Analytical Solution of the Rayleigh – Plesset Equation

M. V. Gasanov ; A. G. Gulkanov ; K. A. Modestov

Russian journal of nonlinear dynamics, Tome 20 (2024) no. 1, pp. 3-13.

Résumé

In this paper, we consider a mathematical model of the dynamics of the behavior of a spher- ically symmetric Rayleigh – Plesset bubble in the van der Waals gas model. The analysis of the model takes into account various isoprocesses without the presence of condensation and a model that takes into account condensation in an isothermal process. In each case, various character- istics are searched for, such as oscillation frequency (linear/small oscillations), damping factor, relaxation time, decrement, and logarithmic decrement. Solutions are found in quadratures for various parameters of the equation. The theoretical results obtained are compared with the results of the numerical solution of the Cauchy problem for various isoprocesses.

Détail

Keywords: nonlinear differential equations, small fluctuations, van der Waals equation, Rayleigh – Plesset equation
Mots-clés : exact solution

Bibliographie

[1] Strutt, J. W. (3rd Baron Rayleigh), “On the Pressure Developed in a Liquid during the Collapse of a Spherical Cavity”, Philos. Mag. (6), 34:200 (1917), 94–98 | DOI

[2] Plesset, M. S. and Prosperetti, A., “Bubble Dynamics and Cavitation”, Annu. Rev. Fluid Mech., 9 (1977), 145–185 | DOI | MR | Zbl

[3] Brennen, Ch. E., Cavitation and Bubble Dynamics, corrected edition of the 1995 original, Cambridge Univ. Press, New York, 2014, xviii, 249 pp. | MR | Zbl

[4] Man, V. H., Li, M. S., Derreumaux, P., and Nguyen, P. H., “Rayleigh – Plesset Equation of the Bubble Stable Cavitation in Water: A Nonequilibrium All-Atom Molecular Dynamics Simulation Study”, J. Chem. Phys., 148:9 (2018), Art. 094505, 8 pp. | DOI | MR

[5] Shah, Y. G., Vacca, A., and Dabiri, S., “Air Release and Cavitation Modeling with a Lumped Parameter Approach Based on the Rayleigh – Plesset Equation: The Case of an External Gear Pump”, Energies, 11:12 (2018), Art. 3472, 28 pp. | DOI

[6] Geng, L., Chen, J., and Escaler, X., “Improvement of Cavitation Mass Transfer Modeling by Including Rayleigh – Plesset Equation Second Order Term”, Eur. J. Mech. B Fluids, 84 (2020), 313–324 | DOI | MR | Zbl

[7] Alehossein, H. and Qin, Z., “Numerical Analysis of Rayleigh – Plesset Equation for Cavitating Water Jets”, Int. J. Numer. Methods Eng., 72:7 (2007), 780–807 | DOI | MR | Zbl

[8] Bogoyavlenskiy, V. A., “Single-Bubble Sonoluminescence: Shape Stability Analysis of Collapse Dynamics in a Semianalytical Approach”, Phys. Rev. E, 62:2 (2000), 2158–2167 | DOI

[9] Lauterborn, W. and Kurz, T., “Physics of Bubble Oscillations”, Rep. Prog. Phys., 73:10 (2010), 106501 | DOI

[10] Doinikov, A. A., Novell, A., Escoffre, J.-M., and Bouakaz, A., “Encapsulated Bubble Dynamics in Imaging and Therapy”, Bubble Dynamics and Shock Waves, Shock Wave Science and Technology Reference Library, 8, ed. C. Delale, Springer, Berlin, 2013, 259–289

[11] Obreschkow, D., Bruderer, M., and Farhat, M., “Analytical Approximations for the Collapse of an Empty Spherical Bubble”, Phys. Rev. E, 85:6 (2012), Art. 066303, 4 pp. | DOI

[12] Amore, P. and Fernández, F. M., “Mathematical Analysis of Recent Analytical Approximations to the Collapse of an Empty Spherical Bubble”, J. Chem. Phys., 138:8 (2013), Art. 084511, 3 pp. | DOI

[13] Kudryashov, N. A. and Sinelshchikov, D. I., “Analytical Solutions of the Rayleigh Equation for Empty and Gas-Filled Bubble”, J. Phys. A, 47:40 (2014), Art. 405202, 10 pp. | DOI | MR

[14] Kudryashov, N. A. and Sinelshchikov, D. I., “Analytical Solutions for Problems of Bubble Dynamics”, Phys. Lett. A, 379:8 (2015), 798–802 | DOI | MR

[15] Kudryashov, N. A. and Sinelshchikov, D. I., “On the Connection of the Quadratic Liénard Equation with an Equation for the Elliptic Functions”, Regul. Chaotic Dyn., 20:4 (2015), 486–496 | DOI | MR | Zbl