Cusped ripples at the plane surface of a viscous liquid

We study the time-evolution of periodical ripples of a viscous liquid at th e plane free surface under the action of a distant pure straining how. We n eglect inertial forces (Stokes how) and include surface tension effects. Th e solutions for a contracting surface and constant strain rate show that th e ripples may develop near-cusps during a stage of the evolution, though la ter the free surface inevitably asymptotically tends to a smooth plane with vanishing ripples due to the action of capillarity. We obtain the conditio n for cusp formation in this intermediate stage in terms of the initial cap illary number and aspect ratio. If the capillary number is kept constant, t he surface tends to shrink through a succession of self-similar trochoidal shapes, whose aspect ratio is given by the capillary number.