Goldobin D S, Pimenova A V, Kovalevskaya K V, Lyubimov D V, Lyubimova T P
Institute of Continuous Media Mechanics, UB RAS, 1 Academik Korolev str., Perm 614013, Russia.
Department of Theoretical Physics, Perm State University, 15 Bukireva str., Perm 614990, Russia.
Phys Rev E Stat Nonlin Soft Matter Phys. 2015 May;91(5):053010. doi: 10.1103/PhysRevE.91.053010. Epub 2015 May 15.
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.
我们研究了在高频水平振动作用下，两种不混溶流体的水平薄液层之间界面处的波动。此前，能量泛函的变分原理可用于处理受振动作用的流体动力学系统中自由界面的准稳态，该原理揭示了此系统中存在驻定周期波和孤子。然而，这种方法没有提供处理演化问题的常规手段：既无法处理稳定性问题，也无法处理与传播波相关的问题。在这项工作中，我们严格推导了该系统中长波的演化方程，结果发现这些方程与正（或好）的Boussinesq方程相同。利用这些方程，可以找到所有与时间无关的剖面孤波（驻定孤子是这些传播波的一种特殊情况），它们存在于线性不稳定阈值以下；驻定孤子和慢孤子总是不稳定的，而快孤子是稳定的。根据初始扰动的不同，不稳定孤子要么以爆炸方式增长，这意味着在有限时间内液层破裂，要么分裂成稳定孤子。这些结果是在长波近似下得出的，因为对平界面状态的线性稳定性分析[D.V. Lyubimov和A.A. Cherepanov，《流体动力学》21，849（1986）]表明，薄液层的不稳定性是长波长的。
