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Dynamics and fracture of ligaments from a droplet on a vibrating surface
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Image of FIG. 1.
FIG. 1.

(a) Schematic of the top view of the static droplet (left figure) and the droplet in the initial radial spreading regime (right figure). The focal plane of the camera passes through the droplet dividing the droplet into two equal halves (one half nearer to the camera and the other half farther from it). (b) Snapshots of different regimes of droplet dynamics and the definition of maximum ligament length (), full-width-at-half-maximum length (), and daughter droplet diameter (). The flow fields inside ( ) and around ( , due to acoustic streaming) an evolving ligament are also depicted. denotes the relative viscosity with respect to water. Physical mechanisms observed are: (i) no breakup, (ii) tip breakup, (iii) base breakup, (iv) tip-base breakup, and (v) Rayleigh instability breakup. (c) Comparison of the force densities at the stable levitated position of a sphere and at the centroid of the undeformed droplet (not drawn to scale). Inset shows the normalized potential field inside the levitator. [Figure 1(c) adapted from E. G. Lierke, Acta Acust. Acust. , 206 (2002). Copyright 2002, S. Hirzel Verlag Stuttgart.] (d) Variation of pressure across the length scales of the ligaments (enhanced online). [URL: http://dx.doi.org/10.1063/1.4817542.1] [URL: http://dx.doi.org/10.1063/1.4817542.2]doi: 10.1063/1.4817542.1.

doi: 10.1063/1.4817542.2.

Image of FIG. 2.
FIG. 2.

The different phases in the evolution of ligaments. stands for the typical length scale of the ligament in the growth phase. For other notations, refer to Sec. III B .

Image of FIG. 3.
FIG. 3.

Cumulative modal energy as a function of the number of modes for glycerol and water.

Image of FIG. 4.
FIG. 4.

The temporal variation of the first 7 modal coordinates of (a) water (representing low viscosity fluids) and (b) glycerol (representing high viscosity fluids). The snapshots given as insets show the evolution of shape the droplet as it spreads.

Image of FIG. 5.
FIG. 5.

First 4 POD modes for (a) water and (b) glycerol. Each mode contains intensity values (given by the color bar) at the two-dimensional array of pixel locations corresponding to the snapshots analyzed.

Image of FIG. 6.
FIG. 6.

A comparison of the contribution from the first 2 POD modes to the initial (at T = 0) and final (at T = 1) droplet shapes in the spreading regime for (a) water and (b) glycerol. T is the non-dimensional time defined as / , where is the measured time scale of the spreading regime.

Image of FIG. 7.
FIG. 7.

Ligament final aspect ratio ( = /) vs. Ohnesorge number ( = /√()). Open symbols and “NB” represent no breakup. Closed symbols and “B” represent breakup. ⟨ ⟩ denotes average across a fluid. The dashed black lines roughly mark the border of the domain of each fluid on the plot. The thick horizontal solid line (green) is = , the critical aspect ratio of a jet undergoing capillary breakup.

Image of FIG. 8.
FIG. 8.

Normalized initial ligament growth rate, as per the scaling given in Eq. (8) vs .

Image of FIG. 9.
FIG. 9.

Distribution of final aspect ratio of all ligaments (disintegrating and non-disintegrating) of different fluids.

Image of FIG. 10.
FIG. 10.

Ligament growth rate and fraction of ligaments that breakup vs . Open and closed symbols represent breakup and no breakup, respectively.

Image of FIG. 11.
FIG. 11.

Time evolution of length of ligaments during the growth phase. Open and closed symbols represent breakup and no breakup, respectively. The experimentally measured initial ligament growth rate, is used to non-dimensionalize time, . The snapshots show examples of both growing and receding phases of non-disintegrating (top row) and disintegrating (bottom row) ligaments. The time interval between two consecutive frames is around 48 s.

Image of FIG. 12.
FIG. 12.

Collapse of ligament growth profile using the theoretical time scale, = √( /(4 )). The inset shows ligament final aspect ratio vs. capillary number ( = μ /σ). Open and closed symbols represent breakup and no breakup, respectively.


Generic image for table
Table I.

Properties of glycerol/water mixtures studied.



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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Dynamics and fracture of ligaments from a droplet on a vibrating surface