This paper is devoted to the asymptotic analysis of a thin film equation which describes the evolution of a thin liquid droplet on a solid support driven by capillary forces. We propose an analytic framework to rigorously investigate the connection between this model and Tanner's law [22] which claims: the edge velocity of a spreading thin film on a pre-wetted solid is approximately proportional to the cube of the slope at the inflection. More precisely, we investigate the asymptotic limit of the thin film equation when the slippage coefficient is small and at an appropriate time scale (see Equation (8)). We show that the evolution of the droplet can be approximated by a moving free boundary model (the so-called quasi-static approximation) and we present some results pointing to the validity of Tanner's law in that regime. Several papers [5, 6, 10] have previously investigated a similar connection between the thin film equation and Tanner's law either formally or for particular solutions. Our main contribution is the introduction of a new approach to systematically study this problem by finding an equation for the evolution of the apparent support of the droplet (described mathematically by a nonlinear function of the solution).

中文翻译：

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关于薄膜方程与坦纳定律的关系

本文致力于薄膜方程的渐近分析，该方程描述了由毛细管力驱动的固体支持物上的薄液滴的演变。我们提出了一个分析框架来严格研究该模型与坦纳定律 [22] 之间的联系，该定律声称：在预润湿的固体上展开薄膜的边缘速度大约与拐点处斜率的立方成正比。更准确地说，我们研究了当滑移系数较小且时间尺度合适时薄膜方程的渐近极限（见方程（8））。我们表明可以通过移动自由边界模型（所谓的准静态近似）来近似液滴的演化，并且我们提出了一些结果，指出了坦纳定律在该制度中的有效性。几篇论文 [5, 6, 10] 之前已经研究了薄膜方程和坦纳定律之间的类似联系，无论是形式上的还是特定解的。我们的主要贡献是引入了一种新方法来系统地研究这个问题，通过找到液滴表观支撑演化的方程（由解的非线性函数在数学上描述）。