In an experimental context, the contamination of an air–liquid interface by ambient pollutants can strongly affect the dynamics and the stability of a given flow. In some configurations, the interfacial flow can even be blocked by surface tension effects. A cylindrical free-surface flow driven by a slow rotating disc is considered here as a generic example of such effects and is investigated both experimentally and numerically. We suggest here a simple numerical model, without any superficial transport of the pollutants, adaptable into any code for single-phase flows. For the stationary axisymmetric base flow, the radial velocity at the interface is set to zero whereas the usual stress-free boundary conditions are retained for the perturbations. The model does not feature any free parameter. For a geometrical aspect ratio of 1/4, known to display ambiguous behaviour regarding stability thresholds, the modal selection as well as a nonlinear stability island found in the experiments are well reproduced by the model, both qualitatively and quantitatively. The robustness of the model has also been validated by replacing the radial velocity profile by a more accurate experimental fit, with very little influence on the stability results.