Swirling of Viscous Fluid Threads in Microchannels

Viscous threads that are swept along in the flow of a less viscous miscible liquid can break up into viscous swirls. We experimentally investigate the evolution of miscible threads that flow off center in microchannels. Thin threads near the walls of a straight square channel become unstable to shear-induced disturbances. The amplification of the undulations transverse to the flow direction ultimately causes the threads to break up and form an array of individual viscous swirls, the miscible counterparts of droplets. This swirling instability provides a means for passively producing discrete diffusive microstructures in a continuous flow regime.

Figure 1

Forming two off-center viscous threads. (a) Two-step hydrodynamic focusing in square channels. The less viscous liquids $L1$ and $L3$ (${\eta }_1={\eta }_3=6\mathrm{cP}$) are identical and injected at volume flow rates, $Q_1$ and $Q_3$, respectively; the more viscous liquid $L2$ (${\eta }_2=500\mathrm{cP}$) is injected at $Q_2$ and forms two threads. (b) Schematic cross-sectional contour plot of the mean flow velocity in the square outlet microchannel. (c),(d) Experimental micrographs with flow rates ($\mu l/\min$): (c) $Q_1=20$, $Q_2=1$, $Q_3=60$; (d) $Q_1=20$, $Q_2=1$, $Q_3=7$.

Figure 2

Thread separation $d/h$ versus the flow rate ratio $Q_1/Q_3$. Numerical computation of the position of the interface between $L1$ and $L3$ predicts a separation $d^{*}$ (solid line). An analytic equation assuming a plane geometry yields a prediction $d^{**}$ (dashed line). Inset: Measured thread width $\epsilon /h$ compared to the diameter of a thread of circular cross section ${\epsilon }_c/h$.

Figure 3

Shear-induced destabilization of viscous threads. (a) Disturbances are amplified as the threads propagate downstream (from left to right). Flow rates ($\mu \mathrm{l}/\min$): $Q_1=40$, $Q_2=5$, $Q_3=20$. (b) Schematic diagram of the onset of swirling instability. (c) Slenderness ratio $\lambda /\epsilon$ versus distance to the sidewall $a/h$. Solid line: $\lambda /\epsilon =2.8\times {10}^{-2}({\eta }_2/{\eta }_1)/(1-2a/h)$. (d),(e) Large slenderness ratios: (d) $\lambda /\epsilon \approx 9.6$, $Q_1=10$, $Q_2=1$, $Q_3=30$; (e) $\lambda /\epsilon \approx 7.8$, $Q_1=10$, $Q_2=5$, $Q_3=40$.

Figure 4

Time series (from top to bottom in each panel) of thread deformations and swirl motion. (a)–(c) Single thread near the lower sidewall (thread’s reference frame). (a) $\lambda /\epsilon \approx 5$, $\Delta t=4\times {10}^{-3}\mathrm{s}$), flow rates ($\mu \mathrm{l}/\min$): $Q_1=20$, $Q_2=2$, $Q_3=20$. (b) $\lambda /\epsilon \approx 4.1$, $\Delta t=3\times {10}^{-3}\mathrm{s}$, $Q_1=20$, $Q_2=4$, $Q_3=20$. (c) $\lambda /\epsilon \approx 3.8$, $\Delta t=3\times {10}^{-3}\mathrm{s}$, $Q_1=15$, $Q_2=3$, $Q_3=15$. (d) Swirls rotate (arrows) in opposite directions as they move downstream (channel’s reference frame), $Q_1=20$, $Q_2=1$, $Q_3=20$ ($\Delta t=2\times {10}^{-3}\mathrm{s}$). Neighboring swirls can pair (*).

Figure 5

Crossover between the swirling instability and the phase-locked multiple folding instability of two threads in a diverging slit microchannel. (a) In the decelerating flow, swirls deform, pile up, and merge to form heterogeneous streams, flow rates ($\mu \mathrm{l}/\min$): $Q_1=20$, $Q_2=1$, and $Q_3=20$. (b) Destabilized thin threads, $Q_1=20$, $Q_2=5$, and $Q_3=40$. (c),(d) Coupled threads, $Q_2=5$ and $Q_3=15$: (c) phase-locked multiple folding, $Q_1=5$; (d) intermingling folding pattern, $Q_1=1$.