Traveling waves along viscous filaments

In this article, deflections of a viscous filament in a classical fiber spinning set-up are analyzed. The deflections are considered in a direction perpendicular to the vertical equilibrium state of the spin line. The result is a traveling wave equation with non-uniform coefficients representing the non-uniform filament velocity and non-uniform tension in the spin line. Under neglect of air drag, the system is conservative with respect to an energy functional, so that its eigenmodes have purely imaginary characteristic values. A numerical analysis of the eigenmodes of the system reveals that deflections propagate from take-up wheel to spinneret, with frequencies being multiples of a basic frequency and amplitudes sinus shaped with the maximum being shifted toward the spinneret. From the numerical results, a formula is derived, which approximates the basic frequency and traveling wave velocity directly in terms of the spinning process parameters. Keywords: Deflection – Eigenmode – Fiber spinning – Frequency – Traveling wave – Viscous filament – Viscous string model