Ultrasonic Waves in Bubbly Liquids: An Analytic Approach
We consider the problem of the propagation of high-intensity acoustic waves in a bubble layer consisting of spherical bubbles of identical size with a uniform distribution. The mathematical model is a coupled system of partial differential equations for the acoustic pressure and the instantaneous ra...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-06-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/11/1309 |
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| author | Pilar Ruiz Gordoa Andrew Pickering |
| author_facet | Pilar Ruiz Gordoa Andrew Pickering |
| author_sort | Pilar Ruiz Gordoa |
| collection | DOAJ |
| description | We consider the problem of the propagation of high-intensity acoustic waves in a bubble layer consisting of spherical bubbles of identical size with a uniform distribution. The mathematical model is a coupled system of partial differential equations for the acoustic pressure and the instantaneous radius of the bubbles consisting of the wave equation coupled with the Rayleigh–Plesset equation. We perform an analytic analysis based on the study of Lie symmetries for this system of equations, concentrating our attention on the traveling wave case. We then consider mappings of the resulting reductions onto equations defining elliptic functions, and special cases thereof, for example, solvable in terms of hyperbolic functions. In this way, we construct exact solutions of the system of partial differential equations under consideration. We believe this to be the first analytic study of this particular mathematical model. |
| first_indexed | 2024-03-10T10:39:26Z |
| format | Article |
| id | doaj.art-cbfb2c4802604803ba3ca3878ff5d1ca |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T10:39:26Z |
| publishDate | 2021-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-cbfb2c4802604803ba3ca3878ff5d1ca2023-11-21T23:06:45ZengMDPI AGMathematics2227-73902021-06-01911130910.3390/math9111309Ultrasonic Waves in Bubbly Liquids: An Analytic ApproachPilar Ruiz Gordoa0Andrew Pickering1Área de Matemática Aplicada, ESCET, Universidad Rey Juan Carlos, C/Tulipán s/n, Móstoles, 28933 Madrid, SpainÁrea de Matemática Aplicada, ESCET, Universidad Rey Juan Carlos, C/Tulipán s/n, Móstoles, 28933 Madrid, SpainWe consider the problem of the propagation of high-intensity acoustic waves in a bubble layer consisting of spherical bubbles of identical size with a uniform distribution. The mathematical model is a coupled system of partial differential equations for the acoustic pressure and the instantaneous radius of the bubbles consisting of the wave equation coupled with the Rayleigh–Plesset equation. We perform an analytic analysis based on the study of Lie symmetries for this system of equations, concentrating our attention on the traveling wave case. We then consider mappings of the resulting reductions onto equations defining elliptic functions, and special cases thereof, for example, solvable in terms of hyperbolic functions. In this way, we construct exact solutions of the system of partial differential equations under consideration. We believe this to be the first analytic study of this particular mathematical model.https://www.mdpi.com/2227-7390/9/11/1309ultrasonic wavesbubble layerexact solutionstraveling wave |
| spellingShingle | Pilar Ruiz Gordoa Andrew Pickering Ultrasonic Waves in Bubbly Liquids: An Analytic Approach Mathematics ultrasonic waves bubble layer exact solutions traveling wave |
| title | Ultrasonic Waves in Bubbly Liquids: An Analytic Approach |
| title_full | Ultrasonic Waves in Bubbly Liquids: An Analytic Approach |
| title_fullStr | Ultrasonic Waves in Bubbly Liquids: An Analytic Approach |
| title_full_unstemmed | Ultrasonic Waves in Bubbly Liquids: An Analytic Approach |
| title_short | Ultrasonic Waves in Bubbly Liquids: An Analytic Approach |
| title_sort | ultrasonic waves in bubbly liquids an analytic approach |
| topic | ultrasonic waves bubble layer exact solutions traveling wave |
| url | https://www.mdpi.com/2227-7390/9/11/1309 |
| work_keys_str_mv | AT pilarruizgordoa ultrasonicwavesinbubblyliquidsananalyticapproach AT andrewpickering ultrasonicwavesinbubblyliquidsananalyticapproach |