Simple indicator of draw resonance instability in melt spinning processes

Abstract

Draw resonance, one of major instabilities in polymer extensional processes such as fiber spinning and film casting, sets in as the drawdown ratio — the ratio of take-up velocity to extrusion velocity — is raised past its critical value. It appears as sustained periodic variations in velocity, mass-flux, crosssection, and spinline tension. There have been many experimental observations and theoretical analyses of this phenomenon during the four decades.1-12 However, there are still unsettled issues about draw resonance, many of them expounded by Petrie.13 In fiber spinning of Newtonian or viscoelastic liquids, the onset of draw resonance is an oscillatory instability or overstability, as termed by Eddington,14 or a stable supercritical Hopf bifurcation, as used in the singularity theory. In an analysis by means of the method of infinitesimal disturbances — linear stability analysis — Schultz et al.6 showed that as the drawdown ratio is raised just beyond the Hopf bifurcation, the frequency of oscillation falls, and, thus, the period of the nascent limit cycle grows. To go beyond linear stability theory and find the nonlinear transient behavior at higher drawdown ratios, some researchers6, 15, 16 solved nonlinear initial value problems with selected initial conditions and found that the solutions at supercritical drawdown ratios approach stable periodic states, or limit cycles. To clarify the physics of the occurrence of the bifurcation and persistence of the supercritical oscillating behavior, Hyun and coworkers5,15-17 focused on kinematic waves that can be envisioned traveling along the spinline. From solutions of the nonlinear transient equations that govern oscillations of a spinline, they identified a combination of three kinds of kinematic waves (waves of spinline throughput, maximum spinline crosssection, and minimum cross-section), and period of oscillation of the spinline, passing through zero at the critical drawdown ratio and, therefore, being an index to the instability. This property of the combination follows from the dominant eigenfunctions, or normal modes, of infinitesimal disturbances at each value of the drawdown ratio, as is established below. Thus, the transition of a spinline from steady to oscillatory behavior as drawdown ratio is raised is a case of well-understood transformation of a stable state to an unstable one and concomitant appearance of a stable limit cycle as a parameter is varied — a stable supercritical Hopf bifurcation.