Weakly nonlinear stability analysis of polymer fibre spinning

Abstract

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.

Non-Newtonian effects on draw resonance in film casting

“…Moreover, an unattainable region for high Deborah numbers and draw ratios is reported, where no steady state solutions exist. Stability studies utilizing the Phan-Thien/Tanner (PTT) [11,12] and the eXtended Pom-Pom model [13,14] constitutive equations reveal in contrast to that a mostly destabilizing effect of increasing elastic properties of the material, which is in qualitative agreement with experimental findings [15,16]. Iyengar and Co [17,18] performed a steady state and a linear stability analysis of onedimensional, infinite width film casting of a modified Giesekus fluid.…”

“…This is similar to the transition from draw resonance to the Rayleigh-Plateau instability when surface tension in fiber spinning is increased [29]. In a weakly nonlinear stability analysis, Gupta and Chokshi [14] found a transition from a supercritical to a subcritical instability if the Deborah number exceeds a threshold value, which is another hint for this hypothesis. However, it has to be noted that this analysis is at least partially wrong, as pointed out by Gallaire et al [34].…”

“…Equations (9a) and (10a) truncated at O(β 2 ) then lead to σ * (0) xy = τ * (0) xy = 0, and therefore Eq. (11c) yields De τ (0) yy + 1 ∂ y u (0) = 0 (14) at leading order. Assuming ∂ y u (0) being non-zero would directly lead to…”

Pushing amplitude equations far from threshold: application to the supercritical Hopf bifurcation in the cylinder wake

“…Interest has recently regrown in the a priori approximation of limit cycles: methods include POD (Noack et al 2003), Central manifold reduction (Carini et al 2015), amplitude equations (Meliga et al 2009, Gupta and Chokshi 2015, semi-linear approaches (Farrell and Ioannou 2003). One of the most prominent and well studied flow serving as archetype of supercritical Hopf bifurcation is the flow around a cylinder.…”