This work investigates the exact general analytical solution of the Rayleigh equation for multielectron bubbles in liquid helium using Kudryashov and Sinelshchikov’s method. We firstly obtain its first integral involving three negative exponential powers, then a specific Sundman transformation is proper established to transform it into an equation for the elliptic functions, and finally the analytical solution expressed by Weierstrass elliptic function is constructed appropriately. As applications, the derived analytical solution is used to test numerical algorithm, and also to construct the analytical expressions of the bubble oscillation period and the derivatives of the bubble radius. Further, the influence of the pressure on the helium is also discussed.
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L. Rayleigh, VIII. On the pressure developed in a liquid during the collapse of a spherical cavity. Philos. Mag. 34, 94–98 (1917)
M.S. Plesset, A. Prosperetti, Bubble dynamics and cavitation. Annu. Rev. Fluid Mech. 9, 145–185 (1977)
C.E. Brennen, Cavitation and Bubble Dynamics, Oxford Engineering Science Series, vol. 44 (Oxford University Press, New York, 1995)
A. Prosperetti, Bubbles. Phys. Fluids 16, 1852–1865 (2004)
M.A. López, Raquel Martínez, A note on the generalized Rayleigh equation: limit cycles and stability. J. Math. Chem. 51, 1164–1169 (2013)
K. Yasui, T. Tuziuti, J. Lee, T. Kozuka, A. Towata, Y. Iida, The range of ambient radius for an active bubble in sonoluminescence and sonochemical reactions. J. Chem. Phys. 128, 184705 (2008)
K. Yasui, T. Tuziuti, Y. Iida, H. Mitome, Theoretical study of the ambient-pressure dependence of sonochemical reactions. Ultrason. Sonochem. 119, 346–356 (2003)
Z. Xu, Numerical simulation of the coalescence of two bubbles in an ultrasound field. Ultrasonics Sonochem. 49, 277–282 (2018)
J. Tempere, I.F. Silvera, S. Rekhi, J.T. Devreese, Sonoluminescence and collapse dynamics of multielectron bubbles in helium. Phys. Rev. B 70, 224303 (2004)
J. Tempere, I.F. Silvera, J.T. Devreese, Multielectron bubbles in helium as a paradigm for studying electrons on surfaces with curvature. Surf. Sci. Rep. 62, 159–217 (2007)
M.M. Salomaa, G.A. Williams, Structure and stability of multielectron bubbles in liquid helium. Phys. Rev. Lett. 47, 1730–1733 (1981)
S.T. Hannahs, G.A. Williams, M.M. Salomaa, Acoustic oscillations of multielectron bubbles in liquid helium, in Proceedings of the IEEE Ultrasonics Symposium, vol. 1 (1995), pp. 635–640
D. Obreschkow, M. Bruderer, M. Farhat, Analytical approximations for the collapse of an empty spherical bubble. Phys. Rev. E 85, 066303 (2012)
A.R. Klotz, Bubble dynamics in \(N\) dimensions. Phys. Fluids 25, 082109 (2013)
F.A. Godínez, M.A. Escobedo, M. Navarrete, Homotopy analysis method for the Rayleigh equation governing the radial dynamics of a multielectron bubble. J. Appl. Math. 2012, 591058 (2012)
Z. Wang, Y.P. Qin, L. Zou, Analytical solutions of the Rayleigh–Plesset equation for N-dimensional spherical bubbles. Sci. China Phys. Mech. Astron. 60, 104721 (2017)
N.A. Kudryashov, D.I. Sinelshchikov, Analytical solutions for problems of bubble dynamics. Phys. Lett. A 379, 798–802 (2015)
S.C. Mancas, H.C. Rosu, Evolution of spherical cavitation bubbles: parametric and closed-form solutions. Phys. Fluids 28, 022009 (2016)
R.A. Van Gorder, Dynamics of the Rayleigh–Plesset equation modelling a gas-filled bubble immersed in an incompressible fluid. J. Fluid Mech. 807, 478–508 (2016)
Y.P. Qin, Z. Wang, L. Zou, M.F. He, Parametric analytical solution for the \(N\)-dimensional Rayleigh equation. Appl. Math. Lett. 76, 8–13 (2018)
Y.P. Qin, Z. Wang, L. Zou, M.F. He, Semi-numerical, semi-analytical approximations of the Rayleigh equation for gas-filled hyper-spherical bubble. Int. J. Comput. Methods 16, 1850094 (2019)
Y.P. Qin, Analytical solution for the collapse motion of an empty hyper-spherical bubble in \(N\) dimensions. Phys. Lett. A 384, 126142 (2020)
N.A. Kudryashov, D.I. Sinelshchikov, On the connection of the quadratic Lienard equation with an equation for the elliptic functions. Regul. Chaot. Dyn. 20, 486–496 (2015)
This work is supported by the Key Scientific Research Projects of the Higher Education Institutions of Henan Province (20B410001, 19B460003), the Key Scientific and Technological Project of Henan Province (192102210222), and the Doctoral Fund of Henan Institute of Technology (KQ1860).
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Qin, Y., Lou, Q., Wang, Z. et al. Kudryashov and Sinelshchikov’s method for solving the radial oscillation problem of multielectron bubbles in liquid helium. J Math Chem (2020). https://doi.org/10.1007/s10910-020-01145-y
- Rayleigh equation
- Multielectron bubble in liquid helium
- Analytical solution
- Sundman transformation
- Weierstrass elliptic function