Marangoni modulation of coupled Rayleigh-Taylor and Faraday instabilities in vertically oscillated liquid films

Abstract

We investigate the Marangoni modulation of coupled Rayleigh-Taylor and Faraday instabilities in a vertically oscillated Newtonian liquid film carrying insoluble surfactants. Linear stability analysis using Floquet theory reveals that an increasing Marangoni number (Ma) selectively suppresses subharmonic modes, driving the system into a harmonic-dominated regime. The interfacial response is found to be highly frequency-dependent. At low forcing frequencies, increasing Ma causes adjacent harmonic tongues to merge into a novel surfactant mode that migrates towards long wavelengths, ultimately coalescing with the RTI branch and fragmenting the dynamically stable window. Conversely, at high frequencies, surfactants monotonically elevate the harmonic instability threshold, significantly widening the stable parameter space. To uncover the underlying mechanisms, a long-wave asymptotic analysis is performed, demonstrating that the critical forcing amplitude factorizes into a static capillary-gravity margin and a dynamic elasto-inertial modulation, yielding a scaling law for the critical mode balance. Finally, nonlinear simulations based on a rigorous weighted-residual reduced model are utilized to dissect the spatial work performed by individual forces, which shows that surfactants modulate stability through phase-controlled Marangoni transport. In the RTI regime, increasing Ma reverses the transport direction and drives fluid into the peaks, inducing a transition from stabilization to destabilization. In the Faraday instability (FI) regime, the response exhibits a strong frequency dependence, governed by Marangoni transport that redistributes fluid away from interfacial peaks at high frequencies but toward them at low frequencies, thereby suppressing or enhancing the instability accordingly.