A quantitative model to describe capillary rheometer experiments is presented. The model can generate ‘two-branched' discontinuous flow curves and the associated pressure oscillations. Polymer compressibility in the barrel, incompressible axisymmetric generalised Newtonian flow in the die, and a switch curve that defines a stick-slip boundary condition are key elements in the model. Well defined experimental results of high density polyethylene are used to validate the model. The calculations compare well with the experiments. The only fit parameters are those related to the slip law. The present model can be made fully predictive by relating the slip law to fundamental polymer-wall interaction properties. The model offers a mathematical framework to validate potential slip laws.