This work builds on the foundation laid by Benney & Timson (1980), who
examined the flow near a contact line and showed that, if the contact
angle is 180 degrees, the usual contact-line singularity does not arise.
Their local analysis, however, does not allow one to determine the
velocity of the contact line and their expression for the shape of the
free boundary involves undetermined constants - for which they have been
severely criticised by Ngan & Dussan V. (1984). As a result, the ideas
of Benny & Timson (1980) have been largely forgotten.
The present work shows that the criticism of Ngan & Dussan V. (1984)
was, in fact, unjust. We consider a two-dimensional steady Couette flow
with a free boundary, for which the local analysis of Benney & Timson
(1980) can be complemented by an analysis of the global flow (provided
the slope of the free boundary is small, so the lubrication
approximation can be used). We show that the undetermined constants in
the solution of Benney & Timson (1980) can all be fixed by matching
their local solution to the global one. The latter also determines the
contact line's velocity, which we compute among other characteristics of
the global flow.