Instability of a thin viscous film flowing under an inclined substrate: steady patterns
Gaétan Lerisson, Pier Giuseppe Ledda, Gioele Balestra, François Gallaire
Abstract
The flow of a thin film coating the underside of an inclined substrate is studied. We measure experimentally spatial growth rates and compare them to the linear stability analysis of a flat film modeled by the lubrication equation. When forced by a stationary localized perturbation, a front develops that we predict with the group velocity of the unstable wave packet. We compare our experimental measurements with numerical solutions of the non-linear lubrication equation with complete curvature. Streamwise structures dominate and saturate after some distance. We recover their profile with a 1D lubrication equation suitably modified to ensure an invariant profile along the streamwise direction and compare them with the solution of a purely two dimensional pendent drop, showing overall a very good agreement. Finally, those different profiles agree also with a 2D simulation of the Stokes equations.