A homotopy continuation approach for analysing finite-time singularities in thin liquid films
A homotopy continuation approach for analysing finite-time singularities in thin liquid films
复制标题
DOI：
10.1093/imamat/hxt021
复制
发表时间：
2013-08
影响因子：
1.2
通讯作者：
D. Tseluiko;J. Baxter;U. Thiele
中科院分区：
文献类型：
--
作者：
D. Tseluiko;J. Baxter;U. Thiele
我们认为自相似的解决方案有关的固体基板上的薄液体膜的破裂，发展下的表面张力和有效的货车德瓦耳斯之间的自由表面的膜和基板的相互作用的不稳定的影响下的稳定的影响。这种解决方案已经在文献中进行了分析，并提出了各种数值方法来获得这样的解决方案。这种方法是基于射击或有限差分格式，并需要精心选择的初始猜测的解决方案。我们提出了一种替代的数值方法，这是基于同伦方法和连续技术，并允许一个达到自相似的解决方案，从分析已知的小振幅稳定的解决方案的相关薄膜方程。我们认为，这种方法是更强大的比以前提出的方法，并不需要初始猜测，以获得解决方案。虽然本研究的重点是特殊情况下的自相似的解决方案有关的平面破裂，具有平方根远场行为，我们的方法也可以用来获得平面的解决方案，具有不同的远场行为和径向对称的自相似的解决方案所考虑的薄膜方程。我们希望这种方法也适用于其他类似类型的方程，显示出一个亚临界的主要分歧和有限时间奇点。
We consider self-similar solutions related to rupture of thin liquid films on a solid substrate that evolve solely under the stabilizing influence of surface tension and the destabilizing influence of effective van der Waals interactions between the free surface of the film and the substrate. Such solutions have been previously analysed in the literature and various numerical approaches to obtain such solutions have been proposed. Such approaches are based either on shooting or finite-difference schemes and require well-chosen initial guesses for solutions. We propose an alternative numerical method, which is based on a homotopy approach and continuation techniques and allows one to reach self-similar solutions from analytically known small-amplitude steady solutions of the related thin-film equation. We argue that this method is more robust than previously proposed methods and does not require initial guesses to obtain solutions. Although the present study focuses on the particular case of self-similar solutions related to planar rupture that have square-root far-field behaviour, our approach can also be used to obtain planar solutions having a different far-field behaviour and radially symmetric self-similar solutions for the considered thin-film equation. We expect the approach to be also valid for other equations of similar type that show a subcritical primary bifurcation and finite-time singularities.