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http://dx.doi.org/10.4134/JKMS.j160219

MULTI-HARMONIC MODELS FOR BUBBLE EVOLUTION IN THE RAYLEIGH-TAYLOR INSTABILITY  

Choi, Sujin (Department of Mathematics Gangneung-Wonju National University)
Sohn, Sung-Ik (Department of Mathematics Gangneung-Wonju National University)
Publication Information
Journal of the Korean Mathematical Society / v.54, no.2, 2017 , pp. 663-673 More about this Journal
Abstract
We consider the multi-harmonic model for the bubble evolution in the Rayleigh-Taylor instability in two and three dimensions. We extend the multi-harmonic model in two dimensions to a high-order and present a new class of steady-state solutions of the bubble motion. The growth rate of the bubble is expressed by a continuous family of two free parameters. The critical point in the family of solutions is identified as a saddle point and is chosen as the physically significant solution. We also present the multi-harmonic model in the cylindrical geometry and find the steady-state solution of the axisymmetric bubble. Validity and limitation of the model are also discussed.
Keywords
Rayleigh-Taylor instability; multi-harmonic model; bubble; steady-state solution;
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1 S.-I. Sohn, Effects of surface tension and viscosity on the growth rates of Rayleigh-Taylor and Richtmyer-Meshkov instabilities, Phys. Rev. E 80 (2009), 055302.   DOI
2 S.-I. Sohn, Asymptotic bubble evolutions of the Rayleigh-Taylor instability, Commun. Nonlinear Sci. Numer. Simul. 17 (2012), no. 11, 4017-4022.   DOI
3 J. P. Wilkinson and J. W. Jacobs, Experimental study of the single-mode three-dimensional Rayleigh-Taylor instability, Phys. Fluids 19 (2007), 124102.   DOI
4 Q. Zhang, Analytical solutions of Layzer-type approach to unstable interfacial fluid mixing, Phys. Rev. Lett. 81 (1998), 3391-3394.   DOI
5 Y. B. Zudin, Analytical solution of the problem of the rise of a Taylor bubble, Phys. Fluids. 25 (2013), 053302.   DOI
6 S. I. Abarzhi, K. Nishihara, and J. Glimm, Rayleigh-Taylor and Richtmyer-Meshkov instabilities for fluids with a finite density ratio, Phys. Lett. A 317 (2003), 470-476.   DOI
7 S. I. Abarzhi, K. Nishihara, and R. Rosner, Multiscale character of the nonlinear coherent dynamics in the Rayleigh-Taylor instability, Phys. Rev. E 73 (2006), 036310.   DOI
8 G. Birkhoff and D. Carter, Rising plane bubbles, J. Math. Mech. 6 (1957), 769-779.
9 Y. G. Cao, H. Z. Guo, Z. F. Zhang, Z. H. Sun, and W. K. Chow, Effect of viscosity on the growth of Rayleigh-Taylor instability, J. Phys. A 44 (2011), no. 27, 275501, 8 pp.
10 P. R. Garabedian, On steady-state bubbles generated by Taylor instability, Proc. R. Soc. London Ser. A 241 (1957), 423-431.   DOI
11 J. Glimm, D. H. Sharp, T. Kaman, and H. Lim, New directions for Rayleigh-Taylor mixing, Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci. 371 (2013), no. 2003, 20120183, 19 pp.
12 K. O. Mikaelian, Explicit expressions for the evolution of single-mode Rayleigh-Taylor and Richtmyer-Meshkov instabilities at arbitrary Atwood numbers, Phys. Rev. E 67 (2003), 026319.   DOI
13 V. N. Goncharov, Analytical model of nonlinear, single-mode, classical Rayleigh-Taylor instability at arbitrary Atwood numbers, Phys. Rev. Lett. 88 (2002), 134502.   DOI
14 H. Jin, X. F. Liu, T. Lu, B. Cheng, J. Glimm, and D. H. Sharp, Rayleigh-Taylor mixing rates for compressible flow, Phys. Fluids 17 (2005), 024104.   DOI
15 D. Layzer, On the instability of superposed fluids in a gravitational field, Astrophys. J. 122 (1955), 1-12.   DOI
16 B. Rollin and M. J. Andrews, Mathematical model of Rayleigh-Taylor and Richtmyer-Meshkov instabilities for viscoelastic fluids, Phys. Rev. E 83 (2011), 046317.   DOI
17 K. O. Mikaelian, Limitations and failures of the Layzer model for hydrodynamic instabilities, Phys. Rev. E 78 (2008), 015303.   DOI
18 P. Ramaprabhu and G. Dimonte, Single-mode dynamics of the Rayleigh-Taylor instability at any density ratio, Phys. Rev. E 71 (2005), 036314.   DOI
19 L. Rayleigh, Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density, Proc. London Math. Soc. 14 (1883), 170-177.
20 S. I. Abarzhi, Stable steady flows in Rayleigh-Taylor instability, Phys. Rev. Lett. 81 (1998), 337-340.   DOI
21 S. I. Abarzhi, Review of theoretical modelling approaches of Rayleigh-Taylor instabilities and turbulent mixing, Phil. Trans. R. Soc. A 368 (2010), 1809-1828.   DOI
22 S. I. Abarzhi, J. Glimm, and A.-D. Lin, Dynamics of two-dimensional Rayleigh-Taylor bubbles for fluids with a finite density contrast, Phys. Fluids 15 (2003), no. 8, 2190-2197.   DOI
23 D. Sharp, An overview of Rayleigh-Taylor instability, Phys. D 12 (1984), 3-10.   DOI
24 S.-I. Sohn, Vortex model and simulations for Rayleigh-Taylor and Richtmyer-Meshkov instabilities, Phys. Rev. E 69 (2004), 036703.   DOI
25 S.-I. Sohn, Density dependence of a Zufiria-type model for Rayleigh-Taylor and Richtmyer-Meshkov bubble fronts, Phys. Rev. E 70 (2004), 045301.
26 J. Hecht, U. Alon, and D. Shvarts, Potential flow models of Rayleigh-Taylor and Richtmyer-Meshkov bubble fronts, Phys. Fluids 6 (1994), 4019-4030.   DOI