  The surface of a liquid near a moving contact line is highly curved owing to
diverging viscous forces. Thus, microscopic physics must be invoked at the
contact line and matched to the hydrodynamic solution farther away. This
matching has already been done for a variety of models, but always assuming the
limit of vanishing speed. This excludes phenomena of the greatest current
interest, in particular the stability of contact lines. Here we extend
perturbation theory to arbitrary order and compute finite speed corrections to
existing results. We also investigate the impact of the microscopic physics on
the large-scale shape of the interface.
