dorsal/arxiv
View SchemaCoalescence of Liquid Drops
| Authors | Jens Eggers, John R. Lister, Howard A. Stone |
|---|---|
| Categories | |
| ArXiv ID | physics/9903017 |
| URL | https://arxiv.org/abs/physics/9903017 |
| DOI | 10.1017/S002211209900662X |
Abstract
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 $2\pi \rmn$ and width $\Delta\ll\rmn$ 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 $\Delta \propto \rmn^3$ and $\rmn \sim (t\gamma/\pi\eta)\ln [t\gamma/(\eta R)]$; 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 $\Delta \propto \rmn^{3/2}$ at the meniscus and $\rmn \sim (t\gamma/4\pi\eta) \ln [t\gamma/(\eta R)]$. 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 $\rmn^{7/4}$.
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"abstract": "When two drops of radius $R$ touch, surface tension drives an initially\nsingular motion which joins them into a bigger drop with smaller surface area.\nThis motion is always viscously dominated at early times. We focus on the\nearly-time behavior of the radius $\\rmn$ of the small bridge between the two\ndrops. The flow is driven by a highly curved meniscus of length $2\\pi \\rmn$ and\nwidth $\\Delta\\ll\\rmn$ around the bridge, from which we conclude that the\nleading-order problem is asymptotically equivalent to its two-dimensional\ncounterpart. An exact two-dimensional solution for the case of inviscid\nsurroundings [Hopper, J. Fluid Mech. ${\\bf 213}$, 349 (1990)] shows that\n$\\Delta \\propto \\rmn^3$ and $\\rmn \\sim (t\\gamma/\\pi\\eta)\\ln [t\\gamma/(\\eta\nR)]$; and thus the same is true in three dimensions. The case of coalescence\nwith an external viscous fluid is also studied in detail both analytically and\nnumerically. A significantly different structure is found in which the outer\nfluid forms a toroidal bubble of radius $\\Delta \\propto \\rmn^{3/2}$ at the\nmeniscus and $\\rmn \\sim (t\\gamma/4\\pi\\eta) \\ln [t\\gamma/(\\eta R)]$. This basic\ndifference is due to the presence of the outer fluid viscosity, however small.\nWith lengths scaled by $R$ a full description of the asymptotic flow for\n$\\rmn(t)\\ll1$ involves matching of lengthscales of order $\\rmn^2, \\rmn^{3/2}$,\n\\rmn$, 1 and probably $\\rmn^{7/4}$.",
"arxiv_id": "physics/9903017",
"authors": [
"Jens Eggers",
"John R. Lister",
"Howard A. Stone"
],
"categories": [
"physics.flu-dyn"
],
"doi": "10.1017/S002211209900662X",
"title": "Coalescence of Liquid Drops",
"url": "https://arxiv.org/abs/physics/9903017"
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