Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-20T01:44:23.722Z Has data issue: false hasContentIssue false
  • English
  • Français

Hamiltonian theory for motions of bubbles in an infinite liquid

Published online by Cambridge University Press:  21 April 2006

T. Brooke Benjamin
T. Brooke Benjamin
Affiliation:
Mathematical Institute, 24/29 St Giles, Oxford OX1 3LB, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

The general dynamical problem for bubbles moving in an infinite expanse of perfect liquid is discussed from the standpoint of Hamiltonian theory, which is appreciated as a basis for linking symmetries with conservation laws and for identifying variational principles that describe steady motions. Allowance is made for surface tension and for an arbitrary gas law relating the pressure and volume of the bubble contents, but particular attention is paid to models where the volume is constant.

In §, the most detailed part of the paper, a comprehensive theory is developed which represents the free surface parametrically and so applies globally in time. Conservation laws for energy and for linear and angular components of impulse are shown to follow simply from respective symmetries; consequences of Galilean in variance and of a scaling symmetry are also explored. Finally in §, variational characterizations of steady translational, spinning and spiralling motions are explained. In §3 a formally simpler Hamiltonian theory is shown to derive from the mildly restrictive assumption that the free surface can be represented in an orthogonal coordinate system; and some special details attending the use of cylindrical coordinates are noted. For bubbles steadily translating along an axis of symmetry, approximate calculations supported by Rayleigh's principle are presented in §4.1. Steadily spiralling motions are treated in §5; estimates based on spheroidal approximations to shape are presented in §5.1; and some speculations about stability are discussed in §5.2. A brief account of generalizations dealing with multiply connected bubbles is given in §6.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 181 , August 1987 , pp. 349 - 379
Copyright
© 1987 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Benjamin, T. B. 1972 The stability of solitary waves. Proc. R. Soc. Lond. A 328, 153–183.Google Scholar
Benjamin, T. B. 1974 Lectures on nonlinear wave motion. In Nonlinear Wave Motion, Lectures in Appl. Math. vol. 15, pp. 3–47. American Mathematical Society.
Benjamin, T. B. 1976 The alliance of practical and analytical insights into the nonlinear problems of fluid mechanics. Lecture Notes in Mathematics, vol. 503, pp. 8–29. Springer.
Benjamin, T. B. 1984 Impulse, flow force and variational principles. IMA J. Appl. Maths 32, 3–68.Google Scholar
Benjamin, T. B. 1986a On the Boussinesq model for two-dimensional wave motions in heterogeneous fluids. J. Fluid Mech. 165, 445–474.Google Scholar
Benjamin, T. B. 1986b Note on added mass and drift. J. Fluid Mech. 169, 251–256.Google Scholar
Benjamin, T. B. & Ellis, A. T. 1966 The collapse of cavitation bubbles and the pressures thereby produced against solid boundaries. Phil. Trans. R. Soc. Lond. A 260, 221–240.Google Scholar
Benjamin, T. B. & Graham-Eagle, J. 1985 Long gravity—capillary waves with edge constraints. IMA J. Appl. Maths 35, 91–114.Google Scholar
Benjamin, T. B. & Olver, P. J. 1982 Hamiltonian structure, symmetries and conservation laws for water waves. J. Fluid Mech. 125, 137–185.Google Scholar
Birkhoff, G. 1950 Hydrodynamics. Princeton University Press (Dover edition 1955).
El Sawi, M. 1974 Distorted gas bubbles at large Reynolds number. J. Fluid Mech. 62, 163–183.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
Kochin, N. E., Kibel, I. A. & Roze, N. V. 1964 Theoretical Hydromechanics. Interscience.
Lamb, H. 1932 Hydrodynamics, 6th ed. Cambridge University Press (Dover edition 1945).
Lewis, D., Marsden, J., Montgomery, R. & Ratiu, T. 1986 The Hamiltonian structure for dynamic free boundary problems. Physica 18D, 391–404.Google Scholar
Miksis, M., Vanden-Broek, J.-M. & Keller, J. B. 1981 Axisymmetric bubble or drop in a uniform flow. J. Fluid Mech. 108, 89–100.Google Scholar
Milder, D. M. 1977 A note regarding ‘On Hamilton's principle for surface waves’. J. Fluid Mech. 83, 159–161.Google Scholar
Miles, J. W. 1977 On Hamilton's principle for surface waves. J. Fluid Mech. 83, 153–158.Google Scholar
Milne-Thomson, L. M. 1968 Theoretical Hydrodynamics, 5th edn. Macmillan.
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
Olver, P. J. 1980 On the Hamiltonian structure of evolution equations. Math. Proc. Camb. Phil. Soc. 88, 71–88.Google Scholar
Olver, P. J. 1983 Conservation laws of free boundary problems and the classification of conservation laws for water waves. Trans. Am. Math. Soc. 277, 353–380.Google Scholar
Ramsey, A. S. 1935 A treatise on hydromechanics. Part II. Hydrodynamics, 4th edn. Bell.
Ryskin, G. & Leal, L. G. 1984 Numerical solution of free-boundary problems in fluid mechanics. Part 2. Buoyancy-driven motion of a gas bubble through a quiescent liquid. J. Fluid Mech. 148, 19–35.CrossRefGoogle Scholar
Saffman, P. G. 1956 On the rise of small bubbles in water. J. Fluid Mech. 1, 249–275.Google Scholar
Saffman, P. G. 1967 The self-propulsion of a deformable body in a perfect fluid. J. Fluid Mech. 28, 385–389.Google Scholar
Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. Zh. Prikl. Tekh. Mekh. Fiz. 9, 86–94 (English transl. J. Appl. Mech. Tech. Phys. 2, 190).Google Scholar