Stability of the interface between two thin liquid layers under tangential high frequency vibrations
DOI:
https://doi.org/10.7242/1999-6691/2019.12.4.31Keywords:
frozen wave, vibrations, oscillatory flow, mean flow, interface, oscillatory Kelvin-Helmholtz instability, “shallow water” approximationAbstract
In this paper, a system of two equally thin layers of immiscible incompressible isothermal ideal liquids under high frequency horizontal harmonic vibrations is considered theoretically. The vessel containing the liquids is assumed to be closed, of rectangular form with weakly-deformable side borders and infinitely long in horizontal direction. Previous studies showed that, for significantly thin layers, the main instability in the system, oscillatory Kelvin-Helmholtz instability, should be the long-wave instability. Therefore, the problem was solved analytically using “shallow water” approximation. For all equations, a formal expansion with respect to two small parameters was used: one associated with a small ratio of the vertical to horizontal scale and another with small perturbations of the flat interface. Evolutionary equations were derived for the interface in the main order of expansion for vibration intensity less than a threshold value for oscillatory Kelvin-Helmholtz instability (subcritical area). The solutions found for these evolutionary equations correspond to traveling waves with soliton or cnoidal interfacial surface. The soliton profile is a limiting case of the cnoidal profile. The maximum speed of these waves was determined. It was demonstrated that the obtained solutions exist only in the subcritical area of parameters and from the critical level subcritical bifurcation emerges. For special case of traveling waves in a quasi-stationary mode (i.e. with static interface) also referred to as a “frozen wave”, a numerical analysis of linear stability was performed using a Fourier series expansion in a horizontal coordinate. The instability of quasi-stationary modes to small disturbances was demonstrated.
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References
Wolf G.H. The dynamic stabilization of rayleigh-taylor instability and corresponding dynamic equilibrium. Z. Physik, 1969, vol. 227, pp. 291-300. https://doi.org/10.1007/BF01397662">https://doi.org/10.1007/BF01397662
Bezdenezhnykh N.A., Briskman V.A., Lapin A.Y., Lyubimov D.V., Lyubimova T.P., Tcherepanov A.A., Zakharov I.V. The influence of high frequency tangential vibrations on the stability of the fluid interface in microgravity. Int. J. Microgravity Res. Appl., 1991, vol. 4(2), pp. 96-97; Sauer R. Einführung in die theoretische Gasdynamik [Introduction to theoretical gas dynamics]. Springer-Verlag, 1960. 214 p. https://doi.org/10.1007/978-3-642-92790-4">https://doi.org/10.1007/978-3-642-92790-4
Bezdenezhnykh N.A., Briskman V.A., Lapin A.Y., Lyubimov D.V., Lyubimova T.P., Tcherepanov A.A., Zakharov I.V. The influence of high frequency tangential vibrations on the stability of the fluid interface in microgravity. Microgravity Fluid Mechanics, ed. H.J. Rath. Springer, 1992. P. 137-144. https://doi.org/10.1007/978-3-642-50091-6_14">https://doi.org/10.1007/978-3-642-50091-6_14
Ivanova A.A., Kozlov V.G., Evesque P. Interface dynamics of immiscible fluids under horizontal vibration. Fluid Dyn., 2001, vol. 36, pp. 362-368. https://doi.org/10.1023/A:1019223732059">https://doi.org/10.1023/A:1019223732059
Talib E., Jalikop S.V., Juel A. The influence of viscosity on the frozen wave instability: theory and experiment. J. Fluid Mech., 2007, vol. 584, pp. 45-68. https://doi.org/10.1017/S0022112007006283">https://doi.org/10.1017/S0022112007006283
Lyubimov D.V., Cherepanov A.A. Development of a steady relief at the interface of fluids in a vibrational field. Fluid Dyn., 1986, vol. 21, pp. 849-854. https://doi.org/10.1007/BF02628017">https://doi.org/10.1007/BF02628017
Khenner M.V., Lyubimov D.V., Belozerova T.S., Roux B. Stability of plane-parallel oscillatory flow in a two-layer system. Eur. J. Mech. B Fluid., 1999, vol. 18, pp. 1085-1101. https://doi.org/10.1016/S0997-7546(99)00143-0">https://doi.org/10.1016/S0997-7546(99)00143-0
Yoshikawa H.N., Wesfreid J.E. Oscillatory Kelvin-Hemlholtz instability. Part 1. A viscous theory. J. Fluid Mech., 2011, vol. 675, pp. 223-248. https://doi.org/10.1017/S0022112011000140">https://doi.org/10.1017/S0022112011000140
Talib E., Juel A. Instability of a viscous interface under horizontal oscillation. Phys. Fluids, 2007, vol. 19, 092102. https://doi.org/10.1063/1.2762255">https://doi.org/10.1063/1.2762255
Lyubimov D.V., Ivantsov A.O., Lyubimova T.P., Khilko G.L. Numerical modeling of frozen wave instability in fluids with high viscosity contrast. Fluid Dyn. Res., 2016, vol. 48, 061415. https://doi.org/10.1088/0169-5983/48/6/061415">https://doi.org/10.1088/0169-5983/48/6/061415
Lyubimov D.V., Khilko G.L., Ivantsov A.O., Lyubimova T.P. Viscosity effect on the longwave instability of a fluid interface. J. Fluid Mech., 2017, vol. 814, pp. 24-41. https://doi.org/10.1017/jfm.2017.28">https://doi.org/10.1017/jfm.2017.28
Goldobin D.S., Kovalevskaya K.V., Lyubimov D.V. Elastic and inelastic collisions of interfacial solitons and integrability of a two-layer fluid system subject to horizontal vibrations. EPL, 2014, vol. 108, 54001 https://doi.org/10.1209/0295-5075/108/54001">https://doi.org/10.1209/0295-5075/108/54001
Goldobin D.S., Pimenova A.V., Kovalevskaya K.V., Lyubimov D.V., Lyubimova T.P. Running interfacial waves in two-layer fluid system subject to longitudinal vibrations. Phys. Rev. E, 2015, vol. 91, 053010. https://doi.org/10.1103/PhysRevE.91.053010">https://doi.org/10.1103/PhysRevE.91.053010
Lyubimov D.V., Lyubimova T.P., Cherepanov A.A. Dinamika poverkhnostey razdela v vibratsionnykh polyakh [The dynamics of the interface in vibration fields]. Moscow, Fizmatlit, 2003. 216 p.