When a liquid film drains on a vertical plate, the film becomes nonuniform near the vertical} edge. Here we experimentally report the three-dimensional (3D) self-similar shape of this film. Based on the well-known 2D self-similar solution of a draining film far from the edge, we identify a new 3D self-similar scaling, which converts the PDE for the film thickness with three independent variables into an ODE. Interferometry is performed to measure the film thickness as a function of position and time, and the results are in excellent agreement with the theoretical predictions.

中文翻译：

---

垂直边缘附近的自相似排水

当液体薄膜在垂直板上排出时，薄膜在垂直边缘附近变得不均匀。在这里，我们通过实验报告了该薄膜的三维（3D）自相似形状。基于远离边缘的排水膜的著名2D自相似解，我们确定了一种新的3D自相似缩放，该缩放将具有三个自变量的膜厚度的PDE转换为ODE。进行干涉测量以测量膜厚度随位置和时间的变化，其结果与理论预测非常吻合。