Elastic instabilities of entangled polymer melts are common in industrial processes but the physics responsible is not well understood. We present a numerical linear stability study of a molecular based constitutive model which grants us physical insight into the underlying mechanics involved. Two constriction flows are considered - one shear dominated, the other extension dominated - and two distinct instabilities are found. The influence of the molecular structure and the behaviour of the polymer dynamics are investigated and in both cases chain relaxation and orientation play a crucial role. This suggests a molecular-based physical interpretation of the underlying mechanisms responsible for flow instabilities. © 2013 Elsevier B.V.