Nonlinear periodic and solitary rolling waves in falling two-layer viscous liquid films

We investigate nonlinear periodic and solitary two-dimensional rolling waves in a falling two-layer liquid film in the regime of nonzero Reynolds numbers. At any flow rate, a falling two-layer liquid film is known to be linearly unstable with respect to long-wave deformations of the liquid-air surface and liquid-liquid interface. Two different types of zero-amplitude neutrally stable waves propagate downstream without growing or shrink-ing: a zigzag surface mode and a thinning varicose interface mode. Using a boundary-layer reduction of the Navier-Stokes equation, we investigate the onset, possible bifurcations, and interactions of nonlinear periodic traveling waves. Periodic waves are obtained by continuation as stationary periodic solutions in the comoving reference frame starting from small-amplitude neutrally stable waves. We find a variety of solitary waves that appear when a periodic solution approaches a homoclinic loop. Similar to falling one-layer films, we find two families of homoclinics, each family containing countably many solutions that can be characterized by the number of the major humps or dips in their profiles. Solitary waves with humps can be identified with droplets and travel faster than neutrally stable waves, while solitary waves with dips resemble localised depression regions, or holes, and travel slower than neutrally stable waves. Wave interactions are studied using direct numerical simulations of the boundary-layer model. We reveal that in the early stages of temporal evolution coarsening is dominated by an inelastic collision and merging of waves that travel at different speeds. Eventually, coarsening becomes arrested when the waves have reached a nearly homoclinic solution with a single major hump. In the mixed regime, when both mode types are unstable, the temporal dynamics becomes highly irregular due to the competition between a faster-traveling zigzag mode and a slower-traveling varicose mode. A quintessentially two-layer dynamical regime is found, which corresponds to a ruptured second layer. In this regime, the first layer adjacent to the solid wall acts as a conveyor belt, transporting isolated rolling droplets made of the second fluid downstream.