Hydroelastic response of a floating plate on the falling film: A stability analysis

The role of a floating elastic plate on the hydrodynamic instability of a gravity-driven flow down an inclined plane is examined in the linear and nonlinear regimes. Linear instability of the system with respect to infinitesimal disturbances is captured using normal-mode analysis. The critical conditions for instability are obtained analytically utilizing the small film aspect ratio. The bifurcation of the nonlinear evolution equation is analyzed using weakly nonlinear stability analysis. The time evolution of the surface elevation is analyzed using the nonlinear analysis. The Orr–Sommerfeld boundary value problem corresponding to the perturbed flow is derived, and it is solved numerically using the spectral collocation method. The behavior of the marginal stability curves and temporal growth of the unstable waves are portrayed for a range of dimensionless flow parameters. Moreover, the pressure acting on the surface are calculated and analyzed for various structural parameters, applicable to different flow configurations. The study reveals that the structural parameters such as rigidity and mass per unit length play a crucial role in suppressing and facilitating the unstable surface waves of the flow. However, the compressive force acting on the plate results in the flexural destabilization. Thus, the plate parameters are more efficient in damping the shorter wave disturbances. Numerical observations imply that the floating elastic plate assists in stabilizing the free-falling flow and dampens the high amplitude waves.