Steady wind-generated gravity-capillary waves on viscous liquid film flows

Steady gravity-capillary periodic waves on the surface of a thin viscous liq- uid film supported by an air stream on an inclined wall are investigated. Based on lubrication approximation and thin air-foil theory, this problem is reduced to an integro-differential equation. The small-amplitude analysis is carried out to obtain two analytical solutions up to the second order. Numer- ical computation shows there exist two distinct primary bifurcation branches starting from infinitesimal waves, which approach solitary wave configura- tion in the long-wave limit when the values of physical parameters are above certain thresholds. New families of solutions manifest themselves either as secondary bifurcation occurring on primary branches or as isolated solution branches. The limiting configurations of the primary solution branches with the increase of two parameters are studied in two different cases, where one and two limiting configurations are obtained respectively. For the latter case, the approximation of the configurations is given.