Calculation of a key function in the asymptotic description of moving contact lines. Issue 4 (10th October 2020) - Scott, Julian F
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 ( α ) by Hocking and Rivers (The spreading of a drop by capillary action, J. Fluid Mech. 121 (1982) 425–442), where 0 < α < π is the contact angle of the interface with the wall. Q_i ( α ) 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 ( α ), 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 α = π . We also discuss the limiting cases α → 0 and α → π . The leading-order terms of Q_i ( α ) 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 α → 0, and we believe his results for the next order as α → π to be incorrect. Numerically, we find that the next-order terms are O( α ^2 ) for α → 0 and O( 1 ) as α → π . 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 ( α ) will help those wanting to use the asymptotic theory of contact-line dynamics in future theoretical and numerical work.