The Cole-Hopf Transformation for Solving Rayleigh-Plesset Equation in Bubble Dynamics

The Cole-Hopf Transformation for Solving Rayleigh-Plesset Equation in Bubble Dynamics

Ali F. Abu-BakrAly M. Abourabia 

Mathematics and Computer Science Department, Menoufia University, Shebin El-Kom 32511, Egypt

Theoretical and Mathematical Physics Department, Ural Federal University, Ekaterinburg 620083, Russia

Corresponding Author Email: 
alibakrm@yahoo.com
Page: 
82-87
|
DOI: 
https://doi.org/10.18280/ama_a.550207
Received: 
20 March 2018
| |
Accepted: 
18 May 2018
| | Citation

OPEN ACCESS

Abstract: 

In this paper, we investigate the solution of the behaviour of bubble growth in Newtonian fluid by using extended Rayleigh-Plesset equation. The transformation of Cole-Hopf has been applied on the nonlinear ordinary differential equation of the non-dimensional extended Rayleigh-Plesset equation in order to obtain the exact solution of bubble radius. We have also introduced the study of phase portrait of growth problem. The results studied analytically and indicated in graphics.

Keywords: 

Cole–Hopf transformation, Rayleigh-Plesset equation, phase portrait, bubble dynamics

1. Introduction
2. Mathematical Model
3. The Cole-Hopf Transformation and the Analytical Solution of Rayleigh-Plesset Equation
4. The Phase Portrait of Extended Rayleigh-Plesset Equation
5. Results and Discussion
6. Conclusions
  References

[1] Yang X, Church C. (2005). A model for the dynamics of gas bubbles in soft tissue. The Journal of the Acoustical Society of America 118: 3595-3606. http://dx.doi.org/10.1121/1.2118307

[2] Hilgeneedt S, Brenner M, Grossmann S, Lohse D. (1998). Analysis of Rayleigh-Plesset dynamics for sonoluminescing bubbles. Journal of Fluid Mechanics 365: 171-204. https://doi.org/10.1017/S0022112098001207

[3] Mohammadein SA. Abu-Bakr AF. (2010). The growth of vapour bubble in a superheated liquid between two-phase turbulent flow. Canadian Journal of Physics 88(5): 317-324. http://dx.doi.org/10.1139/P10-022

[4] Guzman DN, Friczek T, Reetz C, Sun C, Lohse D, Ahlers G. (2016). Vapour-bubble nucleation and dynamics in turbulent Rayleigh_Bénard convection. Journal of Fluid Mechanics 795: 60-95. https://doi.org/10.1017/jfm.2016.178

[5] Cui B, Ni B, Wu Q. (2016). Bubble–bubble interaction effects on dynamics of multiple bubbles in a vortical flow field. Advances in Mechanical Engineering 8(2): 1–12. https://doi.org/10.1177/1687814016631708

[6] Alehossein H, Qin Z. (2007). Numerical analysis of Rayleigh–Plesset equation for cavitating water jets. International Journal Numerical Mechanics Engineering 72: 780–807. https://doi.org/10.1002/nme.2032

[7] Rayleigh L. (1917). On the pressure developed in a liquid during the collapse of a spherical cavity. Philosophical Magazine Series 6, 34: 94–98. http://dx.doi.org/10.1080/14786440808635681

[8] Plesset MS. (1949). The dynamics of cavitating bubbles. Journal of Applied Mechanics 16: 228–231. http://resolver.caltech.edu/CaltechAUTHORS:20140808-114249321

[9] Plesset MS, Zwick SA. (1954). The growth of vapor bubbles in superheated liquids. Journal of Applied Mechanics 25(4): 493–450. http://dx.doi.org/10.1063/1.1721668

[10] Prosperetti A. (1982). A generalization of the Rayleigh-Plesset equarion of bubble dynamics. Physics Fluids 25(3): 409-410. https://doi.org/10.1063/1.863775

[11] Klaseboer E, Khoo BC. (2006). A modified Rayleigh–Plesset model for a non-spherically symmetric oscillating bubble with applications to boundary integral methods. Engineering Analysis with Boundary Elements 30: 59–71. https://doi.org/10.1016/j.enganabound.2005.09.003

[12] Johnsen E, Colonus T. (2009). Numerical simulations of non-spherical bubble collapse. Journal of Fluid Mechanics 629: 231–262. https://doi.org/10.1017/S0022112009006351

[13] Van-Gorder RA. (2016). Dynamics of the Rayleigh-Plesset equation modelling a gas-filled bubble immersed in an incompressible fluid, Journal of Fluid Mechanics 807: 478-508. https://doi.org/10.1017/jfm.2016.640

[14] Mohammadein SA, Shalaby GA, Abu-Bakr AF, Abu-Nab AK. (2017). Analytical solution of gas bubble dynamics between two-phase flow. Results in Physics 7: 2396-2403. http://dx.doi.org/10.1016/j.rinp.2017.07.007

[15] Kudryashov NA, Sinelshchikov DI. (2015). Analytical solutions for problems of bubble dynamics. Physics Letters A 379 (8): 798–802. http://dx.doi.org/10.1016/j.physleta.2014.12.049

[16] Abourabia AM, El Horbaty MM. (2006). On solitary wave solutions for the two-dimensional nonlinear modified Kortweg–de Vries–Burger equation. Chaos, Solitons and Fractals 29: 354–364. https://doi.org/10.1016/j.chaos.2005.08.112 

[17] Gorguis A. (2006). A comparison between Cole–Hopf transformation and the decomposition method for solving Burgers’ equations. Applied Mathematics and Computation, 173: 126–136. https://doi.org/10.1016/j.amc.2005.02.045

[18] Wazwaz A. (2012). A study on the (2+1)-dimensional and the (2+1)-dimensional higher-order Burgers equations. Applied Mathematics Letters 25: 1495–1499. https://doi.org/10.1016/j.aml.2011.12.034