C. Confined fluid thread breakup: Random initial perturbation

It is known that the time of thread breakup depends on the magnitude of the initial perturbations to which the threads are subjected, although there has been limited systematic study of this phenomenon. Kuhn [46] estimated the dependence of the thread breakup time on the amplitude of random perturbations associated with thermal fluctuations. His estimates have not been found to agree quantitatively with measurement (presumably uncontrolled localized impulsive perturbations are one reason for this discrepancy), but they have been of value in rationalizing the existence of initial thread deformations when the data is extrapolated to t = 0 [3, 18].

The delicate interplay between thread breakup by capillary waves all along the thread and drops pinching off successively from the ends of capsules (end-pinch instability), described in Sec. III B, raises questions about how the thread perturbations act in connection with finite size effects. Can the qualitative nature of the thread breakup process depend on the character of the perturbation (e.g., discrete impulsive versus random perturbations along the thread)? To check for this possibility, we subjected the thread to small amplitude perturbations to model perturbations arising from the effect of thermal fluctuations and the thread preparation in the measurements. We find that the nature of the fluid perturbation can indeed have a strong influence on the thread breakup process in the confined regime.

A small spatially random forcing was applied at a single time step throughout the entire volume of fluid in such a way that no net momentum change occurs. The perturbation at each point was a vector of randomly determined direction, having a magnitude randomly chosen from a uniform distribution in the range [0,10⁻⁵]. In addition, we ensure that there is no total momentum change by pairing up lattice points of identical composition and applying a randomly generated perturbation to one member of the pair and the reverse of that perturbation to the other member of that pair. Notably, the amplitude of these random impulses is much smaller than the discrete impulses described in Sec. III B.

We first applied the random perturbations to a thread under weak confinement (Λ > 2.5). The maximum value of the scale of the impulsive deformation relative to the thread radius ε_max is taken to be 0.001 and the ensemble average ε is half as large, = 5 x 10⁻⁴. These random perturbations ("kicks") were applied after the thread composition had relaxed to its coexisting composition value. For comparison, 0.001 is a typical order of magnitude for experimentally estimated values of the initial thread deformation α(t → 0⁺), relative to R_thread for threads not subjected to impulsive perturbations [3] (In measurements, these perturbation magnitudes values are normally too small for direct microscopic observation and are estimated by extrapolating the thread breakup observations to vanishing time.). We find that a change in the magnitude of ε_max for weakly confined threads [as in Fig. 2(a)] over a range of two orders of magnitude has little impact on asymptotic exponential growth rate of the thread breakup, apart from a change in the "induction time" it takes for the growth to approach the exponential regime [see Eq. (7) and the discussion below]. The dependence of the breakup time on the magnitude of the perturbation (ε) accords qualitatively with Kuhn's model [46], i.e., larger amplitude initial perturbations (impulsive or random) generally shorten the breakup time.

Figure 8(a) shows the influence of random perturbations in the case of a highly confined thread (Λ =1.9; = 5 x 10⁻⁴). In this case, we observe that randomness in the initial impulsive perturbation leads to a change in the early stage of thread breakup. We find that the uniform capillary undulations at short times persist to a longer time (randomness seems to stabilize the capillary instability) and the capillary undulations thus grow to a larger scale than in the "tapped" case [Fig. 5(a)]. However, the end-pinch instability ultimately intercedes to rupture the thread. Notably, the wavelength of the capillary undulations before rupture is larger than the bulk case by about 20%, rather than smaller, as found in the case of the "tapped" thread. The rupture of the thread is followed by an end-pinch instability that causes the formation of "peanut shaped" capsules that relax into plugs with "collars" (rings of trapped fluid within the plugs; see Fig. 1). Over time, the collars of the plugs drift to one or the other side of the plug axial face under capillary action, leaving uniformly spaced cylindrical plugs after these transient features disappear. Small plugs sometimes alternatively disappeared through a dissolution process similar to the satellite droplets described above. We also observe that the propagating (end-pinch) instability by which the thread ruptures into capsules is more rapid in the random perturbation case.

The prevalence of thread breakup by capillary instability or end-pinch instability in highly confined threads is evidently sensitive to the character of the perturbations to which the threads are subjected. Further evidence of this "noise sensitivity" in confined threads is found by increasing the confinement to Λ = 1.8 for the case of a random initial thread perturbation ( = 5 x 10⁻⁴). The breakup evolution for this case is shown in Fig. 8(b) where we find that the early-stage Rayleigh-Plateau instability, pronounced in the Λ = 1.9 case [Fig. 8(a)] is now suppressed in relation to the bulge instability and the associated end-pinch instability. Strikingly, the wavelength of the Rayleigh-Plateau instability developing at early times is decreased by about 10%, as in the "tapped" case [Fig. 5(a)]. This effect ultimately leads to a larger number of plugs [seven compared to six in the bulk case shown in Fig. 2(a)]. The transient capsules, formed after the thread ruptures, again develop a "peanutlike" shape and then evolve into plugs with substantial collars. In Fig. 8(b) we see that the plug collar itself ruptures in the late-stage morphology (t_red ≈ 579), thus forming a droplet on the tube wall. Notably, the spacing in the plugs and the collared plug morphologies are more disordered, reflecting a sensitive dependence of the random perturbations at earlier times. Successive runs with different choices of random numbers describing the random thread perturbations led to distinct morphologies with similar characteristics-disorder in the plug spacings, fluctuations in the number of droplets, and odd transient "collars" on the plugs. This variation is illustrated in the last frame in Fig. 8(a) which corresponds to the morphology obtained in a second simulation at t_red = 578.9. Evidently, many runs should be performed to obtain appropriately averaged properties of the thread breakup process in these highly confined fluids. These morphogical fluctuations do not occur under weak confinement conditions so that the finite size constraint amplifies the sensitivity of thread breakup to noise.

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