  In many spinning processes, as for example in dry spinning, solvent
evaporates out of the spun jets and leads to thinning and solidification of the
produced fibers. Such production processes are significantly driven by the
interaction of the fibers with the surrounding airflow. Faced with industrial
applications producing up to several hundred fibers simultaneously, the direct
numerical simulation of the three-dimensional multiphase, multiscale problem is
computationally extremely demanding and thus in general not possible. In this
paper, we hence propose a dimensionally reduced, efficiently evaluable fiber
model that enables the realization of fiber-air interactions in a two-way
coupling with airflow computations. For viscous dry spinning of an uni-axial
two-phase flow, we deduce one-dimensional equations for fiber velocity and
stress from cross-sectional averaging and combine them with two-dimensional
advection-diffusion equations for polymer mass fraction and temperature
revealing the radial effects that are observably present in experiments. For
the numerical treatment of the resulting parametric boundary value problem
composed of one-dimensional ordinary differential equations and two-dimensional
partial differential equations we develop an iterative coupling algorithm.
Thereby, the solution of the advection-diffusion equations is implicitly given
in terms of Green's functions and leads for the surface values to Volterra
integral equations of second kind with singular kernel, which we can solve very
efficiently by the product integration method. For the ordinary differential
equations a suitable collocation-continuation procedure is presented. Compared
with the referential solution of a three-dimensional setting, the numerical
results are very convincing. They provide a good approximation while
drastically reducing the computational time.
