Measurement and computation of hydrodynamic coupling at an air/water interface with an insoluble monolayer

Abstract

The coupling between a bulk vortical flow and a surfactant-influenced air/water interface has been examined in a canonical flow geometry through experiments and computations. The flow in an annular region bounded by stationary inner and outer cylinders is driven by the constant rotation of the floor and the free surface is initially covered by a uniformly distributed insoluble monolayer. When driven slowly, this geometry is referred to as the deep-channel surface viscometer and the flow is essentially azimuthal. The only interfacial property that affects the flow in this regime is the surface shear viscosity, μ^s, which is uniform on the surface due to the vanishingly small concentration gradient. However, when operated at higher Reynolds number, secondary flow drives the surfactant film towards the inner cylinder until the Marangoni stress balances the shear stress on the bulk fluid. In general, the flow can be influenced by the surface tension, σ, and the surface dilatational viscosity, κ^s, as well as μ^s. However, because of the small capillary number of the present flow, the effects of surface tension gradients dominate the surface viscosities in the radial stress balance, and the effect of μ^s can only come through the azimuthal stress. Vitamin K₁ was chosen for this study since it forms a well-behaved insoluble monolayer on water and μ^s is essentially zero in the range of concentration on the surface, c, encountered. Thus the effect of Marangoni elasticity on the interfacial stress could be isolated. The flow near the interface was measured in an optical channel using digital particle image velocimetry. Steady axisymmetric flow was observed at the nominal Reynolds number of 8500. A numerical model has been developed using the axisymmetric Navier–Stokes equations to examine the details of the coupling between the bulk and the interface. The nonlinear equation of state, σ(c), for the vitamin K₁ monolayer was measured and utilized in the computations. Agreement was demonstrated between the measurements and computations, but the flow is critically dependent on the nonlinear equation of state.