Non-adiabatic elongational flows of viscoelastic melts

Elongational flows of viscoelastic melts are frequently encountered in manu facturing processes in the textile industry. The most common stretching flo w is melt-spinning. In this process, a polymeric melt is withdrawn from a r eservoir, axially stretched, and simultaneously cooled down until the melt freezes. This paper addresses the fundamental question of existence of solutions for the system of quasilinear hyperbolic equations governing the melt-spinning process of a Giesekus fluid and a Phan-Thien-Tanner fluid. The problem is originally posed as a free boundary problem. It will be shown that the free boundary problem can be reformulated as an equivalent boundary-initial val ue problem for which we prove the (local in time) existence of classical so lutions.