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We employ lattice-Boltzmann simulations as described in the Experimental and Numerical Methods section in order to gain critical information that is not accessible from the experiments. First, we present results obtained from the p = 1 case where three levels of confinement were considered: least-confined (H/d = 2.1), moderately confined (H/d = 1.45), and strongly confined (H/d = 1.2). (The labels unconfined, moderately confined, and strongly confined are used in anticipation of the results.)
Before giving a quantitative analysis of the results, we first present two plots that give a pictorial representation of our observations. Figure 7 shows end-on cross sections of the thread (x-y plane) at the widest and narrowest points that result from the three levels of confinement considered here. For the case of H/d = 2.1 (Figure 7A) and the moderately confined case H/D0= 1.45 (Figure 7B), we show the cross sections when the amplitude of the fluctuation is large, but before actual breakup has occurred. For the strongly confined case (Figure 7C), we show the cross sections at the same two planes as for Figure 7B, C at the last time step of the simulation. We can immediately make three qualitative observations, namely that for the case of H/D0 = 2.1 (the least-confined case) the distortion is axisymmetric, for the confined case it is nonaxisymmetric, and for the strongly confined case it is stable (over the time scale of the simulation.) These observations agree with the experimental results.
As the kinetics of the moderately confined case is the most interesting, we explore the kinetics of the growth of the fluctuation in further detail in the next three figures. In Figure 8 we show three-dimensional snapshots of the thread as it breaks up. At early times, (t < 100) the thread appears cylindrical, and Figure 8A (t = 73.9) shows the structure when the distortion is first apparent by visible inspection. As the breakup proceeds (Figure 8B,C), we observe that the distortion becomes increasingly asymmetric. Note that the asymmetry is observed in the bulge and that the asymmetry begins appearing well before the thread impinges on the wall.
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| Figure 8. Three-dimensional representation of the moderately confined thread as it breaks up. The times are (A) t = 73.9, (B) 77.1, (C) 78.6, and (D) 80.2 (final breakup). |
Quantification of the nonaxisymmetric breakup is shown in Figure 9. In these figures, the thick dashed line is the radius of the string as a function of axial position in the y direction (perpendicular to plates), and the solid line is that in the x direction (parallel to the plates). The thin dashed line at 112 indicates the gap thickness of this simulation. Figure 9A-C shows profiles at three times leading to breakup. There are three key points to note. First, the asymmetry occurs even when the amplitude of the distortion is low (Figure 9A). Second, the asymmetry between the x and y directions occurs in the bulge (Z = 200), but not in the neck (Z = 600), a trend that increases in magnitude as the distortion increases (Figure 9B). Third, it is clearly possible to obtain distortions in which the bulge in the parallel direction exceeds the gap width (Figure 9C).
The data from the moderately confined case were
Fourier decomposed in order to quantitate the asymmetry of the growth rates. The profiles were fit to the
following equations:
We find that the largest contributions are made by the
n = 1 modes of a
and a
. This is expected because it
describes a growing sinusoidal distortion. Note that in
the logarithmic plot of Figure 10 the difference between
the parallel and perpendicular components of the primary mode is small. As the distortion increases, Figure
9 qualitatively shows that the thread becomes increasingly different from a simple sine wave. This behavior
is captured in the n = 2 component. Surprisingly (see
figure), it is found that, although the first-harmonic
contribution is largest, the growth rate of the second
harmonic (of the cosine term) was faster than the first
harmonic (sine term). Also, while the amplitude of the
first harmonic (sine) was larger in the plane parallel to
the wall, the second harmonic was larger in the plane
perpendicular to the wall. This behavior indicates that
distortion from a pure sine wave is stronger in the
confinement direction than in the unconfined direction,
which is intuitively reasonable.
 |
| Figure 10. Major components of the Fourier transform of the profiles in the moderately confined case. See text for details. |
Finally, in regard to the issue of thread stability for the highly confined case, we note that in the previous experiment by Migler10 where p = 1, the critical confinement for stability was H/D0 = 0.83 ± 0.15. The critical confinement in the viscous mismatched experiments described with p = 0.25 is H/D0 = 1.3 ± 0.1. It is intuitively reasonable that the higher viscosity of the matrix prevents it from being squeezed out, and thus the geometrical confinement has a stronger impact on the thread. In the simulations of the viscous matched case, we find that the critical confinement occurs at H/D0 = 1.2 which is larger than expected when compared to the prior results of Migler. However, because of the computational constraints described earlier, we view the critical value of H/D0 as determined by simulation to be an upper bound to the actual value.
For the purposes of comparison to experiment, we utilize numerical data from the viscosity mismatched simulations. Returning to Figure 3, we construct a plot of dimensionless apparent volume (defined previously) as a function of the amplitude of the distortion. The solid line is the simulation result, and the discrete data points are from the experiments. For the unconfined data, in the case of the experiments we used H/D0 = 10.6 and for the simulations we used H/D0 = 2.1 (for reasons of computational time). However, both these conditions correspond to essentially unconfined behavior. For the case of H/D0 = 2.1, we see that V'app = 1, indicating axisymmetric growth, whereas for the case of H/D0 = 1.4, the upward trend of the curve indirectly indicates the nonaxisymmetric growth. This is quite similar to that observed experimentally, indicating the qualitative agreement between the two approaches.
Next, we extract data from the simulations that is
contained in the (y) direction. In Figure 11, the calculated dimensionless amplitudes under two ratios of the
gap width to the initial thread diameter (H/D0) are
presented as a function of time (solid curves) with the
experimental data of Figure
4 included as well (discrete
points). For the unconfined data, in the case of the
experiments we used H/D0 = 10.6, and for the simulations we used H/D0 = 2.1 for reasons of computational
time. But both these conditions correspond to essentially
unconfined behavior. Note that the experimental data
are shifted horizontally to achieve the best fit, and so
the comparison between experiment and simulation is
relevant to the growth rate rather than the absolute
time of the experiment. Again, we see that the simulation results qualitatively agree with the experimental
results. The amplitude growth rate decreases with
decreasing H/D0. For H/D0 = 2.1, the amplitudes in the
parallel and perpendicular directions are nearly equal,
showing that the thread is axisymmetric. For the case
of H/D0 = 1.4, we find that the growth rate is strongly
suppressed at early times but increases with time. Also,
we find that the growth is no longer axisymmetric. The
amplitude in the fluctuation in the direction perpendicular to the wall is reduced relative to that in the
parallel direction. This is similar to the experimental
result in the confined case. The other noteworthy
feature is that the ratio of 
(amplitude parallel to the
plate) to (amplitude perpendicular to the plate) ratio
is approximately constant during the growth of the
instability. This finding, along with the observation that
the necks are circular, is used in the next section.
