Cusped ripples at the plane surface of a viscous liquid

Summary We study the time-evolution of periodical ripples of a viscous liquid at the plane free surface under the action of a distant pure straining flow. We neglect inertial forces (Stokes flow) and include surface tension effects. The solutions for a contracting surface and constant strain rate show that the ripples may develop near-cusps during a stage of the evolution, though later the free surface inevitably asymptotically tends to a smooth plane with vanishing ripples due to the action of capillarity. We obtain the condition for cusp formation in this intermediate stage in terms of the initial capillary number and aspect ratio. If the capillary number is kept constant, the surface tends to shrink through a succession of self-similar trochoidal shapes, whose aspect ratio is given by the capillary number. Received 23 March 1998, accepted for publication 23 July 1998