论文标题
粘性Marangoni蔓延的确切解决方案
Exact solutions for viscous Marangoni spreading
论文作者
论文摘要
当表面活性分子在液体界面释放时，它们的扩散动力学由Marangoni流控制。尽管以不同的限制进行了研究，但确切的解决方案仍然很少。在这里，我们考虑了沿深流体层界面的不溶性表面活性剂的扩散。对于二维Stokes流，最近显示，非线性传输问题可以精确地映射到复杂的汉堡方程[Crowdy，Siam J. Appl。数学。 81，2526（2021）]。我们首先提出了该方程式的非常简单的推导。然后，我们提供完全显式的解决方案，发现改变初始表面活性剂分布（脉冲，孔或周期性）会导致不同的扩散行为。通过获得基本解决方案，我们还讨论了表面扩散的影响。我们确定可以将扩散描述为有效的扩散过程的情况，但观察到这种近似通常是无效的。最后，简要考虑了具有轴向对称性的三维流动的情况。我们的发现应为Marangoni扩散提供参考解决方案，可以通过荧光或可拍摄的表面活性剂实验测试。
When surface-active molecules are released at a liquid interface, their spreading dynamics is controlled by Marangoni flows. Though such Marangoni spreading was investigated in different limits, exact solutions remain very few. Here we consider the spreading of an insoluble surfactant along the interface of a deep fluid layer. For two-dimensional Stokes flows, it was recently shown that the non-linear transport problem can be exactly mapped to a complex Burgers equation [Crowdy, SIAM J. Appl. Math. 81, 2526 (2021)]. We first present a very simple derivation of this equation. We then provide fully explicit solutions and find that varying the initial surfactant distribution - pulse, hole, or periodic - results in distinct spreading behaviors. By obtaining the fundamental solution, we also discuss the influence of surface diffusion. We identify situations where spreading can be described as an effective diffusion process but observe that this approximation is not generally valid. Finally, the case of a three-dimensional flow with axial symmetry is briefly considered. Our findings should provide reference solutions for Marangoni spreading, that may be tested experimentally with fluorescent or photoswitchable surfactants.
