Global eigenmodes of free-interface vertical liquid sheet flows

The global dynamics of unsteady free-interface vertical liquid sheet flows is studied, where the dynamics is termed global because it refers to the whole fluid system. The formal development of a proper mathematical model is presented initially, which accounts of pressure disturbances produced by the compliant interface in an air enclosure adjacent to the sheet. The linear spectral analysis (here restricted to sinuous disturbances only) shows that the surface tension is globally stabilizing, the spectrum exhibiting two typical branches related to the two characteristics curves of the governing equations. This basic finding is confirmed by means of both computations of the optimal amplifications (i.e. the greatest amplifications over all initial perturbations) of the relevant system energy and direct numerical simulations of the spatio-temporal evolution of an initial disturbance having the form of an interface Gaussian perturbation.