Combined influence of inertia, gravity, and surface tension on the linear stability of Newtonian fiber spinning

The draw resonance effect appears in fiber spinning processes if the ratio of take-up to inlet velocity, the so-called draw ratio, exceeds a critical value and manifests itself in steady oscillations of flow velocity and fiber diameter. We study the effect of surface tension on the draw resonance behavior of Newtonian fiber spinning in the presence of inertia and gravity. Utilizing an alternative scaling makes it possible to visualize the results in stability maps of highly practical relevance. The interplay of the destabilizing effect of surface tension and the stabilizing effects of inertia and gravity lead to nonmonotonic stability behavior and local stability maxima with respect to the dimensionless fluidity and the dimensionless inlet velocity. A region of unconditional instability caused by the influence of surface tension is found in addition to the region of unconditional stability caused by inertia, which was described in previous works [M. Bechert, D. W. Schubert, and B. Scheid, Eur. J. Mech B 52, 68 (2015); Phys. Fluids 28, 024109 (2016)]. Due to its importance for a particular group of fiber spinning applications, a viscous–gravity–surface-tension regime, i.e., negligible effect of inertia, is analyzed separately. The mechanism underlying the destabilizing effect of surface tension is discussed and established stability criteria are tested for validity in the presence of surface tension.

Figure 1

Left: Sketch of the fiber spinning process. The flow direction is indicated by a big arrow and aligned with gravity. Right: Fiber model between inlet and outlet.

Figure 2

(a) Top: Neutral stability curve for the viscous–surface-tension model as calculated by numerical continuation (line) together with the results of Chang et al. [6] (points). Bottom: Dimensionless oscillation frequency at criticality. (b) Empirical fit to the stability curve and corresponding relative fit residual $\left|\delta \right|$. Only a few points of the numerical data are shown for the sake of readability.

Figure 3

Neutral stability curves for the full fiber model. (a) Critical draw ratio vs dimensionless fluidity $F$ for various values of $S$ and $Q$. (b) Critical draw ratio vs dimensionless inlet velocity $Q$ for various values of $S$ and $F$. Symbols indicate extrema, the loci of which are given in Fig. 4.

Figure 4

Stability maps in the $Q\text{−}F$ plane for various values of $S$: (a) $S=0$, (b) $S={10}^{-3}$, (c) $S={10}^{-2}$, (d) $S={10}^{-1}$, (e) $S=1$. Thin solid lines correspond to isolines of constant critical draw ratio, with particular values indicated below the lines. The dashed lines indicate the loci of a stability maximum with respect to $F$ at constant $Q$, as illustrated by the cross in Fig. 3. The dotted black (gray) lines indicate the loci of a stability maximum (minimum) with respect to $Q$ at constant $F$, as illustrated by the open (filled) circle in Fig. 3.

Figure 5

Stability behavior in the viscous–gravity–surface-tension regime. (a) Neutral stability curves ${\text{Dr}}_c$ vs $F/Q$ for various values of $S$. (b) Stability map with isolines of constant critical draw ratio, with particular values indicated on the left of the lines. The thick black line visualizes the transition to unconditional instability, i.e., ${\text{Dr}}_c\le 1$. The thick gray line corresponds to the transition to unconditional instability of the viscous–surface-tension model, neglecting gravity, as discussed in Sec. 4a. The dashed line indicates the loci of a stability maximum with respect to $F/Q$ at constant $S$ and separates the parameter space into regions where increasing $F/Q$ is purely stabilizing (below) and purely destabilizing (above).

Figure 6

Visualization of one half period of draw resonance. The wavy arrows indicate propagation along the spinline. The indices indicate the positions at inlet and outlet, ${\mathrm{in}}^{*}$ denotes a position close to the inlet.

Figure 7

Evolution of the force perturbation waves in space and time. Plots (a), (b), and (c) visualize the force perturbation $\tilde{F}=f/f_s-1$ along the $x$ axis at various time values within a half oscillation period $T/2$ for $1/{\text{Ca}}^{*}=0.1$ (a), 5 (b), and 10 (c), respectively. Time evolves from the thick line $\left(t=0\right)$ to the dashed line $\left(t=T/2\right)$ along the arrow in steps of $T/16$, with each step shown by a dashed-dotted line. (d) Ratio of oscillation amplitudes of force perturbation $\tilde{F}$ at inlet to outlet.

Figure 8

Check of (a) the stability criterion proposed by Kim et al. [13] and (b) the approximation from Jung et al. [14] at criticality. The time is scaled by the oscillation period $T$ and only a few points are shown for $t_L+T/4$ and $2t_L+\mathrm{\Delta }{t}_a$ for the sake of readability.