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.

我们提出了一系列薄膜流动的松弛模型，其中粘度和表面张力效应都是固有的。第一步，给出系统无耗散部分的一阶双曲线近似。该方法基于增强拉格朗日方法，其中使用经典惩罚方法并将拉格朗日中的高阶导数提升为新的自变量，为此寻求双曲闭包方程。然后，我们展示了粘性项可以通过将它们直接插入获得的系统，使其成为双曲抛物线类型，或者将它们转换为渐近等效于前一个公式的近似代数源项来处理。最后，还提出了将该方法扩展到经典的非线性表面张力模型。