  The hydrodynamic stability behaviour of a two-layer falling film is explored
with a floating flexible plate on the top surface. The stress balance at the
surface is modeled using a modified membrane equation. There is an insoluble
surfactant at the liquid-liquid interface of the flow system. The linear
instability of perturbation waves is captured by numerically solving the
generalized Orr-Sommerfeld eigenvalue problem using Chebyshev spectral
collocation technique. Four different types of linear stability modes are
identified: namely surface mode (SM), interface mode (IM), interface surfactant
mode (ISM), and shear mode (SHM). The floating flexible plate has an inferior
impact on all the existing modes in the longwave zone. However, a significant
influence is noticed for the finite wave numbers. The surface and interface
wave instabilities can be suppressed in the smaller wavenumber zone by imposing
higher structural rigidity and uniform thickness of the flexible plate. The
stabilizing nature of the surface mode becomes more powerful when the top layer
viscosity is dominant. A new interfacial instability emerges when the top layer
is less viscous than the lower one. At moderate Reynolds numbers, the behaviour
of 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, the
unstable shear modes induced by the top and bottom layers are detected under
the low inclination angles with a strong inertia force. The occurrence of the
shear modes demands the bottom layer's viscosity and density to be very high
than that of the upper layer. The influence of characteristic parameters of the
flexible plate on the lower layer shear mode is not very sensitive. Finally,
the competition between the different modes for dominance of stability
boundaries is also discussed.
