Waves on a viscous fluid film down a vertical wall

Abstract

Waves on a viscous fluid film flowing down a vertical plane wall are studied theoretically. Stationary solutions are sought for a nonlinear equation that describes the variation of the free surface, and a solitary wave is found as an exact solution. The corrugation of the surface concentrates in a small portion comparable with the film thickness and propagates with a constant speed. There exist various kinds of solitary waves characterized by the number of their humps. To compare these solitary waves with the so-called single waves, solitary waves of the same kind are arranged at regular intervals on the laminar flow. When the nonlinearity is comparatively weak, individual solitary waves are so close to one another that this procedure of finding the periodic solution has to be replaced by the method of expanding the free surface as a Fourier sum and determining its coefficients. The periodic solution of arbitrary wavelength is obtained from these two procedures. There are as many kinds of periodic solutions as those of solitary waves, and a selection rule is proposed as to the problem of which kind of periodic solutions actually appears. Each kind of our periodic solutions is compared with the nearly sinusoidal wave as well as the single wave. It is seen that the wave that was observed experimentally is nothing but a superposition of solitary waves that are mistaken for the solitary waves with only one principal hump. Further, it is expected that solitary waves are still excited, even though there are no artificial perturbations.