Falling film on flexible wall in the limit of weak viscoelasticity

The flow dynamics of an upper-convected-Maxwell (UCM) falling film down a flexible vertical wall is studied in the limit of weak viscoelasticity. A set of Benney-like weakly nonlinear equations for the film thickness and wall deflection, which is valid for small flow rate, is derived based on the long-wave theory. It shows that the unstable role of liquid viscoelasticity is equivalent to that of the flow inertia. A set of asymptotic evolution equations valid for moderate flow rate is obtained based on the integral theory. The linear instability property of the system is examined by using a normal-mode analysis. It shows that the liquid viscoelasticity acts to destabilize the falling film even for the flow with inertia being negligible. The nonlinear evolution equations for the moderate flow rate are solved numerically. The spatio-temporal evolutions of the liquid-air interface and flexible wall are examined. It is concluded that the liquid viscoelasticity plays a role to strengthen the dispersion of the initial imposed perturbation. It can promote the traveling speed of the solitary-like humps and suppress the front-running ripples at the same time. Both the wall damping and wall tension acts to suppress the fluctuations of the flexible wall. However, they play different roles in the evolution of the liquid-air interface.