To model fiber spinning and film casting, a boundary condition on the stress at a chosen synthetic inlet is necessary. However, the exact value of the stress for viscoelastic liquids at the synthetic inlet is a priori unknown. In this article, we present the application of the ＇＇free boundary condition＇＇ to the inlet stress, which avoids the necessity of specifying an a priori unknown value of stress at the synthetic inlet. To apply the free boundary condition, the process must be cast and studied as a two-point boundary value problem by finite elements. To verify the admissibility and accuracy of the free boundary condition, the same process is cast and studied as an initial value problem, directly solvable by a DGEAR subroutine. The initial value problem is cast in a matrix form that allows analytic investigation of admissible solutions: With the upper convected Maxwell model, the fiber can only slim monotonically with the downstream distance, whereas with the Giesekus model there may be cases of initially increasing and subsequently decreasing diameter, i.e., extrudate-swelling.