Fluid Dynamic Instabilities in Drawn Fibers
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FLUID DYNAMIC. INSTABILITIES IN DRAWN FIBERS CHARLES THOMPSON AND ARUN MULPUIR Laboratory for Advanced Computation. Department University of Lowell, Lowell, MA 01854
of Electrical
Engineering
ABSTRACT The nonlinear stability analysis of viscoelastic fibers is presented. The molten fiber is modeled as a Maxwellian viscoelastic fluid and the zeroth order equations governing its behavior given. Linear stability analysis is performed to determine the influence of winder speed and impedance as well as viscosity and elasticity. The results of numerical solution of the nonlinear equations are given.
I. INTRODUC.TION The manufacture of optical, synthetic or textile fibers involves the extrusion of a vety viscous liquid, usually glass or molten polymer, through an orifice. The resulting liquid jet is elongated by pulling it with a winder assembly at a point downstream of the orifice. For sufficiently large winder speeds, the steady fiber flow is affected by a hydrodynamic instability called "draw resonance" [1]. This instability causes periodic variations in the fiber diameter which reduce performanire a.nd (lart disrnpt the spinning process by causing fiber breakage [2]. This phenoinenon often limits the, process speeds in counnercial applications.
Experimental work [3,4] with polymeric materials shows that the most significant variable controlling the onset of draw resonance is the draw ratio, o which is the ratio of the initial to terminal axial velocity of the fiber. Linearized stability analyses [5,6] have confirmed this dependence. The study of the behavior of a Newtonian fluid under isothermal and constant force conditions [7] has been valuable in understanding the basic underlying mechanism. The influence of the radial and axial stresses resulting from the elastic properties of such fiber.s has not been considered in the dynamic model. In this paper we will address the issue of linear and nonlinear stability of a viscoelastic fiber. It will be shown that the elastic forces serve to stabilize disturbances having infinitesimal amplitude. In addition, these elastic forces serve to limit the growth of disturbances having finite amplitude. This results in a supercritical bifurcation from the linear stability result.
II. PROBLEM STATEMENT Consider a thin viscoelastic liquid fiber drawn from an orifice of radius R 0 at an average velocity V as shown in Figure 1. At a distance L., the fiber is wound up at a velocity V . The coordinate position along the fiber is denoted by R and X. The particle wvelocity o:f the fluid is expres•sed in terms of the radial Mat. Res. Soc. Symp. Proc. Vol. 172. 01990 Materials Research Society
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component, V', and axial compo)nent. V', of the particle velocity. At the winder. the time variations in the axial stress on the fiber result from the reaction terminal impedance imposed by the mechanical assembly. If we nondimensionalize the state-variables with respect to the characteristic velocity U . time, L / U and 0 0' 0 stress, p (T2, the dynamic motion within the liquid filament is de
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