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

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.

Figure 1

Two-layer fluid system subject to longitudinal vibrations and the coordinate frame.

Figure 2

Group velocity $v_{\mathrm{gr}}$ of linear waves (black solid line) vs the wave number $k$. The group velocity for all wave packages is larger than the maximal soliton propagation velocity $c=1$. (Soliton stability is discussed in the text below.)

Figure 3

Spectra of eigenvalues $\lambda$ of the problem (46, 47, 48) of linear stability of solitons with speed $c$ specified in plots. The real part of $\lambda$ is the exponential growth rate of perturbations; for $c=0$ and $c=0.4$, one can see existence of one mode with positive $\mathrm{Re}\left(\lambda \right)$, i.e., instability mode, while for $c=0.6$ and $c=0.8$, all the perturbations oscillate without growth. In Fig. 4, the instability mode and the Goldstone mode of neutral stability (invariance to the shifts of soliton in space) are plotted for standing solitons $\left(c=0\right)$.

Figure 4

(a) Exponential growth rate $\mathrm{Re}\left(\lambda \right)$ of perturbations of soliton as a function of the soliton speed $c$. (b) The instability mode and the Goldstone mode of neutral stability (invariance to the shifts of soliton in space) of the standing soliton $\left(c=0\right)$.

Figure 5

(a) The unstable soliton (44) of $c=0.3$ with tiny positive perturbation explodes leading to the formation of a finite amplitude relief, possibly layer rupture. (b) The same unstable soliton $\left(c=0.3\right)$ with tiny negative perturbation falls apart into two fast stable solitons. (c) Falling apart of the unstable soliton with $c=0.1$.

Figure 6

Sample evolution of the dynamic system (34, 35) with arbitrary smooth initial conditions. The dynamics can be viewed as a kinetics of a gas of stable (fast) solitons.