Parametric analytical solution for the \(N\)-dimensional Rayleigh equation. (English) Zbl 1460.76799
Summary: In this letter, we consider the \(N\)-dimensional Rayleigh equation for describing the dynamics of gas-filled spherical bubbles. In the spirit of N. A. Kudryashov and D. I. Sinelshchikov’s work in [J. Phys. A, Math. Theor. 47, No. 40, Article ID 405202, 10 p. (2014; Zbl 1298.76190)], a direct approach is first proposed to construct parametric analytical solution for this equation using trigonometric function. It provides us a simple but efficient way to construct analytical solutions of the bubble radius and period. As its applications, isothermal and adiabatic compressions are studied respectively. We show that both bubble radius and period decrease with the increase in the pressure ratio.
MSC:
| 76T10 | Liquid-gas two-phase flows, bubbly flows |
Keywords:
spherical gas bubble; trigonometric function solution; isothermal compression; adiabatic compressionCitations:
Zbl 1298.76190References:
| [1] | Rayleigh, L., Viii. on the pressure developed in a liquid during the collapse of a spherical cavity, Phil. Mag., 34, 94-98 (1917) · JFM 46.1274.01 |
| [2] | Plesset, M. S.; Prosperetti, A., Bubble dynamics and cavitation, Annu. Rev. Fluid Mech., 9, 145-185 (1977) · Zbl 0418.76074 |
| [3] | Brennen, C. E., (Cavitation and Bubble Dynamics. Cavitation and Bubble Dynamics, Oxford Engineering Science Series, vol. 44 (1995), Oxford University Press: Oxford University Press New York) · Zbl 1302.76002 |
| [4] | Bogoyavlenskiy, V. A., Single-bubble sonoluminescence: shape stability analysis of collapse dynamics in a semianalytical approach, Phys. Rev. E, 62, 2158-2167 (2000) |
| [5] | Klotz, A. R.; Lindvere, L.; Stefanovic, B.; Hynynen, K., Temperature change near microbubbles within a capillary network during focused ultrasound, Phys. Med. Biol., 55, 1549-1561 (2010) |
| [6] | Obreschkow, D.; Bruderer, M.; Farhat, M., Analytical approximations for the collapse of an empty spherical bubble, Phys. Rev. E, 85, 066303 (2012) |
| [7] | Amore, P.; Fernández, F. M., Mathematical analysis of recent analytical approximations to the collapse of an empty spherical bubble, J. Chem. Phys., 138, 084511 (2013) |
| [8] | Klotz, A. R., Bubble dynamics in \(N\) dimensions, Phys. Fluids, 25, 082109 (2013) |
| [9] | Kudryashov, N. A.; Sinelshchikov, D. I., Analytical solutions of the Rayleigh equation for empty and gas-filled bubble, J. Phys. A, 47, 405202 (2014) · Zbl 1298.76190 |
| [10] | Kudryashov, N. A.; Sinelshchikov, D. I., Analytical solutions for problems of bubble dynamics, Phys. Lett. A, 379, 798-802 (2015) |
| [11] | Kudryashov, N. A.; Sinelshchikov, D. I., Analytical solutions of the Rayleigh equation for arbitrary polytropic exponent, AIP Conf. Proc., 1738, 230010 (2016) |
| [12] | Van Gorder, R. A., Dynamics of the Rayleigh-Plesset equation modelling a gas-filled bubble immersed in an incompressible fluid, J. Fluid Mech., 807, 478-508 (2016) · Zbl 1383.76481 |
| [13] | Mancas, S. C.; Rosu, H. C., Evolution of spherical cavitation bubbles: parametric and closed-form solutions, Phys. Fluids, 28, 022009 (2016) |
| [14] | Wang, Z.; Qin, Y. P.; Zou, L., Analytical solutions of the Rayleigh-Plesset equation for \(N\)-dimensional spherical bubbles, Sci. China: Phys. Mech. Astron. (2017), (in press) |
| [15] | Liao, S. J., Beyond Perturbation: Introduction To the Homotopy Analysis Method (2003), Chapman and Hall/CRC Press: Chapman and Hall/CRC Press Boca Raton |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
