A thin film of a non-Newtonian flowing down an inclined plane under the action of gravity is studied focusing on the effects of the fluid rheology on the complex wave dynamics that develops at the free surface. In chapter 3, the properties of the solitary waves, which organize the disordered dynamics of the Newtonian film, are considered. Direct numerical simulations (DNSs) of purely solitary waves have been performed and compared to the results from the four-equation model derived in [112]. The evolution of a power-law laminar film flow is considered in chapter 5 and modeled within the frame of the lubrication approximation by means of the weighted residual approach. Comparisons to Orr-Sommerfeld stability analysis and to DNS show convincing agreement in both linear and nonlinear regimes. A viscoplasctic film is modeled by the Herschel and Bulkley law and is presented in chapter 6. The elasticity of the pseudo-plug region close to the interface is taken into account by an elasto-visco-plastic constitutive relation derived by Saramito [121]. A model is derived in terms of four-equations for the film thickness, local flow rate and amplitudes of the normal and tangential stresses. A linear stability analysis gives values of the critical Reynolds number in remarkable agreement with the Orr-Sommerfeld analysis.
