Onset of nonlinear waves on falling films

Steady waves on a thin liquid film flowing down an inclined plane are analyzed near the critical Reynolds number. A second-order bifurcation analysis of an interface equation, which is valid to third order in the long-wave expansion, reveals two sets of waves near criticality. One set travels faster than twice the interfacial velocity, while the other set is symmetrically slower for the vertical film. Each set contains two families of shocks, one family of periodic waves, and a single solitary wave. Local analytical estimates of the velocity, amplitude, and wavelength are obtained for all waves. These estimates are favorably compared to numerical solutions of the steady waves and to experimental data near criticality. An interesting result is that naturally excited waves with wavelengths close to the maximum growing linear mode should not be studied with a local analysis near the neutral curve. Instead, they are better approximated by a Melnikov perturbation of the solitary wave solution.