Analysis of fiber spinning for the upper-convected Maxwell fluid

The fiber spinning process of a viscoelastic liquid modeled by the constitutive theory of the Maxwell fluid is analyzed. The governing equations are given by one-dimensional mass, momentum and constitutive equations, which arise in the slender body approximation by cross-sectional averaging of the two-dimensional axisymmetric Stokes equations with free boundary. Existence, uniqueness and regularity results are proved by means of fixed point arguments, energy estimates and weak/weak∗∗ convergence methods.The difficulty in this problem lies with the constitutive model of the Maxwell fluid: when both the outflow velocity at the spinneret and the pulling velocity at take up are prescribed, a boundary condition can be imposed for only one of the two elastic stress components at the inlet. The absence of the second stress boundary condition makes the mathematical analysis of the problem hard.