| Abstract: | The falling of a thin viscous fluid layer (film) along a vertical plane under the effect of gravity is accompanied by wave motions in which capillary forces play an essential part. An equation for the film thickness h(x, t) is used extensively in analyses of these motions. This equation, obtained from the Navier—Stokes equations and the boundary conditions under different assumptions, reduces to an ordinary third-order nonlinear differential equation [1–7] for steady plane motions. Periodic solutions of this equation were sought by the methods of asymptotic expansions in the amplitude or by Fourier series expansions [1–7], which assumes a sequential accounting of the nonlinearity as a small perturbation. This limits the validity of the results obtained to the domain of small amplitudes. The case of arbitrary amplitudes is considered in this paper. A solution of the problem, based on an asymptotic expansion in the parameter ε is constructed. In this expansion the equation for the first approximation remains nonlinear but admits of integration, which discloses the class of bounded periodic solutions. Moreover, strict integral relations (for any ε) are obtained, and a variational problem about seeking the lower bound of values of the mean film thickness and other characteristics of the ultimately developed optimal motions is formulated and solved on their basis. The results obtained agree with experiments. |
