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Hamiltonian theory for motions of bubbles in an infinite liquid

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.