Nonlinear Stability of Roll Waves Down an Inclined Falling Film

Kan ZHU, Abdelaziz BOUDLAL, Gilmar MOMPEAN

The present work is concerned with surface instabilities of non-Newtonian liquid films, usually called roll waves (RW). A thin liquid film in which the shear stress is modeled as a power-law is considered to study the stability of nonlinear roll waves down inclined plane walls. In the long wave approximation, depth-integrated continuity and momentum equations are derived by applying Karman's momentum integral method. As the linearized instability analysis of uniform flow only provides a diagnosis of instability, the modulation equations for wave series are derived and a stability criterion depending on two parameters (integro-differential expression) is obtained. The main difficulty to establish the stability domain is due of the presence of singularities near infinitesimal and maximal amplitudes. Numerical calculations are performed using asymptotic formulas near the singularities. The stability diagrams are presented for some values of the flow parameters. They reveal that there are situations wherein at critical values of the flow parameters, where the waves disappear. For the prediction and control of the free-surface profile, it is one of the main reasons for carrying out research in this area, as RW are generally an undesirable phenomenon.