Hyperbolic Relaxation Models for Thin Films Down an Inclined Plane

We present a family of relaxation models for thin films flows where both viscosity and surface tension effects are inherent. In a first step, a first-order hyperbolic approximation to the dissipationless part of the system is presented. The method is based on an augmented Lagrangian approach, where a classical penalty method is used and high-order derivatives in the Lagrangian are promoted to new independent variables, for which hyperbolic closure equations are sought. Then, we show that the viscous terms can be treated either by plugging them directly to the obtained system, making it of the hyperbolic-parabolic type or by casting them into an approximate algebraic source term that is asymptotically equivalent to the former formulation. Finally, the extension of the method to a classical nonlinear surface tension model is also presented. Numerical results, for all the proposed models are shown and compared with experimental results and reference solutions.