Competing Mechanisms at Vibrated Interfaces of Density-Contrast Fluids

TL;DR

This work addresses the interplay between Rayleigh–Taylor and Faraday instabilities in a density-contrast two-fluid interface under vertical vibration. It combines linear Floquet stability analysis with high-fidelity 2D diffuse-interface simulations to reveal a new multi-modal instability region where RT and Faraday mechanisms compete rather than co-activate, and shows that RT growth can nonlinearly suppress Faraday responses even when Faraday initially dominates. A Floquet-based decomposition into modal growth and periodic harmonics captures the initial transient interface dynamics, and DNS validates the predicted transitions and multiscale breakup behavior. The findings have implications for actively controlling interfacial dynamics in vibrated multiphase systems, with potential applications in atomization, spacecraft propellant management, and related engineering processes, and point to avenues for extending the analysis to 3D and weakly nonlinear regimes.

Abstract

Fluid--fluid interfacial instability and subsequent fluid mixing are ubiquitous in nature and engineering. The hydrodynamic instability of fluid interfaces has long centered on the pressure gradient-driven long-wavelength Rayleigh--Taylor instability and the resonance-induced short-wavelength Faraday instability. However, neither instability alone can explain the dynamics when both mechanisms are present. We identify a previously unseen multi-modal instability emerging from their coexistence. When the denser fluid is polydimethylsiloxane, the mixed region at a high density contrast (Atwood number=0.9) spans a vibration amplitude range approximately twice the gravitational acceleration. Using Floquet stability analysis, we show how vibrations govern transitions between the RT and Faraday instabilities, leading to contention between these instabilities rather than resonant enhancement. The initial transient growth is represented by the exponential modal growth of the most unstable Floquet exponent, along with its accompanying periodic behavior. Direct numerical simulations validate these findings and track interface breakup into the multiscale and nonlinear regimes. Specifically, we show that growing RT modes nonlinearly suppress Faraday responses even when the initial growth rate of the Faraday instability is 3.63 times that of RT, so a bidirectional competition hinders their sustained coexistence.

Figures (7)

• Figure 1: The two-fluid interface with long-wavelength RT and short-wavelength Faraday instabilities.
• Figure 2: Analysis of cases with the most unstable wavenumber $k^*/2\pi=5$ for $g^*=1$: (a) Growth rates; (b--e) Temporal evolution of the most unstable waves, with a magnified view showing the first period.
• Figure 3: Interface displacement growth rate for $g^*=-1$ under various parameter combinations, with arrows denoting the transition from $a^*=0$ to $a^*=30$.
• Figure 4: Dominant growth rate across the oscillating frequency-amplitude phase space: (a) $(\mathrm{At},\eta)=(0.9,0.1)$; (b) $(\mathrm{At},\eta)=(0.3,0.1)$. The margins for the Faraday instability and the RT-Faraday transition are highlighted.
• Figure 5: Normalized evolution of Faraday and RT waves with growth rates (insets) under single-mode or Perlin noise excitations: (a) $a^*=16,\,\omega^*=33.9$; (b) $a^*=18,\,\omega^*=27.4$. Both are marked in \ref{['fig:bifurcation_unstableRT']} (a).