We report experimental observations on the flow of molten polyethylene and polypropylene into, within and out of two-dimensional ducts. Using flow birefringence techniques we follow as a function of flow rate, temperature and molecular weight the evolution and relaxation of molecular orientation as the polymer flows through and out of the duct. In particular we are interested in the flow along the centre line of the duct where within the duct we are able to determine readily relaxation times for the polymer melt. Our observations show that the `short-time-scale' relaxation behaviour of the melt depends on the magnitude of the entrance velocity gradient but is essentially independent of both temperature and the `molecular weight' averages of the polymer. The `long-time-scale' relaxation behaviour is found to have a different dependence. To explain our relaxation time and die swell results we propose that the molten polymer forms a composite material that in the quiescent state consists of a molecular network embedded in a matrix of polymer chains of lower molecular weight. The network and matrix respond to the imposed velocity history in different ways.