Viscous Effects on Inertial Drop Formation

A. Deblais, M. A. Herrada, I. Hauner, K. P. Velikov, T. van Roon, H. Kellay, J. Eggers, and D. Bonn

Phys. Rev. Lett. 121, 254501 – Published 21 December 2018

Abstract

The breakup of low-viscosity droplets like water is a ubiquitous and rich phenomenon. Theory predicts that in the inviscid limit one observes a finite-time singularity, giving rise to a universal power law, with a prefactor that is universal for a given density and surface tension. This universality has been proposed as a powerful tool to determine the dynamic surface tension at short time scales. We combine high-resolution experiments and simulations to show that this universality is unobservable in practice: in contrast to previous studies, we show that fluid and system parameters do play a role; notably a small amount of viscosity is sufficient to alter the breakup dynamics significantly.

Received 6 June 2018
Revised 10 September 2018

DOI:https://doi.org/10.1103/PhysRevLett.121.254501

Physics Subject Headings (PhySH)

Drop breakup Interfacial flows Surface tension effects

Fluid Dynamics

Authors & Affiliations

A. Deblais¹, M. A. Herrada², I. Hauner¹, K. P. Velikov^1,3, T. van Roon⁴, H. Kellay⁵, J. Eggers⁶, and D. Bonn¹

¹Van der Waals-Zeeman Institute, Institute of Physics, University of Amsterdam, 1098XH Amsterdam, Netherlands
²Depto. de Mecánica de Fluidos e Ingeniería Aeroespacial, Universidad de Sevilla, E-41092 Sevilla, Spain
³Unilever R&D Vlaardingen, Olivier van Noortlaan 120, 3133 AT Vlaardingen, Netherlands
⁴Technology Centre, University of Amsterdam, 1098XH Amsterdam, Netherlands
⁵Laboratoire Ondes et Matiere d’Aquitaine, UMR 5798 CNRS-U. Bx, Universite de Bordeaux, 351 cours de la Liberation 33405, Talence, France
⁶School of Mathematics, University of Bristol, University Walk, Bristol BS8 1 TW, United Kingdom

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Issue

Vol. 121, Iss. 25 — 21 December 2018

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Images

Figure 1
High-speed image sequence (upper panel) of a drop of water dripping from a faucet of 2 mm in radius. Lower panel: numerical simulation of the same problem using the Basilisk code, showing 2D cuts through the axis. Note the overturning of the upper surface of the main drop in the last two frames. The scale bar represents $R_{0} = 2 mm$ . The dimensionless time to pinch-off $τ = (t_{c} - t) / t^{*}$ is indicated on the top of the figure.
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Figure 2
(a) Prefactor $A$ as a function of the dimensionless time to pinch-off $τ$ for water and decane for different values of the capillary size $R_{0}$ , the Ohnesorge number Oh, and Bond number Bo. Symbols indicate experimental results whereas solid lines were obtained from two different simulation techniques: mapping (red) and Basilisk (black). After a nonmonotonic variation, $A$ reaches a maximum value and starts to decrease on account of viscous effects. The theoretical prediction for $A$ is depicted as a horizontal dashed line. (b) The prefactor $A$ is plotted as a function of the characteristic viscous dimensionless timescale $(t_{c} - t) / t_{ν}$ : all the curves (from basilisk simulation) collapse into a single master curve for late times, showing that the drop of the prefactor results from viscous effects.
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Figure 3
Breakup dynamics of a mercury bridge ( $Oh = 6.0437 \times 10^{- 4}$ ) from the millimeter to the nanometer scale. (a) Electrical resistance of a mercury capillary bridge between two electrodes as function of time. The upper electrode moves upwards, initiating breakup; the bridge’s spatial profile is recorded simultaneously using high-speed imaging. (b) Typical measurement of voltage $V_{s}$ over the divider circuit as a function of time during the breakup process (note the nanometer timescale); at breakup (denoted here as $t = 0$ ) the voltage decreases suddenly. This is used to trigger fast imaging. (c) The shape of the liquid bridge $R (z)$ is computed from a Basilisk simulation, allowing us to calculate the resistance $r$ of the filament in time. (d) The resistance as function of minimum radius, compared to the asymptotic value of the inviscid solution, and the estimate by Burton and Taborek. (e) The measured resistance of the liquid bridge data is well described by the simulations very close to the pinch-off point. (f) The dynamics of thinning of the liquid bridge before overturning recorded by imaging techniques is also well captured by Basilisk simulations. In inset, the prefactor $A$ of mercury is plotted as a function of the dimensionless time to pinch-off $τ$ .
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