Viscous dynamics of drops and bubbles in Hele-Shaw cells: Drainage, drag friction, coalescence, and bursting

Abstract

In this review article, we discuss recent studies on drops and bubbles in Hele-Shaw cells, focusing on how scaling laws exhibit crossovers from the three-dimensional counterparts and focusing on topics in which viscosity plays an important role. By virtue of progresses in analytical theory and high-speed imaging, dynamics of drops and bubbles have actively been studied with the aid of scaling arguments. However, compared with three-dimensional problems, studies on the corresponding problems in Hele-Shaw cells are still limited. This review demonstrates that the effect of confinement in the Hele-Shaw cell introduces new physics allowing different scaling regimes to appear. For this purpose, we discuss various examples that are potentially important for industrial applications handling drops and bubbles in confined spaces by showing agreement between experiments and scaling theories. As a result, this review provides a collection of problems in hydrodynamics that may be analytically solved or that may be worth studying numerically in the near future.

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

“…Some considered that the regime is realized by elastic feature of entangled polymers and termed the regime "viscoelastic" [24][25][26], while others showed that this regime can be reproduced in a purely viscous liquid via numerical simulation and analytical theory [27][28][29].While there still remains a controversy even for nonconfined three-dimensional (3D) bursting, studies have been very limited for bursting of spatially confined films suspended in air despite their fundamental importance. In fact, a recent study on liquid-drop coalescence [8,30] suggests the importance of the study of confined quasi two-dimentional (2D) bursting in air, demonstrating distinct confinement effects on the dynamics. In addition, such a con-fined dynamics should play fundamentally important roles in many practical applications such as related to foams [31][32][33], given that any films in foam are inherently confined by Plateau borders.…”

Rising bubble in a cell with a high aspect ratio cross-section filled with a viscous fluid and its connection to viscous fingering

“…In the singly and doubly confined cases, L is significantly larger and smaller than 2r, respectively. The singly confined case corresponds to experiments using Hele-Shaw cells [13,[15][16][17]. In summary, in this paper, we distinguish the following two cases as illustrated in Figs.…”

“…[13], and the necessary condition in a more general case in which an air bubble could be replaced with a fluid bubble was established in Refs. [15,16]. The underlining physics is the competition between two extreme cases: one case in which the flow velocity gradient develops only inside a fluid drop and the other in which it develops only inside the thin liquid films.…”

“…It is also shown that the shape of the bubble in the viscous regime is almost circular [13], which is not always the case when inertia comes into play, as in the case of a bubble in water or in a moderately viscous oil [14]. In 2016, another scaling regime was found in which the dissipation in the lubricating film governs the dynamics, and the crossover of this regime to the scaling regime found in [13] was clearly shown [15] (see also the review paper [16]). In 2018, another scaling regime closely related to the one found in Ref.…”

Self-similar dynamics of air film entrained by a solid disk in confined space: A simple prototype of topological transitions

“…(1) and (2). The falling velocity U could be determined by the balance between the gravitational energy gain ∆ρgR 2 D 0 U per time and an appropriate viscous dissipation, which should be the most dominant one among the following three [41]: dissipation associated with the Couette flow developed inside thin films between the disk surface and the cell wall ≃ η(U/e) 2 R 2 e and dissipations associated with Poiseuille flows around the disk corresponding to the velocity gradient in the y direction ≃ η(U/D) 2 R 2 D and in the radial direction for the disk ≃ η(U/R) 2 R 2 D. The last dissipation may be smaller than the first two because e, D ≪ R. However, the relative importance of the first and second is delicate; while the volume of dissipation for the first is well described by 2πR 2 D, that for the second can be π(cR) 2 D with a fairly large numerical constant c. In the present case, c seems indeed fairly large and the second dissipation seems the most dominant. This is because the balance of this dissipation and the gravitational energy gain gives U ≃ ∆ρgD 2 /η, which is consistent with Eq.…”