Falling films on flexible substrates

Falling films have a long history and modern advances in the theory have led to accurate modelling for flows down rigid inclines. We now consider the possible effects that can be introduced by flexible substrates. In this work, we derive Benney-like coupled equations for the film thickness and substrate deflection using long-wave theory. Weakly nonlinear equations are also derived, which, in the limit of small substrate deflections, reduce to the Kuramoto-Sivashinsky equation. We also use boundary-layer theory in conjunction with the Karman-Polhausen approximation to derive three strongly coupled evolution equations for the film thickness, substrate deflection and film volumetric flow rate. Inertial, gravtiational, capillary, viscous, substrate tension and damping effects are elucidated via numerical solutions of the above systems of equations.