Direct Calculation of Limit Cycles of Draw Resonance and Their Stability in Spinning Process

Yun, Jin-Mun; Shin, Dong‐Myeong; Lee, Joo Sung; Jung, Hyun Wook; Hyun, Jae Chun

doi:10.1678/rheology.36.133

Cited by 11 publications

(8 citation statements)

References 20 publications

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“…Numerical simulation of fibre spinning of Newtonian fluid has been performed by Cao (1993), who established a clear mechanism of propagation of a cycle of draw resonance (Cao 1991) by capturing spinline profiles at various time instants. For viscoelastic fluids, Yun et al (2008) performed dynamic transient simulations by using the PTT model to understand the supercritical behaviour by resolving the bifurcation diagram. By full numerical simulations, the authors constructed the limit cycles in the linearly unstable region.…”

Section: Introductionmentioning

confidence: 99%

Weakly nonlinear stability analysis of polymer fibre spinning

Gupta

Chokshi

2015

J. Fluid Mech.

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The extensional flow of a polymeric fluid during the fibre spinning process is studied for finite-amplitude stability behaviour. The spinning flow is assumed to be inertialess and isothermal. The nonlinear extensional rheology of the polymer is described with the help of the eXtended Pom-Pom (XXP) model, which is known to exhibit a significant strain hardening effect necessary for fibre spinning applications. The linear stability analysis predicts an instability known as draw resonance when the draw ratio, DR, defined as the ratio of the velocities at the two ends of the fibre in the air gap, exceeds a certain critical value, DR c . The critical draw ratio DR c depends on the fluid elasticity represented by the Deborah number, De = λv 0 /L, the ratio of the polymer relaxation time to the flow time scale, thus constructing a stability diagram in the DR c -De plane. Here, λ is the characteristic relaxation time of the polymer, v 0 is the extrudate velocity through the die exit and L is the length of the air gap for the spinning flow. In the present study, we carry out a weakly nonlinear stability analysis to examine the dynamics of the disturbance amplitude in the vicinity of the transition point. The analysis reveals the nature of the bifurcation at the transition point and constructs a finite-amplitude manifold providing insight into the draw resonance phenomena. The effect of the fluid elasticity on the nature of the bifurcation and the finite-amplitude branch is examined, and the findings are correlated to the extensional rheological behaviour of the polymer fluid. For flows at small Deborah number, the Landau constant, which captures the role of nonlinearities, is found to be negative, indicating supercritical Hopf bifurcation at the transition point. In the linearly unstable region, the equilibrium amplitude of the disturbance is estimated and shows a limit cycle behaviour. As the fluid elasticity is increased, initially the equilibrium amplitude is found to decrease below its Newtonian value, reaching the lowest value for De when the strain hardening effect is maximum. With further increase in elasticity, the material undergoes strain softening behaviour which leads to an increase in the equilibrium amplitude of the oscillations in the fibre cross-section area, indicating a destabilizing effect of elasticity in this regime. Interestingly, at a certain high Deborah number, the bifurcation crosses over from supercritical to subcritical nature. In the subcritical regime, a threshold amplitude branch is constructed from the amplitude equation.

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Section: Introductionmentioning

confidence: 99%

Weakly nonlinear stability analysis of polymer fibre spinning

Gupta

Chokshi

2015

J. Fluid Mech.

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“…There are two solution branches beyond each critical onset. One is unstable steady state (dotted line) and the other is stable limit cycle, implying that periodic draw resonance instability is a stable supercritical Hopf bifurcation (Yun et al, 2008). The existence of inertia force can stabilize the film casting systems such as the low-and high-speed fiber spinning processes (Shin et al, 2005;Shin et al, 2006).…”

Section: Resultsmentioning

confidence: 99%

“…It is worth mentioning again that the bifurcation type could be determined by Floquet multipliers obtained from the Monodromy matrix during the direct calculation of one-periodic limit cycle solution. Details on this numerical method are referred to Yun et al (2008).…”

Section: Resultsmentioning

confidence: 99%

“…In an analysis by the method of infinitesimal disturbances -linear stability analysis - Schultz et al (1984) clarified that as the drawdown ratio rises just beyond the Hopf bifurcation point, the frequency of oscillation falls, and thus, the period of the nascent limit cycle grows . Yun et al (2008) recently substantiated the Hopf bifurcation mode of draw resonance by directly solving one-periodic limit cycle solution in viscoelastic fiber spinning case.…”

Section: Introductionmentioning

confidence: 99%

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Nonlinear dynamics and chaotic motion in the isothermal film casting process

Jung

Lee²

2011

Korea-Aust. Rheol. J.

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In the industrial film casting process with the two-stage cascade (velocity and tension controlled) loop, various aspects of nonlinear dynamics and chaotic motion have been investigated solving simple 1-D viscoelastic model. Most of the extensional deformation processes exhibit sustained-periodic oscillations or stable limit cycles of state variables (called draw resonance instability) beyond the critical onset, even if they are conventionally controlled systems. Besides such well-known instability, some intriguing nonlinear dynamic phenomena such as period-doubling and eventual chaotic motion of film thickness and width could be observed when a disturbance is imposed in the second tension-controlled system. In other words, the bifurcation mode is dramatically changed by a sinusoidal disturbance under the combined velocity-tension control condition in contrast to only constant take-up velocity condition inducing a supercritical Hopf bifurcation of draw resonance.

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“…Stability studies have mainly focused on the draw resonance phenomenon which is distinguished by the self-sustained periodic oscillation of spinline variables like spinline cross-sectional area and spinline tension over the critical onsets. From the linear stability analysis and direct transient simulation, stability windows for various fluids have been established in spinning systems [13][14][15][16][17][18][19][20]. Recently, Yun et al [20] revisited the limit cycles of draw resonance using Hopf bifurcation theory, reporting that draw resonance is a supercritical Hopf bifurcation by directly calculating limit cycles and eigenvalues of monodromy matrix over one period of oscillation.…”

Section: Introductionmentioning

confidence: 99%

Sensitivity of spinning process with flow-induced crystallization kinetics using frequency response method

Yun¹,

Shin

Lee³

et al. 2010

Korean J. Chem. Eng.

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The sensitivity of the low-and high-speed spinning processes incorporated with flow-induced crystallization has been investigated using frequency response method, based on process conditions employed in Lee et al. [1] and Shin et al. [2,3]. Crystallinity occurring in the spinline makes the spinning system less sensitive to any disturbances when it has not reached its maximum onto the spinline in comparison with the spinning case without crystallization. Whereas, the maximum crystallinity increases the system sensitivity to disturbances, interestingly exhibiting high amplitude value of the spinline area at the take-up in low frequency regime. It also turns out that neck-like deformation in the spinline under the high-speed spinning conditions plays a key role in determining the sensitivity of the spinning system.

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Direct Calculation of Limit Cycles of Draw Resonance and Their Stability in Spinning Process

Cited by 11 publications

References 20 publications

Weakly nonlinear stability analysis of polymer fibre spinning

Weakly nonlinear stability analysis of polymer fibre spinning

Nonlinear dynamics and chaotic motion in the isothermal film casting process

Sensitivity of spinning process with flow-induced crystallization kinetics using frequency response method

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