Falling liquid films on longitudinal grooved geometries: Integral boundary layer approach

Falling thin liquid film on a substrate with complex topography is modeled using a three equation integral boundary layer system. Linear stability and nonlinear dynamics of the film in the framework of this model are studied on a topography with sinusoidal longitudinal grooves aligned parallel in the direction of the main flow. The linear stability theory reveals the stabilizing nature of the surface tension force and the groove measure on the film, and the pronounced destabilizing effects of inertia. The evolution of the film thickness is tracked numerically for a vertically falling film on a grooved geometry by choosing wavenumbers corresponding to the unstable mode where the growth rate of instability is maximum. The effect of surface geometry on the temporal evolution of the film dynamics is analyzed on a periodic domain. Numerical investigations agree with the linear stability predictions and show that the longitudinal grooves exert a stabilizing effect on the film and the waviness is suppressed when the steepness of the longitudinal groove measure increases