Title page for ETD etd-110298-102457


Type of Document Dissertation
Author Hagen, Thomas Ch.
URN etd-110298-102457
Title Elongational Flows in Polymer Processing
Degree PhD
Department Mathematics
Advisory Committee
Advisor Name Title
Renardy, Michael J. Committee Chair
Baird, Donald G. Committee Member
Herdman, Terry L. Committee Member
Renardy, Yuriko Y. Committee Member
Rogers, Robert C. Committee Member
Keywords
  • Fiber Spinning
  • Linear Stability
  • Quasilinear Hyperbolic Equations
  • Spectral Determinacy
Date of Defense 1998-12-01
Availability unrestricted
Abstract

The production of long, thin polymeric fibers is a main objective of the textile industry. Melt-spinning is a particularly simple and effective 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 solidification.

We will address the following topics: first 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 fixed-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 justification 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 strongly continuous 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 confirming prior results on the stability of the steady-state solution.

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