When two drops of radius $R$ touch, surface tension drives an initially singular motion which joins them into a bigger drop with smaller surface area. This motion is always viscously dominated at early times. We focus on the early-time behavior of the radius $\rmn$ of the small bridge between the two drops. The flow is driven by a highly curved meniscus of length $2π\rmn$ and width $Δ\ll\rmn$ around the bridge, from which we conclude that the leading-order problem is asymptotically equivalent to its two-dimensional counterpart. An exact two-dimensional solution for the case of inviscid surroundings [Hopper, J. Fluid Mech. ${\bf 213}$, 349 (1990)] shows that $Δ\propto \rmn^3$ and $\rmn \sim (tγ/πη)\ln [tγ/(ηR)]$; and thus the same is true in three dimensions. The case of coalescence with an external viscous fluid is also studied in detail both analytically and numerically. A significantly different structure is found in which the outer fluid forms a toroidal bubble of radius $Δ\propto \rmn^{3/2}$ at the meniscus and $\rmn \sim (tγ/4πη) \ln [tγ/(ηR)]$. This basic difference is due to the presence of the outer fluid viscosity, however small. With lengths scaled by $R$ a full description of the asymptotic flow for $\rmn(t)\ll1$ involves matching of lengthscales of order $\rmn^2, \rmn^{3/2}$, \rmn$, 1 and probably $\rmn^{7/4}$.
这封信提出了两个粘性球形液滴合并的标度理论。一个初始值的问题，制定和解析求解的液滴聚结后形成的液体颈部的半径的演变。初始值问题的两个渐近解再现了在粘性和惯性制度的著名的标度关系。Paulsen等人[Phys. Rev. Lett. 106，114501（2011）]在理论中体现，并且它们的拟合关系（其示出了不同粘度的数据在单个曲线上的塌陷）是对初始值问题的一般解的近似。
Non-coalescence of sessile drops from different but miscible liquids: Hydrodynamic analysis of the twin drop contour as self stabilizing, traveling wave
Unified Viscous-to-inertial Scaling in Liquid Droplet Coalescence
这封信提出了一个理论上的两个球形液滴，最初是固定的合并。在聚结后形成的液体颈部的半径的演变被制定为一个初始值问题，然后解决，以产生一个精确的解决方案，没有自由参数，其两个渐近近似再现众所周知的缩放关系在粘性和惯性制度。Paulsen等人[Phys. Rev. Lett. 106，114501（2011）]也通过该理论恢复，使得不同粘度的数据塌陷到单个曲线上。
The evolution of the liquid bridge formed between two coalescing sessile yield-stress drops is studied experimentally. We find that the height of the bridge evolves similar to a viscous Newtonian fluid, $h_0\sim t$, before arresting at long time prior to minimizing its liquid/gas interfacial energy. We numerically solve for the final arrested profile shape and find it depends on the fluid's yield stress $τ_y$ and coalescence angle $α$, represented by the Bingham number $τ_y h_{drop} / σ$ modified by the drop's height-width aspect ratio. We present a scaling argument for the bridge's temporal e…
Thin liquid or gas films are everywhere in nature, from foams to submillimetric bubbles at a free surface, and their rupture leaves a collection of small drops and bubbles. However, the mechanisms at play responsible for the bursting of these films is still in debate. The present study thus aims at understanding the drainage dynamics of the thin air film squeezed by gravity between a millimetric droplet and a smooth solid or a liquid thin film. Solving coupled lubrication equations and analyzing the dominant terms in the solid and liquid film cases, we explain why the drainage is much faster i…
When two droplets with different surface tensions collide, the shape evolution of the merging droplets is asymmetric. Using experimental and numerical techniques, we reveal that this asymmetry is caused by asymmetric capillary waves, which are the result of the different surface tensions of the droplets. We show that the asymmetry is enhanced by increasing the surface tension difference, and suppressed by increasing the inertia of the colliding droplets. Furthermore, we study capillary waves in the limit of no inertia. We reveal that the asymmetry is not directly caused by Marangoni forces. In…
The breakup and coalescence of drops are elementary topological transitions in interfacial flows. The breakup of a drop changes dramatically when polymers are added to the fluid. With the strong elongation of the polymers during the process, long threads connecting the two droplets appear prior to their eventual pinch-off. Here, we demonstrate how elasticity affects drop coalescence, the complement of the much studied drop pinch-off. We reveal the emergence of an elastic singularity, characterised by a diverging interface curvature at the point of coalescence. Intriguingly, while the polymers …
Oil spills have posed a serious threat to our marine and ecological environment in recent times. Containment of spills proliferating via small drops merging with oceans/seas is especially difficult since their mitigation is closely linked to the coalescence dependent spreading. This inter-connectivity and its dependence on the physical properties of the drop has not been explored until now. Furthermore, pinch-off behavior and scaling laws for such three-phase systems have not been reported. To this end, we investigate the problem of gentle deposition of a single drop of oil on a pool of water,…
We study the effect of surfactants on the dynamics of a drop-interface coalescence using full three-dimensional direct numerical simulations. We employ a hybrid interface-tracking/level-set method, which takes into account Marangoni stresses that arise from surface tension gradients, interfacial and bulk diffusion, and sorption kinetic effects. We validate our predictions against the experimental data of Blanchette and Bigioni [Nat. Phys. 2(4):254-257 (2006)] and perform a parametric study that demonstrates the delicate interplay between the flow fields and those associated with the surfactant…
Recent investigations on the coalescence of polymeric droplets on a solid substrate have reported strong disagreements; the heart of the issue is whether coalescence of polymeric drops is similar to that of Newtonian fluid and is independent of molecular relaxation, or whether the role of entanglement of polymeric chains leads to a transition kinetics different from that of Newtonian fluid. Via this report, we resolve the disagreements through a discussion on the effects of merging method on the dominant forces governing the coalescence process, i.e., inertia, dissipation, and relaxation. Our …
Emory University（埃默里大学）；University of Vermont（佛蒙特大学）；University of Pennsylvania（宾夕法尼亚大学）；University of Michigan（密歇根大学）；Georgia Institute of Technology（佐治亚理工学院）
We report on the collision-coalescence dynamics of drops in Leidenfrost state using liquids with different physicochemical properties. Drops of the same liquid deposited on a hot concave surface coalesce practically at contact, but when drops of different liquids collide, they can bounce several times before finally coalescing when the one that evaporates faster reaches a critical size, of the order of the capillary length. The bouncing dynamics is produced because the drops are not only in Leidenfrost state with the substrate, they also experience Leidenfrost effect between them at the moment…
