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The hydrodynamic stability behaviour of a two-layer falling film is exploredwith a floating flexible plate on the top surface. The stress balance at thesurface is modeled using a modified membrane equation. There is an insolublesurfactant at the liquid-liquid interface of the flow system. The linearinstability of perturbation waves is captured by numerically solving thegeneralized Orr-Sommerfeld eigenvalue problem using Chebyshev spectralcollocation technique. Four different types of linear stability modes areidentified: namely surface mode (SM), interface mode (IM), interface surfactantmode (ISM), and shear mode (SHM). The floating flexible plate has an inferiorimpact on all the existing modes in the longwave zone. However, a significantinfluence is noticed for the finite wave numbers. The surface and interfacewave instabilities can be suppressed in the smaller wavenumber zone by imposinghigher structural rigidity and uniform thickness of the flexible plate. Thestabilizing nature of the surface mode becomes more powerful when the top layerviscosity is dominant. A new interfacial instability emerges when the top layeris less viscous than the lower one. At moderate Reynolds numbers, the behaviourof interface mode is different in two different zones $mr<1$ and $mr>1$, where$m$ and $r$ denote the viscosity and density ratio, respectively. Further, theunstable shear modes induced by the top and bottom layers are detected underthe low inclination angles with a strong inertia force. The occurrence of theshear modes demands the bottom layer's viscosity and density to be very highthan that of the upper layer. The influence of characteristic parameters of theflexible plate on the lower layer shear mode is not very sensitive. Finally,the competition between the different modes for dominance of stabilityboundaries is also discussed.

Impact of a floating flexible plate on the stability of double-layered falling flow