The Shape of a Compressible Drop on a Vibrating Solid Plate

Andrey Ivantsov, Tatyana Lyubimova (), Grigoriy Khilko and Dmitry Lyubimov
Additional contact information
Andrey Ivantsov: Institute of Continuous Media Mechanics of the Ural Branch of the Russian Academy of Sciences Perm, 614013 Perm, Russia
Tatyana Lyubimova: Institute of Continuous Media Mechanics of the Ural Branch of the Russian Academy of Sciences Perm, 614013 Perm, Russia
Grigoriy Khilko: Institute of Continuous Media Mechanics of the Ural Branch of the Russian Academy of Sciences Perm, 614013 Perm, Russia
Dmitry Lyubimov: Theoretical Physics Department, Perm State University, 614068 Perm, Russia

Mathematics, 2023, vol. 11, issue 21, 1-24

Abstract: The influence of high-frequency vibrations on the shape of a compressible drop placed on an oscillating solid substrate is studied in this paper. Due to the significant difference in characteristic temporal scales, the average and pulsating motions of the drop can be considered separately. For nearly hemispherical drop, the solution to the problem of pulsating motion is found in the form of series in Legendre polynomials. Frequencies of natural sound oscillations of hemispherical axisymmetric drop are obtained. Resonances of the acoustic mode of drop oscillations are found. The problem of forced oscillations of hemispherical drop in the limit of weakly compressible liquid is considered. It is found that drop oscillation amplitude grows with vibration intensity according to quadratic law, which is consistent with the solution of the pulsation problem for finite compressibility assumption. A variational principle for calculation of average drop shape is formulated based on minimization of energy functional for the case, so the compressibility of the liquid should be taken into account. It is shown that the functional (the sum of the kinetic and potential energies of the pulsating flow, the kinetic energy of the averaged flow, and the surface tension energy of the drop) decreases and reaches a minimum value at quasi-equilibrium state, in which the average shape of the drop becomes static. The influence of vibrations on the drop shape is studied for small values of the vibrational parameter. The surface of the drop in the absence of vibrations is assumed to be hemispherical. Calculations showed that under vibrations, drop height decreases, while the area of the base increases.

Keywords: vibrations; resonance; natural frequencies; hemispherical drop; drop shape; Legendre polynomials; variational principle (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/11/21/4527/pdf (application/pdf)
https://www.mdpi.com/2227-7390/11/21/4527/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:11:y:2023:i:21:p:4527-:d:1273435

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().