Stochastic Jetting and Dripping in Confined Soft Granular Flows

We report new dynamical modes in confined soft granular flows, such as stochastic jetting and dripping, with no counterpart in continuum viscous fluids. The new modes emerge as a result of the propagation of the chaotic behavior of individual grains—here, monodisperse emulsion droplets—to the level of the entire system as the emulsion is focused into a narrow orifice by an external viscous flow. We observe avalanching dynamics and the formation of remarkably stable jets—single-file granular chains—which occasionally break, resulting in a non-Gaussian distribution of cluster sizes. We find that the sequences of droplet rearrangements that lead to the formation of such chains resemble unfolding of cancer cell clusters in narrow capillaries, overall demonstrating that microfluidic emulsion systems could serve to model various aspects of soft granular flows, including also tissue dynamics at the mesoscale.

Figure 1

Scheme of the microfluidic system for generation of a quasi-2D granular medium (water-in-oil emulsion) and its fragmentation under flow focusing with an external third immiscible phase (fluorinated fluid). Dimensions as measured by profilometry are $W_0=2$, $W=1$, $L=2$, $H=0.11$, $w=0.11$, and $h=0.08\mathrm{mm}$.

Figure 2

Dynamical modes observed in the system upon varying $Q_c/Q_d$ $Q_d=0.5\mathrm{mL}/\mathrm{h}$. $Q_c/Q_d$ decreases from top to bottom (note different values for simulations and experiment). Smaller snapshots in the column on the right show flow patterns observed in the experiment with the emulsion replaced by a simple viscous liquid, either water (blue) or oil (transparent) for the same $Q_c/Q_d$ as in the experiment with the emulsion. The color bar refers to simulation snapshots, with $U$ being the local velocity in lattice units. Experimental snapshots were taken from movies SM1 ($Q_c/Q_d=1$), SM2 ($Q_c/Q_d=2$), SM3 ($Q_c/Q_d=8$), SM 4 ($Q_c/Q_d=16$); simulation-based snapshots are taken from movies SM5 ($Q_c/Q_d=0.5$), SM6 ($Q_c/Q_d=2.5$), SM7 ($Q_c/Q_d=5$), and SM8 ($Q_c/Q_d=10$) in the Supplemental Material [32].

Figure 3

(a) Fluctuations of the minimum jet width $w_{\min }(t)$ (b) Fluctuations of jet width $w_0(t)$ at the entrance to the constriction. The avalanching events are marked with arrows. (c) Number of grains $N$ in clusters generated in the dripping and irregular dripping regimes ($Q_c/Q_d=16$ and 8) vs the index of cluster $i$ in order of generation, and (d)–(e) the corresponding histograms $n(N)$. The histogram in (f) shows analogous data for a system with a much thinner and longer orifice, yet with $Q_c/Q_{c,\mathrm{match}}$ very close to the case in (e) (see main text and Supplemental Material [32] for further discussion).

Figure 4

(a)–(b) Numerical (at $Q_c/Q_d=3.5$) and experimental (at $Q_c/Q_d=8$) snapshots (see SM14 and SM15 [32], respectively, for the full movies) visualizing (a) doublet unfolding upon entry into the constriction and (b) a failure of unfolding, leading to an increased velocity gradient $\nabla U$ around the doublet (pink marker in the simulation panel) and breakup of the jet. (c) The order of entry of circulating tumor cells (CTCs) forming a cluster in a narrowing channel under co-flow (figure adapted with permission from Au et al. [20]; copyright National Academy of Sciences 2016) and (d) the order of entry of droplets into the constriction in our experiment, at $Q_c/Q_d=8$ (see SM16 [32] for the full movie).