The spreading of insoluble surfactant at the free surface of a deep fluid layer

Abstract

The unsteady spreading of an insoluble monolayer containing a fixed mass of surface-active material over the initially horizontal free surface of a viscous fluid layer is investigated. A flow driving the spreading is induced by gradients in surface tension, which arise from the nonuniform surfactant distribution. Distinct phases in the flow's dynamics are distinguished by a time T = H02/v, where H0 is the fluid depth and v its viscosity. For times t [Lt ] T, i.e. before the lower boundary has any significant influence on the flow, a laminar sub-surface boundary-layer flow is generated. The effects of gravity, capillarity, surface diffusion or surface contamination may be weak enough for the flow to drive a substantial unsteady displacement of the free surface, upward behind the monolayer's leading edge and downward towards its centre. Similarity solutions are identified describing the spreading of a localized planar monolayer strip (which spreads like t1/2) or an axisymmetric drop (which spreads like t3/8); using the Prandtl transformation, the associated boundary-layer problems are solved numerically. Quasi-steady sub-layers are shown to exist at the centre and at the leading edge of the monolayer; that due to surface contamination, for example, may eventually grow to dominate the flow, in which case spreading proceeds like t3/4. Once t = O(T), vorticity created at the free surface has diffused down to the lower boundary and the flow changes character, slowing appreciably. The dynamics of this stage are modelled by reducing the problem to a single nonlinear diffusion equation. For a spreading monolayer strip or drop, the transition from an inertia-dominated (boundary-layer) flow to a viscosity-dominated (thin-film) flow is predicted to be largely complete once t ≈ 85 T.