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

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.