Elongational Flows in Polymer Processing by

The production of long, thin polymeric bers is a main objective of the textile industry. Melt-spinning is a particularly simple and eective technique. In this work, we shall discuss the equations of melt-spinning in viscous and viscoelastic flow. These quasilinear hyperbolic equations model the uniaxial extension of a fluid thread before its solidication. We will address the following topics: rst we shall prove existence, uniqueness, and regularity of solutions. Our solution strategy will be developed in detail for the viscous case. For non-Newtonian and isothermal flows, we shall outline the general ideas. Our solution technique consists of energy estimates and xed-point arguments in appropriate Banach spaces. The existence result for a simple transport equation is the key to understanding the quasilinear case. The second issue of this exposition will be the stability of the unforced frost line formation. We will give a rigorous justication that, in the viscous regime, the linearized equations obey the \Principle of Linear Stability". As a consequence, we are allowed to relate the stability of the associated C0 semigroup to the numerical resolution of the spectrum of its generator. By using a spectral collocation method, we shall derive numerical results on the eigenvalue distribution, thereby conrming prior results on the stability of the steady-stat