# Laboratoire de Mécanique des Fluides et d’Acoustique - UMR 5509

LMFA - UMR 5509
Laboratoire de Mécanique des Fluides et d’Acoustique
Lyon
France

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Article dans Q. J. Mech. Appl. Math. (2020)

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

Julian Scott

An important element of the asymptotic description of flows having a moving liquid/gas interface which intersects a solid boundary is a function denoted $Q_i(\alpha)$ 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. $Q_i(\alpha)$ 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 $Q_i(\alpha)$, 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 $\alpha\rightarrow 0$ and $\alpha\rightarrow\pi$. The leading-order terms of $Q_i(\alpha)$ 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 $\alpha\rightarrow 0$⁠, and we believe his results for the next order as $\alpha\rightarrow\pi$ to be incorrect. Numerically, we find that the next-order terms are $O(\alpha^2)$ for $\alpha\rightarrow 0$ and $O(1)$ as $\alpha\rightarrow\pi$⁠. The latter result agrees with Hocking, but the value of the $O(1)$ 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 $Q_i(\alpha)$ will help those wanting to use the asymptotic theory of contact-line dynamics in future theoretical and numerical work.

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