THE LEADING-EDGE OF AN OIL-SLICK, SOAP FILM, OR BUBBLE STAGNANT CAP IN STOKES-FLOW

Trace impurities often collect on the upstream side of an obstacle in the surface of flowing liquid. The transition from practically free surface to surface sufficiently clogged to be treated as stationary can be quite sharp. The viscous flow underneath is nonlinearly coupled to the convective mass transfer of surface-active material. For two-dimensional flow at high Reynolds number the first observations were due to Thoreau, Langton and Reynolds over 100 years ago, and the theory was given by Harper & Dixon in 1974. If the whole problem is considered from a frame of reference moving with the stream instead of fixed to the downstream surface film, the solution refers to the leading edge of a slowly spreading oil slick. The present work gives the theory corresponding to Harper & Dixon's foil low Reynolds numbers (Stokes flow), for which there is a very simple leading approximation near the transition for a soluble surfactant, and a more complicated one, which can still be found exactly, for an insoluble surfactant which spreads onto clear liquid by surface diffusion. In both cases the surface remains flat: the ridge often observed is not, a Stokes flow phenomenon. The results are used to clarify the circumstances in which Savic's stagnant-cap approximation is useful for a bubble rising in a viscous liquid: the rear stagnation point now plays the role of the obstacle in the surface, and the flow near the surface transition can be treated locally as if it were two-dimensional instead of axisymmetric.