Results of an experimental and computational study of the flow in an annular region bounded by stationary inner and outer cylinders and driven by the rotation of the floor are presented. The top is a flat air/water interface, covered by an insoluble monolayer. We develop a technique to determine the surface shear viscosity from azimuthal velocity measurements at the interface which extends the range of surface shear viscosity that can be measured using a deep-channel viscometer in the usual Stokes flow regime by exploiting flow inertia. A Navier–Stokes-based model of bulk flow coupled to a Newtonian interface that has surface shear viscosity as the only interfacial property is developed. This is achieved by restricting the flow to regimes where the surface radial velocity vanishes. The use of inertia results in an improved signal-to-noise ratio of the azimuthal velocity measurements by an order of magnitude beyond that available in the Stokes flow limit. Measurements on vitamin K1 and stearic acid monolayers were performed, and their surface shear viscosities over a range of concentrations are determined and found to be in agreement with data in the literature.