Folding of Viscous Threads in Diverging Microchannels

We study the folding instability of a viscous thread surrounded by a less viscous miscible liquid flowing from a square to a diverging microchannel. Because of the change in the flow introduced by the diverging channel, the viscous thread minimizes viscous dissipation by oscillating to form bends rather than by simply dilating. The folding frequency and the thread diameter can be related to the volume flow rates and thus to the characteristic shear rate. Diffusive mixing at the boundary of the thread can significantly modify the folding flow morphologies. This microfluidic system enables us to control the bending of the thread and to enhance mixing between liquids having significantly different viscosities.

Figure 1

Schematics of the experimental setup. (a) Diagram of a diverging microchannel module (height, $h=100\mu \mathrm{m}$). Simultaneous injection of silicone oils L1 (more viscous) and L2 (less viscous) yields a thread of L1 in the square channel. The thread enters a diverging channel characterized by the angle $\alpha$. (b) Flow profiles during the folding instability (side view). Arrows indicate velocities.

Figure 2

Viscous folding of a thread in diverging microchannels for different channel angles $\alpha$ and flow rate ratios $\phi$. The viscosities are fixed at ${\eta }_1=500\mathrm{cP}$ and ${\eta }_2=6\mathrm{cP}$ ($\chi =83$). (a)–(d) The thread flow rate is fixed at $Q_1=5\mu \mathrm{l}/\min$. (a) $\phi =Q_1/Q_2=0.4$, $\alpha =\pi /2$. (b) $\phi =0.2$, $\alpha =\pi /2$. (c) $\phi =0.4$, $\alpha =\pi$. (d) $\phi =0.2$, $\alpha =\pi$. (e) $\phi =0.03$ ($Q_1=1\mu \mathrm{l}/\min$), $\alpha =\pi /2$. (f) $\phi =0.02$ ($Q_1=1\mu \mathrm{l}/\min$), $\alpha =\pi$.

Figure 3

Folding frequency, $f$, vs the characteristic shear rate, $\gamma$, for different $\chi$ and $\alpha$. Line: $f=\gamma /4$. Open symbols, $\alpha =\pi /2$; filled symbols, $\alpha =\pi$. Viscosities (in cP) are: ${\eta }_1=100$, ${\eta }_2=6$ (●); ${\eta }_1=200$, ${\eta }_2=6$ (▽); ${\eta }_1=500$, ${\eta }_2=10$ (△); ${\eta }_1=500$, ${\eta }_2=6$ (◇); ${\eta }_1=500$, ${\eta }_2=6$ (◆); ${\eta }_1=100$, ${\eta }_2=0.82$ (□); ${\eta }_1=500$, ${\eta }_2=0.82$ (○); ${\eta }_1=500$, ${\eta }_2=0.5$ (⊳). Inset: dimensionless thread diameter, $\epsilon (\phi )$; line:$\epsilon ={\phi }^{0.6}$.

Figure 4

Diffusive-to-convective folding micrograph diagram: $Q_1$ vs $\phi$. Thread viscosity: ${\eta }_1=500\mathrm{cP}$, outer viscosity: ${\eta }_2=0.5\mathrm{cP}$ ($\chi =1000$). (a)–(c) Folding with strong diffusive mixing ($Q_1=5\mu \mathrm{l}/\min$). (d)–(f) Folding with weak diffusive mixing ($Q_1=20\mu \mathrm{l}/\min$). Heterogeneous viscous mixtures are formed at low $\phi$ and high $Q_1$.

Figure 5

Subfolding of a viscous thread. The folding thread oscillates alternatively between the upper and lower sides of the folding envelope, creating subfolds within folds (${\eta }_1=200\mathrm{cP}$, ${\eta }_2=0.5\mathrm{cP}$, $\chi =400$). These folds are stretched in the extensional shear flow, producing structures resembling branches downstream. Flow rates ($\mu \mathrm{l}/\min$): (a) two apparent branches: $Q_1=12$, $Q_2=50$. (b) Four apparent branches: $Q_1=20$, $Q_2=100$.