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.