In the preparation of fluid threads for observation of their breakup, it is common to subject the threads to intentional local perturbations that can influence subsequent thread breakup [30]. These perturbations are distinct from perturbations of the thread arising from equilibrium interface fluctuations associated with the thermal energy of the fluid and can have a very large impact on the time scale of the thread breakup. Alternatively, there are instances under processing conditions where we wish to stimulate thread breakup through the application of some localized external perturbation such as an acoustic or other (electric, magnetic depending on the nature and responsiveness of the fluid) field, mechanical force or laser pulse to a particular part of the fluid thread. We thus consider the influence of finite size effects on thread breakup in the case where the thread has been subjected to a localized impulsive perturbation ("tap"). Specifically, a localized impulse is directed towards the center of the thread for the duration of ten LB time steps was applied over five consecutive lattice spaces along the thread surface and parallel to the thread orientation. The magnitude of the impulse was quantified by ε, the extent of thread deformation induced at the time of its application relative to Rthread. Figure 5(a) illustrates the progressive breakup of a liquid thread having an initial radius of 9.47 (lattice units) and confined within a tube of radius 19 so that Λ = Rtube / Rthread is equal to 1.9. The length of the tube is 600, again as in Fig. 2(a), but in this case we show the entire tube. The arrow in figure indicates the position where the localized impulsive perturbation was applied and the magnitude of the impulsive perturbation equals, ε = 0.1, which is a typical value for measurements on threads subjected to large and localized "taps" (e.g., ε in the range 0.05–0.6 are investigated in [3]). This figure indicates the full progression of the thread breakup process and new features evidently arise because of confinement. The response of the thread to "tapping" is nearly sinusoidal and rotationally symmetric about the fluid thread with a wavelength δ ≈ 0.57 ± 0.03 (uncertainty estimate same as described above). This is again in close accord with bulk thread breakup [11] (see Sec. II). At these later times, we observe the growth of fluid "bulges" where the wall fluid thickens at the expense of the fluid thread. At intermittent points, this thickening becomes large enough to rupture the thread to form "plugs." (This phenomenon has been observed in liquid thread breakup in highly confined fluid threads of water in an oil matrix [10].) The thread pinch-off appears to be a nucleationlike process, corresponding to essentially random points along the thread where the thread happens to grow to a scale sufficient to lead to rupture [43]. There is a correlation, however, with the rupture point and the tapping position when the tapping amplitude is sufficiently large (see. Fig. 5). The relation between the λmax of the early-stage thread undulations to the spacing between the plugs is unclear, however. The somewhat larger distance between the plugs, relative to the initial wavelength of the instability, might give the impression of an effectively longer "instability wavelength," but this conclusion is questionable. Once the thread ruptures, we see a propagating ("end pinch") instability that grows from the ruptured thread ends towards the capsule center. A near periodic array of plugs forms as the capsule shortens through progressive fission of droplets from the capsule ends. (Propagating instabilities of this kind have also been observed in the breakup of the confined liquid capsules [10], and highly extended droplets and vesicles under flow conditions [44(a), 44(b)].) Notably, satellite formation is suppressed in confined thread breakup, relative to the weak confinement case shown in Fig. 2(a). This is apparently due to the "flattening" of the thread undulation bulges of the thread due to confinement. This flattening (see Fig. 11 below where this effect is clearly illustrated for parallel plate confinement) leads to a more gradual tapering of the connecting threads between the thread bulges, which then do not so readily break up into satellite droplets by capillary instability. We also find that the spacing of the plugs becomes regular in the late-stage of the capillary breakup, leading to a pattern wavelength comparable to the bulk thread breakup process. Figure 5(b) shows the case of Λ = 1.8 where the confinement effect is enhanced further. It is evident that this apparently slight increase in confinement leads to a strong slowing down of the rate of thread breakup and an increase in the number of plugs.
FIG. 5. Thread breakup of a strongly confined fluid threads subjected to a tapping perturbation. (a) LB simulation of a thread of radius 9.47 (in units of lattice spacing) confined to a tube of radius 18(Λ = 1.9) and having a length 600. The arrow indicates the position along thread where impulsive force was applied. (b) Thread breakup evolution for a tapped thread with increased confinement (Λ = 1.9). Satellite droplets disappear by "dissolving" into the surrounding fluid. The arrow indicates the position along the thread where impulsive force was applied.
From these observations, we conclude that morphological
evolution of highly confined thread breakup is qualitatively
different from unconfined or weakly confined threads (Λ
2.5). With increasing
confinement, thread breakup is predominated
by non-periodic and sparse thread rupturing
events. Extensive collective motion develops from these rupture
points through a propagating wave developing along the
thread which ultimately leads to a string of plugs in the tube.
Once formed, the plugs are highly persistent and their coalescence
is slow. This phenomenon is commonly encountered
in liquid plugs formed in mercury thermometers and is
appropriately named the "Jammin effect" [45].
Next, we consider the crossover between the highly confined thread breakup process in Figs. 5(a) and 5(b) and the weakly confined thread breakup process [Fig. 2(a)]. In Fig. 6, we show the late-stage morphology of the thread breakup where the tube length is fixed [see Fig. 5(a)], but Λ is varied from 1.80 to 2.54. The thread radius is near constant with small variations coming from the concentration relaxation. All the runs correspond to thread lengths well above the Rayleigh-Plateau length. The impulsive perturbation is the same as in Fig. 2(a) (ε = 0.1). We observe a transition from plugs to droplets occurs for Λ ≈ 2 + θ, where θ is on the order of the interfacial width w relative to the tube radius, w/Rtube where w ≈ 3-4 lattice spacings in the present calculations. (Further measurements over a range of quench depths will be required to verify the generality of this finding.) At this scale of relative confinement, the droplets resulting from the thread breakup are just small enough to form without appreciable distortion from a spherical shape in the enclosing tube. We can appreciate the physical origin of this crossover scale by a simple geometrical argument. Assuming that the volume of the droplets formed from the ruptured thread equals the volume of a section of the thread having length λmax, implies that the radius of the droplets is equal to Rdroplet = (3π / δ)1/3R thread. For viscosity-matched fluids (p = 1), this implies, Rdroplet /Rthread ≈ 2.03, which is close to the observed plug-droplet transition in Fig. 5. We also observe that the interdrop and interplug length scale of the late-stage pattern is not strongly sensitive to Λ. The maximum growth rate wavelength λmax of the capillary instability at early stages for highly confined threads (Λ ⊬ 1) is predicted to equal, λmax ≈ 23/2πRthread(δ = 2−1/2 = 0.707) [11]. This corresponds to a 26% decrease of the instability wavelength relative to breakup in the bulk matrix [18]. Although the early-stage results do not evidently apply to the morphology at long times, we note that the inter-drop spacing in the late-stage morphology is about 10% smaller than λmax for the bulk, a trend consistent with the early-stage capillary instability theory.
FIG. 6. Influence of tube confinement on late-stage thread breakup morphology. Image shows a cross sectional view of tube where the white fluid initially forms a cylinder at the center of tube. The initial thread radii vary from 9.44 to (in units of lattice spacing), while the tube radius varies from 17 to 25, so that Λ varies from 1.8 to 2.54.
A further morphological transition in the thread breakup evolution is apparent at an intermediate stage of the thread breakup for moderate confinement. Figure 7 shows an earlier stage evolution (0 < tred < 58.3) of the image shown in Fig. 6 for Λ = 2.09. This thread breakup process is akin to the Rayleigh-Plateau instability in bulk (Fig. 2), although the anisotropic capsules become unstable in a late-stage of the instability and form a regular array of plugs, as in the highly confined case [Figs. 5(a) and 5(b)]. The aspect ratio of the plugs diminishes with increasing Λ, up to the plug-droplet transition range (Λ = 2 + θ) where the droplets become nearly spherical. Note that the satellite drops "dissolve" into the surrounding fluid matrix at this moderately deep quench depth.
FIG. 7. Rayleigh-Plateau instability distorted by finite size effects. LB simulation of thread of initial radius 9.58 (in units of lattice spacing) confined to a tube of radius 20(Λ = 2.09) and having length 600. The corresponding late-stage morphology is indicated in Fig. 5.