Secondary instability in thin film flows under an inclined plane: growth of lenses on spatially developing rivulets

The response of a thin film flowing under an inclined plane, modelled using the lubrication equation, is studied. The flow at the inlet is perturbed by the superimposition of a spanwise-periodic steady modulation and a decoupled temporally periodic but spatially homogeneous perturbation. As the consequence of the spanwise inlet forcing, the so-called rivulets grow downstream and eventually reach a streamwise-invariant state, modulated along the direction perpendicular to the flow. The linearized dynamics in the presence of a time-harmonic inlet forcing shows the emergence of a time-periodic flow characterized by drop-like structures (so-called lenses) that travel on the rivulet. The spatial evolution is rationalized by a weakly non-parallel stability analysis. The occurrence of the lenses, their spacing and thickness profile, is controlled by the inclination angle, flow rate, and the frequency and amplitude of the time-harmonic inlet forcing. The faithfulness of the linear analyses is verified by nonlinear simulations. The results of the linear simulations with inlet forcing are combined with the computations of nonlinear travelling lenses solutions in a double-periodic domain to obtain an estimate of the dripping length, for a large range of conditions.