In this paper we tackle the problem of surfactant spreading on a thin liquid film in the framework of isogeometric analysis. We consider a mathematical model that describes this phenomenon as an initial boundary value problem (IBVP) that includes two coupled fourth order partial differential equations (PDEs), one for the film height and one for the surfactant concentration. In order to solve this problem numerically, it is customary to transform it into a mixed problem that includes at most second order PDEs. However, the higher-order continuity of the approximation functions in Isogeometric Analysis (IGA) allows us to deal with the weak form of the fourth order PDEs directly, without the need of resorting to mixed methods. We demonstrate numerically that the IGA solution is able to reproduce results obtained before with mixed approaches. Complex phenomena such as Marangoni-driven fingering instabilities triggered by perturbations are easily captured.

翻译：本文在等几何分析框架下研究了表面活性剂在薄液膜上的铺展问题。我们采用了一个数学模型，将该现象描述为包含两个耦合的四阶偏微分方程的初始边值问题（IBVP），分别控制液膜高度和表面活性剂浓度。为数值求解该问题，通常需将其转化为至多包含二阶偏微分方程的混合问题。然而，等几何分析（IGA）中近似函数的高阶连续性使我们能够直接处理四阶偏微分方程的弱形式，而无需借助混合方法。我们通过数值实验证明，IGA解能再现先前混合方法获得的结果。由扰动引发的马兰戈尼驱动指进不稳定性等复杂现象可被轻松捕捉。