Experimental observation of local rearrangements in dense quasi-two-dimensional emulsion flow

We experimentally study rearranging regions in slow athermal flow by observing the flow of a concentrated oil-in-water emulsion in a thin chamber with a constricting hopper shape. The gap of the chamber is smaller than the droplet diameters, so that the droplets are compressed into quasi-two-dimensional pancakes. We focus on localized rearrangements known as “T1 events” where four droplets exchange neighbors. Flowing droplets are deformed due to forces from neighboring droplets, and these deformations are decreased by nearby T1 events, with a spatial dependence related to the local structure. We see a tendency of the T1 events to occur in small clusters.

Figure 1

A typical case of a T1 event: two droplets move apart and are no longer neighbors, and the other two come together to become neighbors. The field of view is $1.76\times 1.44{\mathrm{mm}}^2$.

Figure 2

A sketch of the experimental setup. (a) Side view: oil droplets are compressed into disks between parallel glass plates and are driven by a syringe pump. (b) Top view: the spacers are cut into a contracting shape. The microscope field of view is indicated. The right is an experimental image, $11.2\times 8.5{\mathrm{mm}}^2$, and flow direction is to the right. Note that at the left side of the image, the parallel edges are due to the limited field of view; the walls are still diagonal in this region.

Figure 3

(a) The distribution of droplets shaded by the magnitude of deformation $D$, with the gray scale indicating the relative $D$ values. Inset: sketch of $r$ measured from the center of the droplet, used to define deformation in Eq. (1). (b) The mean value of the largest contact length on a droplet $l_c$ as a function of its deformation $D$, normalized by the average $\langle D\rangle$. The correlation coefficient between the raw data ($l_c$ and $D$) is $\approx 0.5$. The error bars indicate the standard deviation of the data.

Figure 4

(a) An example of the time-averaged velocity profile. The data are from Run 1, at a location where the channel width is $w\approx 8$ mm. The fit curve is Eq. (2). Inset: the width $w\left(x\right)$ is the range that droplet centers can reach, which is $\langle r\rangle$ away from the physical walls. (b) The parameters $\alpha \left(x\right)$ (circles) and $\beta \left(x\right)$ (triangles) obtained from fitting the flow profile with Eq. (2). Here the fit curves are from Eq. (3) with $k_{\alpha }=1.23$ and $k_{\beta }=2.73$. (c) $k_{\alpha }$ and $k_{\beta }$ for all the six runs versus different flux rates. The dashed lines indicate the mean values $k_{\alpha }=1.24$ and $k_{\beta }=2.87$.

Figure 5

Temporal fluctuations of the global deformation $D_G$ from Run 1. The data are smoothed by a time window of 0.33 s to reduce noise. The red time windows indicate two events ($93.9<t<95.6$ s and $115.0<t<116.4$ s), corresponding to the T1 events shown in Fig. 8. The top-right inset is the power spectrum of the data, with the line showing the power law $P\left(\omega \right)\sim {\omega }^{-1}$. The bottom inset shows the definitions of $\mathrm{\Delta }{D}_G$ and $\tau$.

Figure 6

(a) Scatter plot of the time $\tau$ between local extrema of $D_G\left(t\right)$, as a function of the change $\mathrm{\Delta }{D}_G$ between the extrema; see the lower inset of Fig. 5 for definitions of these two variables. The horizontal bands are due to our limited time resolution. The colors indicate the flux rate $\tilde{A}$ in units of ${\mathrm{mm}}^2$/s. (b) Probability distribution function of $\mathrm{\Delta }{D}_G$. The dashed line is a Gaussian fit with width 0.044. (c) The T1 frequency $f_{T1}$ normalized by the flux rate $\tilde{A}=A/\pi {\langle r\rangle }^2$, plotted as as a function of the size of the deformation change. The error bars are the standard error of the mean. For the point at far left, there is only one observation with this magnitude of $\mathrm{\Delta }{D}_G$, so no error bar is shown. For panels (b) and (c) the six experimental runs are averaged together.

Figure 7

(a) Spatial-temporal map of the local deformation values around a T1 event, averaged over 186 T1 events during a 194-s duration movie (Run 1). The distance $\mathrm{\Delta }R$ is defined in the frame of reference comoving with the center of the four droplets undergoing T1 rearrangements, at a speed of roughly one mean radius per second. $\mathrm{\Delta }t=0$ is the moment when the T1 event occurs, indicated by the horizontal dashed line. The color indicates the mean magnitude of the deformation $D$ on individual droplets compared to the global mean (white), where smaller values are blue and larger values are red, as indicated by the color bar. (b) The droplet number density is indicated by brightness. The bright spot at $\mathrm{\Delta }R/\langle r\rangle =1.5,\mathrm{\Delta }t=0$ corresponds to the four droplets undergoing the T1 event, as $\mathrm{\Delta }t=0$ is defined when they are all equidistant from the center of the event. Qualitatively similar images are seen for the different experiments; we show here Run 1, which has the most T1 events observed.

Figure 8

(a) Spatial distribution of consecutive T1 events during the large deformation drops corresponding to the time windows of Fig. 5. The colored circles show the positions of the T1 events (at the center of the four droplets) at $t=94.1$, 94.5, 95.3, and 95.5 s and $t=115.7$, 116.1, and 116.3 s for each event. The color ordering is dark blue (first) to red (last). The background images are taken at $t=95$ s and $t=116$ s. The sketch on the right shows an example of clusters of T1 events. Each point indicates the center of a T1 event, and so the number of T1 events is $N_{T1}=7$ in this case. Black bonds indicate those with separations less than $6\langle r\rangle$. In this case, the seven T1 events are composed of three distinct clusters, and the largest cluster has $N_{\mathrm{cluster}}=4$. (b) The mean largest cluster size $N_{\mathrm{cluster}}$ as a function of the nondimensional flux rate $\tilde{A}$. The error bars indicate the standard deviation. (c) The probability distribution of ratio $N_{\mathrm{cluster}}/N_{T1}$ for data with $N_{T1}\ge 3$.