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.
