Abstract
This paper discusses theoretical and computational issues regarding viscous flows which have a free surface. A number of mathematical models for a particular flow are described and compared, both with one another and with some physical experiments. We consider some approximate models based both on lubrication theory and finite element methods. The importance of the choice of boundary conditions in modeling practical flow phenomena is discussed, and some related open theoretical questions regarding the well-posedness of mathematical models for such phenomena are presented. The discussion also touches upon the role that surface tension has so far played in the mathematical theory of free-surface flows and in many numerical calculations. Briefly outlined is some preliminary work related to convergence estimates for finite-element methods for free-boundary problems.
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