Calculation of a key function in the asymptotic description of moving contact lines

IF 0.8 4区 工程技术 Q3 MATHEMATICS, APPLIED
J. Scott
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引用次数: 2

Abstract

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.
移动接触线渐近描述中一个关键函数的计算
具有与固体边界相交的移动液/气界面的流动的渐近描述的一个重要元素是由hockking和Rivers表示为$Q_i \left( \alpha \right)$的函数(由毛细作用引起的液滴扩散,J.流体力学。121(1982) 425-442),其中$0 < \alpha < \pi$为接触面与壁面的接触角。$Q_i \left( \alpha \right)$是由这些作者介绍的内渐近区域和中间渐近区域的匹配产生的,是渐近理论应用中所必需的。本文描述了一种新的计算$Q_i \left( \alpha \right)$的数值方法,由于它明确地允许控制积分方程核的对数奇异性,并在被积函数中使用非奇异因子的二次插值,因此比霍金和里弗斯采用的方法更精确。尽管如此,我们的结果与他们的结果非常一致,然而,在$\alpha = \pi $附近有明显的偏差。我们还讨论了极限情况$\alpha \to 0$和$\alpha \to \pi $。$Q_i \left( \alpha \right)$在两个极限处的首阶项与霍金(A)运动流体界面的分析一致。第2部分。滑移流对力奇点的去除[j] .流体力学。79(1977) 209-229)。下一阶项也被考虑在内。霍金没有超越$\alpha \to 0$的领先顺序,我们认为他的下一个顺序$\alpha \to \pi $的结果是不正确的。数值上,我们发现$\alpha \to 0$的下一阶项为$O\left( {\alpha ^2} \right)$, $O\left( 1 \right)$为$\alpha \to \pi $。后一种结果与霍金的结论一致,但$O\left( 1 \right)$常数的值却不一致。希望给出关于$Q_i \left( \alpha \right)$的数值方法的细节和更精确的信息,包括数值和其极限行为,将有助于那些想要在未来的理论和数值工作中使用接触线动力学的渐近理论的人。
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来源期刊
CiteScore
1.90
自引率
11.10%
发文量
14
审稿时长
>12 weeks
期刊介绍: The Quarterly Journal of Mechanics and Applied Mathematics publishes original research articles on the application of mathematics to the field of mechanics interpreted in its widest sense. In addition to traditional areas, such as fluid and solid mechanics, the editors welcome submissions relating to any modern and emerging areas of applied mathematics.
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