaging operators introduced in two-phase flow modeling. Proc. 1st OECD/NEA Specialists meeting on Transient two-phase flow, Toronto, vol. 1, p. 5.
Drew, D. A. & Wood, R. T. 1985 Overview and Taxonamy of Models and Methods. International Workshop on Two-Phase Flow Fundamentals, National Bureau of Standards, Maryland, September 1985.
Geurst, J. A. 1985 Virtual mass in two-phase flow. Physica 129 A, 233.Google Scholar
Geurst, J. A. 1986 Variational principles and two-fluid hydrodynamics of bubbly liquid/gas mixtures. Physica 135 A, 455.Google Scholar
Jeffrey, D. J. 1973 Conduction through a random suspension of spheres. Proc. R. Soc. Lond. 335, 355.Google Scholar
Kok, J. B. W. 1988 Hydrodynamic equations and effective properties of bubbly flows. In Transient Phenomena in Muliphase Flow (ed. N. Afgan), p. 127. Hemisphere.
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press (Dover edition 1945.)
Levich, V. G. 1962 Physiochemical Hydrodynamics. Prentice Hall.
Lhuillier, D. 1985 Phenomenology of inertia effects in a dispersed solid-fluid mixture. Intl J. Muliphase Flow 11, 427.Google Scholar
McPhedran, R. C. & McKenzie, D. R. 1978 The conductivity of lattices of spheres I. The simple cubic lattice. Proc. R. Soc. Lond. A 359, 45.Google Scholar
Pauchon, C. & Banerjee, S. 1986 Interphase momentum interaction effects in the averaged multifield model. Part I: void propagation in bubbly flows. Intl J. Multiphase Flow 12, 559.Google Scholar
Pauchon, C. & Smereka, P. 1992 Momentum interactions in dispersed flow: An averaging and a variational approach. Intl J. Multiphase Flow 18, 65.Google Scholar
Rayleigh, R. S. 1892 On the influence of obstacles arranged in a rectangular order upon the properties of a medium. Phil. Mag. 34, 484.Google Scholar
Singh, P. & Joseph, D. D. 1993 Chaos and structure in two-dimensional beds of spheres fluidized by water. Intl J. Multiphase Flow (submitted.)Google Scholar
Smereka, P. & Milton, G. 1991 Bubbly flow and its relation to conduction in composites. J. Fluid Mech. 233, 65.Google Scholar
Wallis, G. B. 1989 Inertial coupling in two-phase flows: Macroscopic properties of suspensions in an invicid fluid. Multiphase Sci. Tech. 5, 239.Google Scholar
Wallis, G. B. 1990 On Geurst's equations for inertial coupling in two-phase flow. In Two Phase Waves in Fluidized Beds, Flowing Composites and Granular Media, Institute for Maths and its Applications, University of Minnesota.
Wijngaarden, L. van 1976 Hydrodynamical interaction between gas bubbles in liquid. J. Fluid Mech. 77, 27.Google Scholar
Wijngaarden, L. van 1982 Bubble interactions in liquid/gas flows in liquid. Appl. Sci. Res. 38, 331.Google Scholar
Wijngaarden, L. van & Kapteyn, C. 1990 Concentration waves in dilute bubble/liquid mixtures. J. Fluid Mech. 212, 111.Google Scholar