A rigorous mathematical treatment of the shape of a dissipative rope fountain

This is a theoretical and numerical study of a model of a rope fountain subject to a drag force that depends linearly on the rope velocity. A precise, analytical description of the long-term shape adopted by the rope is given, and various consequences are derived from it. Using parameters that naturally appear in the model, we distinguish between cases wherein energy is conserved by means of a constant tension far from the rope source (the ‘free’ case) and where energy conservation is a consequence of a non-constant tension (the ‘braked’ case). The model is used, among other things, to generate rope fountain shapes based on approximate experimental estimates of the parameter values and a careful numerical treatment.