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

International audienceAn 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 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