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Goldobin DS, Pimenova AV, Kovalevskaya KV, et al. Running interfacial waves in a two-layer fluid system subject to longitudinal vibrations. Phys Rev E Stat Nonlin Soft Matter Phys. 2015;91(5):053010doi: 10.1103/PhysRevE.91.053010.
Goldobin, D. S., Pimenova, A. V., Kovalevskaya, K. V., Lyubimov, D. V., & Lyubimova, T. P. (2015). Running interfacial waves in a two-layer fluid system subject to longitudinal vibrations. Physical review. E, Statistical, nonlinear, and soft matter physics, 91(5), 053010. https://doi.org/10.1103/PhysRevE.91.053010
Goldobin, D S, et al. "Running interfacial waves in a two-layer fluid system subject to longitudinal vibrations." Physical review. E, Statistical, nonlinear, and soft matter physics vol. 91,5 (2015): 053010. doi: https://doi.org/10.1103/PhysRevE.91.053010
Goldobin DS, Pimenova AV, Kovalevskaya KV, Lyubimov DV, Lyubimova TP. Running interfacial waves in a two-layer fluid system subject to longitudinal vibrations. Phys Rev E Stat Nonlin Soft Matter Phys. 2015 May;91(5):053010. doi: 10.1103/PhysRevE.91.053010. Epub 2015 May 15. PMID: 26066252.
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Phys Rev E Stat Nonlin Soft Matter Phys. 2015 May;91(5):053010. doi: 10.1103/PhysRevE.91.053010. Epub 2015 May 15.

Running interfacial waves in a two-layer fluid system subject to longitudinal vibrations.

Physical review. E, Statistical, nonlinear, and soft matter physics

D S Goldobin, A V Pimenova, K V Kovalevskaya, D V Lyubimov, T P Lyubimova

Affiliations

  1. Institute of Continuous Media Mechanics, UB RAS, 1 Academik Korolev str., Perm 614013, Russia.
  2. Department of Theoretical Physics, Perm State University, 15 Bukireva str., Perm 614990, Russia.
  3. Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, United Kingdom.

PMID: 26066252 DOI: 10.1103/PhysRevE.91.053010

Abstract

We study the waves at the interface between two thin horizontal layers of immiscible fluids subject to high-frequency horizontal vibrations. Previously, the variational principle for energy functional, which can be adopted for treatment of quasistationary states of free interface in fluid dynamical systems subject to vibrations, revealed the existence of standing periodic waves and solitons in this system. However, this approach does not provide regular means for dealing with evolutionary problems: neither stability problems nor ones associated with propagating waves. In this work, we rigorously derive the evolution equations for long waves in the system, which turn out to be identical to the plus (or good) Boussinesq equation. With these equations one can find all the time-independent-profile solitary waves (standing solitons are a specific case of these propagating waves), which exist below the linear instability threshold; the standing and slow solitons are always unstable while fast solitons are stable. Depending on initial perturbations, unstable solitons either grow in an explosive manner, which means layer rupture in a finite time, or falls apart into stable solitons. The results are derived within the long-wave approximation as the linear stability analysis for the flat-interface state [D.V. Lyubimov and A.A. Cherepanov, Fluid Dynamics 21, 849 (1986)] reveals the instabilities of thin layers to be long wavelength.

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