Hamiltonian description of bubble dynamics
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AL, NONLINEAR, AND SOFT MATTER PHYSICS
Hamiltonian Description of Bubble Dynamics A. O. Maksimov Il’ichev Pacific Oceanological Institute, Far East Division, Russian Academy of Sciences, ul. Baltiœskaya 43, Vladivostok, 690041 Russia e-mail: [email protected] Received January 9, 2007
Abstract—The dynamics of a nonspherical bubble in a liquid is described within the Hamiltonian formalism. Primary attention is focused on the introduction of the canonical variables into the computational algorithm. The expansion of the Dirichlet–Neumann operator in powers of the displacement of a bubble wall from an equilibrium position is obtained in the explicit form. The first three terms (more specifically, the second-, third-, and fourth-order terms) in the expansion of the Hamiltonian in powers of the canonical variables are determined. These terms describe the spectrum and interaction of three essentially different modes, i.e., monopole oscillations (pulsations), dipole oscillations (translational motions), and surface oscillations. The cubic nonlinearity is analyzed for the problem associated with the generation of Faraday ripples on the wall of a bubble in an acoustic field. The possibility of decay processes occurring in the course of interaction of surface oscillations for the first fifteen (experimentally observed) modes is investigated. PACS numbers: 47.55.dd, 47.10.Df, 43.25.Rq DOI: 10.1134/S1063776108020143
1. INTRODUCTION The nonlinear dynamics of a gas inclusion (a bubble) in a low-viscosity liquid under the effect of an acoustic field has been extensively investigated for a long time. The vast majority of previous studies have been aimed at examining the behavior of a bubble as a sound source. However, it should be noted that, for the solution of this problem, volume pulsations are of particular interest. Since shape distortions, i.e., surface oscillations, have proved to be a substantially weaker sound source (at least in the linear approximation) and the behavior of a spherically symmetric bubble is more amenable to theoretical treatment, it is this model that provides the basis for the majority of theoretical investigations into the dynamics of gas inclusions. At the same time, gas bubbles very frequently have a nonspherical initial shape or become nonspherical because of the development of various instabilities. However, the behavior of these nonspherical bubbles has been described in the literature to a considerably lesser extent [1]. The key circumstance that determines a number of advantages of the use of the Hamiltonian formalism in the description of the dynamics of nonspherical bubbles is that the nonlinear boundary conditions on the surface of a bubble can be represented in terms of functional derivatives of the energy. This formalism was proposed by Zakharov [2] for the description of surface waves. Subsequently, Benjamin and Olver [3] advanced the above approach proposed by Zakharov and established a relation between the Hamiltonian structure of equations, symmetry, and conservation laws. In our study
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