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International Journal of Fluid Mechanics Research
ESCI SJR: 0.22 SNIP: 0.446 CiteScore™: 0.5

ISSN Print: 2152-5102
ISSN Online: 2152-5110

International Journal of Fluid Mechanics Research

DOI: 10.1615/InterJFluidMechRes.v32.i4.50
pages 454-487

Planimetry of Vibrocapillary Equilibria at Small Wave Numbers

A. N. Timokha
National Academy of Sciences of Ukraine, Institute of Mathematics, Tereschenkivska 3, 01601 Kiev, Ukraine


Time-averaged geometrical shapes (vibroequilibria) of a limited volume of ideal liquid in a rectangular vessel which performs high-frequency reciprocating vibrations have been analyzed for a two-dimensional potential flow. A concept of quasi-potential energy and an assumption of smallness of the wave numbers is used in the research. Particular exact analytical solutions are presented. The general case study is based on straight numerical minimization of a quasi-potential energy functional. An auxiliary boundary-value problem for the wave function, being a constraint, is solved by the modified Nystrom- Kress method. A theoretical description is given for the experimental phenomena of “flattening” and vibrostabilization of the free liquid surface, “overturn” ( “reorientation” of the liquid, its localization near one of the vertical walls), and “dip” (an even spreading of the liquid between the walls with a “cavity” appearing in the center) which occur under horizontal vibrations of the vessel. Numerical results for the vibroequilibria in the conditions of the Earth gravitation (large Bond's numbers) and zero gravity (lack of mass forces) are discussed. The solution ambiguity and vibroequilibrium dependence on transitional processes are pointed out. It has been confirmed theoretically that an “overturn” is more probable with a small depth, while a “dip” is typical of non-small depths of the liquid. Preliminary theoretical results describing the “flattening” and vibrostabilization of a drop hanging on a vibrating plate have been obtained, including the case of a negligibly small surface-tension (large Bond's numbers).

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