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Numerical simulation of Stokes flow down an inclined plane

A boundary-integral equation method is applied to the problem of evolving two-dimensional flow of a viscous liquid with a free surface down an inclined plane wall. The flow is assumed to have a sufficiently low Reynolds number that Stokes flow is a good approximation; the stream function then satisfies the biharmonic equation. Numerical solutions are found by using a boundary integral equation applied to both harmonic functions that appear in the Almansi biharmonic representation. Different approaches to the linear algebraic scheme and different time-stepping routines are discussed. The motion of the free surface with, or without, surface tension is then modelled by a simple time-stepping routine. The numerical scheme is shown to give accurate representations of the free surface up to and beyond the point of overturning. One example is illustrated in which the overturning evolves until the free surface is close to forming a cusp. Solutions are also illustrated by two spatially periodic wave profiles: one is close to a time periodic flow and the other becomes a steep nearly steady wave.