We investigate the dynamics of a three-dimensional thin laminar liquid film flowing down an inclined plane, influenced by both gravity and wind forces. The gas phase is modelled using a quasi-laminar approximation, in which viscosity is neglected. We expand Miles’ theory of wind-induced pressure, which previously focused on a one-dimensional interface, to a two-dimensional framework. From this, we derive evolution equations for the liquid film, including Benney-type and Nepomnyashchy-type models that capture the effects of wind on flow instability and wave dispersion. Our findings reveal two families of travelling-wave solutions and their bifurcation structures through numerical continuation. We analyse secondary instabilities using the centre-manifold method and Floquet theory, discovering finite-wavenumber bands in which oblique perturbations are more unstable than longitudinal ones. Based on these secondary-instability characteristics, we propose a classification framework for various flow regimes. Time-dependent numerical simulations support our predictions of secondary instabilities and reveal a range of nonlinear phenomena that occur after their onset. These phenomena include checkerboard-symmetric patterns, strip-like structures, phase-type instabilities and localised wave packets, some of which have been observed experimentally in previous studies.
Xie et al. (Tue,) studied this question.