Summary

An important element of the asymptotic description of flows having a moving liquid/gas interface which intersects a solid boundary is a function denoted $Q_i \left( \alpha \right)$ by Hocking and Rivers (The spreading of a drop by capillary action, J. Fluid Mech. 121 (1982) 425–442), where $0 < \alpha < \pi$ is the contact angle of the interface with the wall. $Q_i \left( \alpha \right)$ arises from matching of the inner and intermediate asymptotic regions introduced by those authors and is required in applications of the asymptotic theory. This article describes a new numerical method for the calculation of $Q_i \left( \alpha \right)$, which, because it explicitly allows for the logarithmic singularity in the kernel of the governing integral equation and uses quadratic interpolation of the non-singular factor in the integrand, is more accurate than that employed by Hocking and Rivers. Nonetheless, our results show good agreement with theirs, with, however, noticeable departures near $\alpha = \pi $. We also discuss the limiting cases $\alpha \to 0$ and $\alpha \to \pi $. The leading-order terms of $Q_i \left( \alpha \right)$ in both limits are in accord with the analysis of Hocking (A moving fluid interface. Part 2. The removal of the force singularity by a slip flow, J. Fluid Mech. 79 (1977) 209–229). The next-order terms are also considered. Hocking did not go beyond leading order for $\alpha \to 0$, and we believe his results for the next order as $\alpha \to \pi $ to be incorrect. Numerically, we find that the next-order terms are $O\left( {\alpha ^2} \right)$ for $\alpha \to 0$ and $O\left( 1 \right)$ as $\alpha \to \pi $. The latter result agrees with Hocking, but the value of the $O\left( 1 \right)$ constant does not. It is hoped that giving details of the numerical method and more precise information, both numerical and in terms of its limiting behaviour, concerning $Q_i \left( \alpha \right)$ will help those wanting to use the asymptotic theory of contact-line dynamics in future theoretical and numerical work.

中文翻译：

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移动接触线渐近描述中关键函数的计算

概要

Hocking and Rivers所表示的函数$ Q_i \ left（\ alpha \ right）$（通过毛细作用分散液滴，J.流体机械 121（1982）425–442），其中$ 0 <\ alpha <\ pi $是界面与墙的接触角。$ Q_i \ left（\ alpha \ right）$来自那些作者介绍的内部和中间渐近区域的匹配，在应用渐近理论时是必需的。本文介绍了一种用于计算$ Q_i \ left（\ alpha \ right）$的新数值方法，因为它明确允许控制积分方程的核中为对数奇异性，并使用非奇异的二次插值比Hocking and Rivers所采用的方法更精确。尽管如此，我们的结果显示出与他们的一致，但是，在$ \ alpha = \ pi $附近有明显的偏离。我们还将讨论极限情况$ \ alpha \到0 $和$ \ alpha \ to \ pi $。J.流体机械。 79（1977）209–229）。还考虑了下一项。霍金没有超越$ \ alpha \至0 $的领先顺序，我们认为他针对下一订单$ \ alpha \至\ pi $的结果是不正确的。从数字上看，我们发现下一阶项是$ O \ left（{\ alpha ^ 2} \ right）$对于$ \ alpha \ to 0 $和$ O \ left（1 \ right）$作为$ \ alpha \到\ pi $。后一个结果与Hocking一致，但是$ O \ left（1 \ right）$常量的值不同。希望提供有关数值方法的详细信息以及关于数值和限制行为的更精确的信息（涉及数值Q_i \ left（\ alpha \ right）$）将有助于那些希望使用接触线渐近理论的人未来理论和数值工作的动力学。