Bursting dynamics of viscous film without circular symmetry: the effect of confinement

We experimentally investigate the bursting dynamics of confined liquid film suspended in air and find a viscous dynamics distinctly different from the non-confined counterpart, due to lack of circular symmetry in the shape of expanding hole: the novel confined-viscous bursting proceeds at a constant speed and a rim formed at the bursting tip does not grow. We find a confined-viscous to confined-inertial crossover, as well as a nonconfined-inertial to confined-inertial crossover, at which bursting speed does not change although the circular symmetry in the hole shape breaks dynamically.

(b) Bursting dynamics in the viscous regime- Figure 1(c) though (e) demonstrate experimental results on the dynamics of bursting tips in the viscous regime (see Supplemental Material (SM) movie 1 [35]). Snapshots of a whole bubble and a magnified tip taken after bursting starts from the left edge are shown in Fig. 1(c) and (d), respectively. It is demonstrated by the three snapshots taken at a regular interval in Fig. 1(d) and by the relation between the tip position r and elapsed time t given in Fig. 1(e) that the bursting velocity is constant, while the velocity depends on the film thickness as shown in Fig. 1(e). Although a rim exists at the tip, the rim does not grow as the bursting proceeds, as indicated in As shown in Fig. 2(a), we add particles (techpolymer MBX-20, SEKISUI PLASTICS) into oil to see the flow inside the film (the number of the added particles are made small to minimize the disturbance of the flow by them). The movement of particles implies that the y and z components of the flow are practically zero (see movie 2 [35] and Sec. B of SM [36]) and that the remaining x component is constant in the z direction (see Sec. C of SM [36]). In addition, the length of the disturbed region L increases with D, which is justified in the following manner. In Fig. 2(b), the x component v of flow velocity normalized by bursting velocity V is given as a function of the distance from the tip x on the basis of the particle-tracking analysis. As revealed in the inset of Fig. 2 As for the shape of a bursting tip, our analysis indicates that the radius of curvature at the bursting tip R is determined by h. Figure 2(d) shows that R, which is determined by fitting the shape of the rim to a parabolic function, as a function of the film thickness h for different D and η. This indicates that R increases with h and is independent of D or η.
Theoretically, the observed bursting velocity can be explained by a global balance between the surface and dissipative energies under an assumption consistent with Fig. 2(c). The bursting is driven by decrease in surface energy of the film, which is dimensionally estimated as d(γDr)/dt = γDV per unit time. The viscous dissipation is estimated as follows. As  Fig. 2(c). The velocity gradient V /D originating from these two origins is developed in a volume of the order of hDL with L ∼ D. Thus, the viscous energy dissipation is described as η(V /D) 2 hD 2 . From the balance of the two energies, (c) Bursting dynamics in the inertial regime-As the Reynolds number Re increases, the confined-viscous regime exhibits a crossover to the confined-inertial regime at Re ∼ 1 (Re ∼ 10 −3 in Fig. 1 ∼ to 3). Here, Re is estimated as ρV D/η by considering the ratio of the inertial force F i = d(MV )/dt = ρDhV 2 (with the mass of a rim M = ρrDh) to the viscous force F v ∼ ηV hD 2 /D 2 (see Sec. F of SM for another derivation of the expression for Re, which supports L ∼ D suggested in Fig. 2(c) [36]). Note that the bursting in the confinedinertial regime is assumed to proceed at a constant speed (V = dr/dt) as experimentally observed.
The bursting velocity U i in the confined-inertial regime is derived from the balance between capillary force 2γD and inertial force F i , which results in U i = 2γ/(ρh). This velocity is the same with that in the nonconfined 3D case including the numerical coefficient, i.e., U i = U C .
The confined-inertial bursting predicted above can be observed in experiment for Re ≫ 1 (see Fig. 4, in which Re ∼ 20). In Fig. 4(a), the expanding hole grows at first with maintaining the circular shape as in the case of the nonconfined 3D film and then changes its shape to the quasi-rectangular one, but the bursting speed is unchanged throughout bursting as shown in Fig. 4(b). This observation is consistent with the above prediction in the following manner. First, the observed initial 3D bursting is in the inertial regime because, if a bursting proceeds at a constant speed for a 3D film, the bursting velocity is U C , as we mentioned in the second introductory paragraph. Second, since Re ≫ 1 is satisfied, the observed rectangular bursting is expected to be in the confined-inertial regime and thus proceeds with the velocity U i , which is observed to be equal to the nonconfinedinertial velocity U C , as predicted. Finally, the rim growth, which we suppose in the deviation of U i , is consistent with experimental observation. This is because, at the bursting tip drop generation can be seen, which is caused by fragmentation of an amply growned rim into small droplets (see SM movie 3 [35]).
Discussion.-At very short times after the nucleation of an initial hole for bursting before the confined-viscous regime sets in, the film seems to rupture at a velocity dozens times higher than U η . If we could capture this initial stage of bursting, we would see the change in the shape of the expanding hole from circular (nonconfined 3D) to quasirectangular (confined quasi-2D) in the viscous regime as observed in the inertial bursting.
However, this ultrafast regime is difficult to capture and requires a separate study. This is because the control of the point where bursting starts is technically difficult and the time scales for the ultrafast initial regime and the viscous regime are extremely different.
In Ref. [37], the authors confined a film between two needles and punctured the film by another needle to measure the bursting velocity and reported a bursting velocity different from ours. This difference may originate from difference between the needle and Hele-Shaw geometries and/or significant difference in characteristic length scales of the two experiments.
The present work could make a significant contribution to the field of bursting film in air, given that only a few scaling regimes have been known despite the long history of research.
For example, this work provides fundamentally important knowledge for understanding the dynamics of foams in general [33]. Controlling the rim growth with the aid of Eq. (1) may also be useful for environmental problems or industrial applications associated with generation of droplets [19,20,38]. Furthermore, Eq. (1) and the remarkable invariance of the bursting speed (U i = U C ) at the 3D to quasi-2D crossover in the inertial regime could be useful for measuring the thickness of confined liquid film in a wide range from micronto nanometer-scales.