Segmented flows of viscous threads in microchannels

The passage of microfluidic droplets in miscible thread-forming flows is shown to improve the mixing efficiency of low- and high-viscosity fluids at the small scale. Hydrodynamic interactions between recirculating flow patterns and viscous core-annular flows are experimentally investigated in square microchannels. The deformation of thick segmented flows injected in thin fluids is examined through the evolution of droplet size, spacing, and velocity along a square microchannel. Droplets also help in probing the various states of lubrication of the viscous central stream in the bulging, tubing, and threading regimes at various capillary and Péclet numbers. A range of intriguing flow phenomena is revealed using a dual approach based on thread and droplet mutual behaviors, including droplet breakup, formation of low-viscosity currents, thread splitting, viscous fingering, and viscous buckling instabilities. The thread-forming ability of miscible fluids having large viscosity contrasts is discussed in conjunction with mixing applications where a thinner or a viscosifier is continuously added to a viscosity-differing fluid in confined microsystems.

Figure 1

(a) Schematics of injection scheme with two-step hydrodynamic focusing section in square microchannels. (b) Initial droplet size $d_0/h$ as a function of downstream flow rate $Q_3$ for various initial droplet regimes, fluid group M2, flow rates in $\mu \mathrm{l}/\min$: (i) diluted, $d_0/h\approx 0.97$, and $\left(Q_1,Q_2\right)=\left(5,50\right)$ with $\mathrm{C}{\mathrm{a}}_0\approx 0.38$; (ii) moderate, $d_0/h\approx 1.71$, and (25, 50) with $\mathrm{C}{\mathrm{a}}_0\sim 0.48$; and (iii) concentrated, $d_0/h\approx 3.86$, and (20, 10) with $\mathrm{C}{\mathrm{a}}_0\approx 0.19$. (c) Initial flow morphologies with spatial extent of average initial droplet size $d_0/h$ and wavelength ${\lambda }_0$ at various initial droplet fraction ${\alpha }_1$. Solid lines: see main text. (d) Evolution of initial $d_0/L_0$ as a function of ${\phi }_1$ for both fluid pairs. Solid line: $d_0/L_0=1.5{\phi }_0$. Inset: linear representation of initial segmented flows.

Figure 2

Droplet- and thread-based phase diagrams. (a) Top: Droplet-based flow map showing breakup and nonbreakup regimes classified as a function of initial droplet size $d/h$ and modified capillary number $\mathrm{C}{\mathrm{a}}_{\mathrm{M}}$. ${\alpha }_1=0.09(◯)$, 0.33 ($\square$), 0.67 ($▵$), groups D3 (blue) and M2 (black), droplet breakup (open symbol), and nonbreakup (closed symbols). Solid line: $d_0/h=0.33{\mathrm{Ca}}_{\mathrm{M}}^{-0.3}$. Bottom: Micrographs of droplet regimes for group M3. (b) Top: Flow map based on thread with inner $Q_1+Q_2$ and outer $Q_3$ with bulging ($▵$), tubing ($\square$), and threading ($◯$) regimes, group M2. Solid line ${\phi }_T=3$ (bulging-tubing) and ${\phi }_T=0.48$ (tubing-threading). Inset: Schematic of thread cross section downstream of a droplet. Bottom: Micrograph of thread regimes labeled on top map for ${\alpha }_1=0.09$. The arrow indicates location where regimes are determined.

Figure 3

Evolution of segmented flow dynamics. (a) Spatial distribution of normalized droplet velocities $V_D/V_0$ for (i) diluted ${\alpha }_1=0.09$, (ii) moderate ${\alpha }_1=0.33$, and (iii) concentrated ${\alpha }_1=0.67$ droplet flows for the weakly diffusive fluid. Colors correspond to thread regimes based on ${\phi }_T$ as $Q_3$ is varied for fixed $Q_1$ and $Q_2$. (b) Relative mobility coefficient $k/k_0$ as a function of ${\phi }_T$ for various concentrations ${\alpha }_1$. (c) Measurements of final normalized length $d_{\mathrm{F}}/d_0$ and spacing $L_{\mathrm{F}}/L_0$ of droplets for fixed $Q_1=20$ and $Q_2=10\mu \mathrm{l}/\min$ versus fluid-mixing fraction ${\alpha }_{\mathrm{F}}=Q_3/\left(Q_2+Q_3\right)$. (d),(e) Evolution of droplet size and spacing in the concentrated regime with micrographs of evolving droplet from $x/h\sim 0$ to 8 (top to bottom), $\left(Q_1,Q_2,Q_3\right)=\left(20,10,30\right)\mu \mathrm{l}/\min$ for fluid group (d) D3, and (e) M2.

Figure 4

Deformation of segmented flows. (a) Spatial evolution of wavelength λ for cases shown in Figs. 3 (top) and 3 (bottom). (b) Evolution of normalized final wavelength ${\lambda }_{\mathrm{F}}/{\lambda }_0$ as function of thread flow rate ratio ${\phi }_{\mathrm{T}}=\left(Q_1+Q_2\right)/Q_3$ for various ${\alpha }_1=0.09(◯)$, 0.33 ($\square$), 0.67 ($▵$), and fluid groups M3 (blue) and D2 (black). Solid line: ${\lambda }_{\mathrm{F}}/{\lambda }_0=1+1/\left(k_o{\phi }_{\mathrm{T}}\right)$, with $k_0=1.8$. Inset: schematic of linear dilution model. (d) Spatial development of aspect ratio $d/L$ for cases shown in Figs. 3 (top) and 3 (bottom). (d) Final aspect ratio $d_{\mathrm{F}}/L_{\mathrm{F}}$ as a function of droplet flow rate ratio ${\phi }_{\mathrm{D}}=Q_1/\left(Q_2+Q_3\right)$. Solid line: $d_{\mathrm{F}}/L_{\mathrm{F}}=a{\phi }_{\mathrm{D}}$, with $a=1.5$. Inset: schematics of segmented flows having fixed $d_{\mathrm{F}}/L_{\mathrm{F}}$ and various ${\lambda }_{\mathrm{F}}$.

Figure 5

Bulging regime. Flow rates in $\mu \mathrm{l}/\min$. (a) Time series of droplet entering the second junction and pulling low-viscosity bulges for [$\left(Q_1,Q_2,Q_3\right)=\left(1,10,1\right)$, group M2] and $\mathrm{\Delta }t=30\mathrm{ms}$. (b) Spatial evolution of $V_{\mathrm{D}}/V_0$ at low ${\alpha }_1$ and large $\mathrm{C}{\mathrm{a}}_0$, symbols correspond to top micrographs for [(5, 50, 2), M2] (c) Evolution of $V_D^{*}/V_0$ with $Q_3$ for group M2 at low ${\alpha }_1$ and large $\mathrm{C}{\mathrm{a}}_0$. (d) Spatial evolution of droplet showing bulges bypass at the tubing transition, $x/h\approx 2$, 5, and 8 (from top to bottom). (e) Time series of ligament entrainment in droplet reference frame for [(5, 50, 2), D3] and $\mathrm{\Delta }t=10\mathrm{ms}$. (f) Time series of viscous sheet formation in droplet reference frame for [(20, 10, 1), M2] and $\mathrm{\Delta }t=20\mathrm{ms}$. (g) Flowering pattern of recirculating threads in droplet reference frame for [(20, 10, 4), D3] and $\mathrm{\Delta }t=20\mathrm{ms}$.

Figure 6

Tubing regime. Flow rates in $\mu \mathrm{l}/\min$ (a) Time-series of viscous fingering in the droplet reference frame, [$\left(Q_1,Q_2,Q_3\right)=\left(1,10,2\right)$, group D3], and $\mathrm{\Delta }t=40\mathrm{ms}$. Side schematics show upstream and downstream cross-sections. (b) Evolution of finger width $f$ as a function of time $t-t_0$. Inset: $J_{\mathrm{Diff}}$ vs. time, dashed-line: $J_{\mathrm{Diff}}=2.4\mathrm{mm}/\mathrm{s}$. (c) Time-series of merging diffusive fingers in the droplet reference frame, [(1,10,3), D3], and $\mathrm{\Delta }t=65\mathrm{ms}$. (d) Time-series showing convective distortion of fingers in the droplet reference frame [(1,10,4), D3], and $\mathrm{\Delta }t=60\mathrm{ms}$. (e) Spatial evolution of $V_{\mathrm{D}}/V_0$ for low ${\alpha }_1$ and large $\mathrm{C}{\mathrm{a}}_0$. Symbols correspond to micrographs for [$\left(Q_1,Q_2\right)=\left(5,50\right)$, group M2]. (g) Evolution of $V_D^{*}$ with $Q_3$ from tubing to threading regime for [(5,50), M2].

Figure 7

Threading regime. Flow rates in μl/min. (a) Spatial evolution of $V_{\mathrm{D}}/V_0$ for low ${\alpha }_1$ and large $\mathrm{C}{\mathrm{a}}_0$, group M2. Symbols correspond to micrographs where $\left(Q_1,Q_2\right)=\left(5,50\right)$. Inset: schematics of droplet and thread velocities. (d) Examples of droplet partial envelopment with L2 for $\left(Q_1,Q_2\right)=\left(5,10\right)$, and $Q_3=300$, 200, and 125 (from top to bottom) with group D3. (c) Normalized thread size ${\varepsilon }_0^{*}/h$ as a function of ${\phi }_T$, solid line: ${\varepsilon }_0/h={({\phi }_{\mathrm{T}}/2)}^{1/2}$. Top inset: example of micrograph used to measure ${\varepsilon }_0^{*}$. Bottom insets: schematics with plug velocity profile. (d) Spatial evolution of ${\varepsilon }_{\mathrm{F}}/{\varepsilon }_0$ and ${\varepsilon }_R/{\varepsilon }_0$ for low ${\alpha }_1$ and large $\mathrm{C}{\mathrm{a}}_0$ with $\left(Q_1,Q_2\right)=\left(5,50\right)$ and $Q_3=125(◯)$, 175 ($\square$), 200 ($▵$) with group M2. Inset: schematic of measurements. (e) Thread buckling modes. From left to right: folding [$\left(Q_1,Q_2,Q_3\right)=\left(20,10,25\right)$, group M2], subfolding [(25, 50, 175), M2]; thread thinning [(20, 10, 125), M2]; dissolving thread [(1, 10, 150), M2], internal folding [(5, 50, 200), D3], folding collapse [(5, 10, 125), D3], and alternating shooting threads, bottom thread (left) followed by top thread (right) [(1, 10, 50), D3]. (f) Pile buckling. From top to bottom: structural buckling [(25, 50, 150), M2], edge buckling [(20, 10, 14), M2], diffusion-induced buckling [(20, 10, 15), D3] and [(20,10,25), D3].