Interfacial velocity measurements were performed in an optical annular channel, consisting of stationary inner and outer cylinders, a floor rotating at a constant rate, and a flat free surface on which an insoluble monolayer was initially spread. Measurements for essentially inviscid monolayers and some viscous monolayers on water show good agreement with numerical predictions for a Newtonian interface (Boussinesq-Scriven surface model) coupled to a bulk flow described by the Navier-Stokes equations. Here, we consider in detail a viscous monolayer, namely hemicyanine, and find that above a certain concentration, the monolayer does not behave Newtonian at a Reynolds number of about 250. We show that the discrepancies between the measurements and predicted Newtonian behavior are not due to compositional effects (i.e., nonuniform monolayer distribution), Reynolds number (i.e., inertia and/or secondary flows), or surface dilatational viscosity (which does not play any role in the parameter regime investigated). We show prima facie evidence that the observed shear thinning nature of the velocity profile is associated with a phase transition at C approximate to 0.9 mg/m(2) at low Reynolds numbers. At large Reynolds numbers (Re = 8500), hemicyanine is found to flow like a viscous Newtonian monolayer on the air/water interface, with viscosity dependent only on the local concentration.