Nonlinear stability of liquid films over an inclined plane

Given a film of viscous heavy liquid with upper free boundary over an inclined plane, a steady laminar motion develops parallel to the flat bottom ofthe layer. We name this motion\emph{ Poiseuille Free Boundary} PFBflow because of its (half) parabolic velocity profile. In flowsover an inclined plane the free surface introduces additionalinteresting effects of surface tension and gravity. These effectschange the character of the instability in a parallel flow, see{Smith} [1]. \par\noindentBenjamin [2], and Yih [3], have solved the linear stabilityproblem of a uniform film on a inclined plane. Instability takesplace in the form of an infinitely long wave, however\emph{surface waves of finite wavelengths are observed}, see e.g.Yih [3]. Up to date direct nonlinear methods for the study ofstability seem to be still lacking.
Aim of this talk is the investigation of nonlinear stability ofPFB providing \emph{ a rigorous formulation of the problem by theclassical direct Lyapunov method assuming periodicity in theplane}, when above the liquid there is a uniform pressure due tothe air at rest, and the liquid is moving with respect to the air.Sufficient conditions on the non dimensional Reynolds, Webernumbers, on the periodicity along the line of maximum slope, onthe depth of the layer and on the inclination angle are computedensuring Kelvin-Helmholtz \emph{nonlinear stability}. We use\emph{a modified energy method, cf. [4],[5], which providesphysically meaningful sufficient conditions ensuring nonlinearexponential stability}. The result is achieved in the class ofregular solutions occurring in simply connected domains havingcone property.\par\noindentNotice that the linear equations, obtained by linearization of ourscheme around the basic Poiseuille flow, do coincide with theusual linear equations, cf. {Yih} [3]. \\ {\bf References}\\ [1]  M.K. Smith, \textit{The mechanism for the long-waveinstability in thin liquid films} J. Fluid Mech., \textbf{217},1990, pp.469-485.
\\ [2]  Benjamin T.B., \textit{Wave formation in laminar flow down aninclined plane}, J. Fluid Mech. \textbf{2}, 1957, 554-574.
\\ [3]  Yih Chia-Shun, \textit{Stability of liquid flow down aninclined plane}, Phys. Fluids, \textbf{6}, 1963, pp.321-334.
\\ [4] Padula M., {\it On nonlinear stability of MHD equilibriumfigures}, Advances in Math. Fluid Mech., 2009, 301-331.
\\ [5] Padula M., \textit{On nonlinear stability of linear pinch},Appl. Anal.  90 (1), 2011, pp. 159-192.