Kinetic model for the motion of compressible bubbles in a perfect fluid |
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| Institution: | 1. Lavrentyev Institute of Hydrodynamics, Lavrentyev prospect 15, Novosibirsk 630090, Russia;2. Laboratoire de Modélisation en Mécanique et Thermodynamique, Faculté des Sciences de Saint-Jérôme, Case 322, Avenue Escadrille Normandie–Niemen, 13397 Marseille cedex 20, France;1. Division of Neurology, Djavad Mowafaghian Centre for Brain Health, University of British Columbia, Vancouver, BC V6T 2B5, Canada;2. Faculty of Dentistry, Université de Montréal, Montréal, QC, Canada;3. Centre Hospitalier de L''Université de Montréal, Montréal, QC, Canada;4. Parkinson Canada, Toronto, ON, Canada;5. Department of Neurology, Montréal General Hospital, Montréal, QC, Canada;1. Faculty of Mechanical Engineering, University of Campinas (UNICAMP), CP 6066, CEP 13081-970, Campinas, SP, Brazil;2. Centro Nacional de Investigación y Desarrollo Tecnológico (CENIDET-TNM-SEP), Prol. Av. Palmira S/N. Col. Palmira, Cuernavaca, Morelos, CP. 62490, Mexico |
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| Abstract: | Collective behavior of compressible gas bubbles moving in an inviscid incompressible fluid is studied. A kinetic approach is employed, based on an approximate calculation of the fluid flow potential and formulation of Hamilton's equations for generalized coordinates and momenta of bubbles. Kinetic equations governing the evolution of a distribution function of bubbles are derived. These equations are similar to Vlasov's equations. Conservation laws which are direct consequences of the kinetic system are found. It is shown that for a narrowly peaked distribution function they form a closed system of hydrodynamical equations for the mean flow parameters. The system yields the analogue of Rayleigh–Lamb's equation governing oscillations of bubbles. A variational principle for the hydrodynamical system is established and the linear stability analysis is performed. |
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