A mathematical model and numerical framework are presented for computing multi-physics multi-layer coating flow dynamics, with applications to the leveling of multi-layer paint films. The algorithm combines finite difference level set methods and high-order accurate sharp-interface implicit mesh discontinuous Galerkin methods to capture a complex set of multi-physics, incorporating Marangoni-driven multi-phase interfacial flow and the transport, mixing, and evaporation of multiple dissolved species. In particular, we develop several numerical methods for this multi-physics problem, including: high-order local discontinuous Galerkin methods for Poisson problems with Robin boundary conditions on implicitly-defined domains, to capture solvent evaporation; finite difference surface gradient methods, to robustly and accurately incorporate Marangoni stresses; and a coupled multi-physics time stepping approach, to incorporate all the different solvers at play including quasi-Newtonian fluid flow. The framework is applicable to an arbitrary number of layers and dissolved species; here, we apply it in a variety of settings, including multi-solvent evaporative paint dynamics, the flow and leveling of multi-layer automobile paint coatings in both 2D and 3D, and an examination of interfacial turbulence within a multi-layer matter cascade. Our results reproduce several phenomena observed in experiment, such as the formation of Marangoni plumes and Bénard cells. We also use the model to study the impact of long-wave deformational surface modes on immersed interfaces as well as the emergence of the final multi-layer film profile.

中文翻译：

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多层涂层流的混合有限差分水平集-隐式网格间断伽辽金法

提出了用于计算多物理多层涂层流动动力学的数学模型和数值框架，并应用于多层漆膜的流平。该算法结合有限差分水平集方法和高阶精确锐界面隐式网格不连续伽辽金方法来捕获一组复杂的多物理场，结合Marangoni驱动的多相界面流以及多相的传输、混合和蒸发溶解的物种。特别是，我们为这个多物理问题开发了几种数值方法，包括：用于隐式定义域上具有罗宾边界条件的泊松问题的高阶局部不连续伽辽金方法，以捕获溶剂蒸发；有限差分表面梯度法，稳健且准确地结合马兰戈尼应力；以及耦合的多物理场时间步进方法，以合并所有不同的求解器，包括准牛顿流体流。该框架适用于任意数量的层和溶解的物种；在这里，我们将其应用于各种设置，包括多溶剂蒸发涂料动力学、2D 和 3D 多层汽车油漆涂层的流动和流平，以及多层物质级联内的界面湍流检查。我们的结果再现了实验中观察到的几种现象，例如马兰戈尼羽流和贝纳德细胞的形成。我们还使用该模型研究长波变形表面模式对浸入界面的影响以及最终多层薄膜轮廓的出现。