Weakly nonlinear lower-branch stability of fully developed and developing free-surface flows

We examine the weakly nonlinear stability of both fully developed and of developing liquid layers. The study of these free-surface flows is more complicated than that of many other flows owing to the fully nonlinear boundary conditions present. The scalings used for the two problems follow from the work of J. S. B. Gajjar, who described their linear stability properties. We use the technique given by F. T. Smith to derive amplitude equations of the type presented by J. T. Stuart and J. Watson. Both flows are found to be supercritically stable in general and a variety of asymptotic cases are considered.