Stability of Stationary Velocity Profiles in Fiber Spinning

Abstract

We consider a process of fiber spinning, where a viscous (but not necessarily Newtonian) fluid is being pushed through a narrow "spinning hole" and, upon exit, is being stretched, the latter step in order to obtain an acceptable alignment of the (polymeric) molecules in the fluid. This alignment is necessary for the final mechanical properties of the fiber. After leaving the spinning hole, the fluid passes through a layer of air (the "air gap") and enters a bath, in which it solidifies almost instantaneously. Somewhere inside this bath, the fiber is being drawn by a wheel, which delivers the force, necessary for the stretching process.

When the speed of the drawing wheel is set to high, it is impossible to obtain a uniform fiber: one clearly observes variations in the fiber diameter. This phenomenon is called draw resonance. Experimental evidence suggests that the draw ratio, that is, the ratio between the speeds at the wheel and at the exit of the spinning hole, is the unique parameter to steer the onset of draw resonance, and for Newtonian fluids this is known to be true. In the Newtonian case, the onset of draw resonance can be shown to be a Hopf bifurcation.

The question, asked to the Study Group, was to extend the results on Newtonian fluids to fluids with more general rheologies, with power law fluids as a first choice. What came out of the Study Group was not yet a final solution to this problem, but rather an attempt to come to an easier formulation of the model equations. At the workshop, we thought that we had succeeded, but afterwards we found that there was a hidden mistake, which we could not easily correct. This mistake in itself is worth mentioning, because the approach we tried may be successful in other cases, and this mistake may easily slip into the considerations there. In this note, we restrict ourselves to the Newtonian case, because it is easy to describe the mistake in this setting, and the power law setting would not add anything.

As a final remark, we note that the "power law case" has been solved now, by using similar methods as we did in the Newtonian case. (A publication is in preparation.)