A Lagrangian-Eulerian simulation technique has been applied to model v
iscoelastic extrusion flow. The essential feature of our method is to
track explicitly the motion of fluid elements in a Lagrangian fashion
and to solve fluid fields on a comoving grid in an Eulerian manner at
each physical time step. The method is able to track the details of fl
uid, e.g. deformation and stream line, and also to advance free surfac
es in a physical way. Results are presented for the time-dependent flo
w from rest of both Oldroyd-B and PTT (Phan-Thien and Tanner) fluids,
through a planar 2:1 abrupt contraction and in free jet flows.