The capillary Bow of molten polymers has received much attention becau
se at high Bow rates the extrusion of the polymer melt is commonly acc
ompanied by instabilities. These manifest themselves as surface distor
tions, called melt fracture, in the final plastic product. We have mod
eled this Row, incorporating a stick-slip boundary condition at the ca
pillary wall. The boundary condition is enforced by a model for the lo
cal state of the polymer, which undergoes a phase transition controlle
d by the local shear. We numerically solve the model. coupled to the h
ydrodynamic Row and to a Maxwell model for viscoelasticity. In various
regimes, the model exhibits steady flow, periodic oscillations, and m
ore complicated spatiotemporal structures, which explain the sharkskin
texturing observed in melt fracture.