This work builds on the foundation laid by Benney & Timson (1980, Stud. Appl. Maths 63, 93), who examined the flow near a contact line and showed that, if the contact angle is 180°, the usual contact-line singularity does not arise. Their local solution, however, involves an undetermined constant, which was claimed by Ngan & Dussan V. (1984 Phys. Fluids 24, 2785) to indicate that "there is something inherently wrong" with Benney & Timson's work, and it was largely forgotten. I consider two-dimensional Couette flows with a free boundary, for which the local analysis of Benney & Timson can be complemented by an analysis of the global flow. It is shown that the undetermined constant in the solution of Benney & Timson can be fixed by matching the local and global solutions. The latter also determines the contact line's velocity. It is shown that, in some cases, the flow involves brief intermittent periods of rapid acceleration of contact lines.