Nonlinear waves in a rivulet falling down a vertical plate

Nonlinear waves in a rectilinear rivulet flowing down a vertical plate are studied based on the developed theoretical model. The model equations are derived by the weighted residual method through projecting the Navier–Stokes equations onto the constructed system of basic orthogonal polynomials. Stability of the rivulet flow is analyzed and dispersion dependences for linear waves are obtained. Nonlinear wave regimes of a rivulet flow are numerically studied within the framework of two different problems, namely, the problem of stationary traveling waves with a given wavelength and the problem of spatial development of forced waves with a given frequency. Characteristics of nonlinear quasi-two-dimensional stationary traveling waves are obtained, and spatial development of forced waves is studied. Waves of various families are identified. It is shown that in a certain narrow range of excitation frequency, there are no stationary traveling waves, but a pulsating regime of flow occurs. •A semi-analytical theoretical model for a wave rivulet flow has been developed.•The existence of waves of various families is revealed.•Peculiarity of nonlinear quasi-two-dimensional stationary traveling waves are obtained.•The spatial development of excited waves is studied.