Bubble deformations and segmented flows in corrugated microchannels at large capillary numbers

We experimentally investigate the interaction between individual bubble deformations and collective distortions of segmented flows in nonlinear microfluidic geometries. Using highly viscous carrier fluids, we study the evolution of monodisperse trains of gas bubbles from a square to a smoothly corrugated microchannel characterized with a series of extensions and constrictions along the flow path. The hysteresis in the bubble shape between accelerating and decelerating flow fields is shown to increase with the capillary number. Measurements of instantaneous bubble velocities reveal the presence of a capillary pull that produces a nonmonotonic behavior for the front velocity in accelerating flow regions. Functional relationships are developed for predicting the morphology and dynamics of viscous multiphase flow patterns at the pore scale.

Figure 1

Segmented flows in corrugated microchannels. (a) Schematics of microfluidic module with focusing section where bubbles are regularly generated in a square channel connected to a corrugated channel. (b) Geometrical detail of corrugated channels made with disks of radius $h$ equally disposed along the square microchannel. (c) Examples of micrographs showing distorted arrangements of air bubbles in viscous silicone oil $(\nu ={10}^3$ cS) from dilute flows with small bubbles to concentrated flows with large bubbles: (i) $\mathrm{Ca}\approx 3.8\times {10}^{-1}$, (ii) $\mathrm{Ca}\approx 2.3\times {10}^{-1}$, and (iii) $\mathrm{Ca}\approx 9.9\times {10}^{-1}$.

Figure 2

Morphological evolution of dilute and concentrated segmented flows from straight to corrugated channels. (a) Micrographs of microfluidic multiphase flows with capillary number (i) $\mathrm{Ca}\approx 1.3\times {10}^{-1}$, (ii) $\mathrm{Ca}=1.9\times {10}^{-1}$, and (iii) $\mathrm{Ca}=3.3\times {10}^{-1}$. (b) Temporal evolution of bubbles length $d/h$ and spacing $L$/$h$ at the transition between square and corrugated channels, dashed lines correspond to $x/h=0$ for the reference bubble front. (c) Relationship between normalized droplet spacing $L$/$h$ and length $d/h$ in corrugated channel describing orbitals. (d) Temporal evolution of spacing-to-length ratio $L$/$d$. (e) Relationship between maximal spacing-to-length ratio in corrugation and square channel, solid line: ${\left(L/d\right)}_{\mathrm{MAX}}=0.81{\left(L_0/d_0\right)}^{0.8}$. (f) Evolution of minimum ratio ${\left(L/d\right)}_{\mathrm{MIN}}$ with maximum ratio ${\left(L/d\right)}_{\mathrm{MAX}}$, solid line ${\left(L/d\right)}_{\mathrm{MIN}}=0.15{\left(L/d\right)}_{\mathrm{MAX}}^{1.8}$, dashed line ${\left(L/d\right)}_{\mathrm{MAX}}=0.4$.

Figure 3

Lateral deformation and bubble path in corrugated microchannels. (a) Time series of experimental micrographs showing bubbles passing through a constriction for low $\mathrm{Ca}\approx 3.9\times {10}^{-2}$ with oil viscosity $\nu ={10}^1$ cS (left) and high $\mathrm{Ca}\approx 3.6\times {10}^{-1}$ with oil viscosity $\nu ={10}^3$ cS (right). (b) Superimposition of bubble contour in a pore for low $\mathrm{Ca}\approx 4.7\times {10}^{-2}\left(\nu ={10}^1\mathrm{cS},\mathrm{\Delta }t=2.5\times {10}^{-3}\mathrm{s}\right)$ (top), and high $\mathrm{Ca}\approx 7.1\times {10}^{-1}(\nu ={10}^3$ cS, $\mathrm{\Delta }t=2.5\times {10}^{-3}\mathrm{s})$ (bottom). (c) Contour superimposition of front (top) and rear (bottom) bubble cap in a constriction, $\mathrm{Ca}=7.1\times {10}^{-1},\mathrm{\Delta }t=5\times {10}^{-4}\mathrm{s}$. (d) Spatial evolution of lateral bubble amplitude $w$ for various bubble sizes, (i) low Ca ranging between $2.7\times {10}^{-2}$ and $7.7\times {10}^{-2}(\nu ={10}^1$ cS), (ii) high Ca ranging between $2.9\times {10}^{-1}$ and $7.1\times {10}^{-1}(\nu ={10}^3$ cS).

Figure 4

Characterization of shape hysteresis of convected bubbles in corrugated channels. (a) Spatial evolution of bubble aspect ratio $d/w$ as a function of channel width $W_{\mathrm{C}}/h$ for low and large capillary number Ca over a spatial period λ. (b) Influence of Ca on normalized hysteretic area $A_{\mathrm{H}}$ for low-viscosity $\nu =10$ cS (□) and high-viscosity $\nu ={10}^3$ cS (○) fluids, solid line: $A_{\mathrm{H}}=29\mathrm{C}{\mathrm{a}}^{1/2}$. (c) Evolution of average bubble roundness $\langle R\rangle$ as a function of mean bubble cross-sectional area $\langle A_{\mathrm{A}}\rangle$; low-viscosity oil $\nu =10$ cS (□) with solid line $\langle R\rangle =1+0.1\langle {\mathrm{A}}_{\mathrm{A}}\rangle$ and high-viscosity oil $\nu ={10}^3$ cS (○) with solid line $\langle R\rangle =1+0.2\langle {\mathrm{A}}_{\mathrm{A}}\rangle$; dashed lines $\langle R\rangle =1$ (circle), 1.27 (square), and 1.65 (equilateral triangle).

Figure 5

Lateral and longitudinal amplitudes of bubble deformation at large Ca in corrugated, high-viscosity oil $\nu ={10}^3$ cS. (a) Evolution of maximal width $w_{\mathrm{MAX}}$ as a function of length $d_{\mathrm{MAX}}$ for $\nu ={10}^3$ cS from bubbly to segmented flows. Gas: air $(\circ )$ or $\mathrm{C}{\mathrm{O}}_2(\diamond )$, solid black line: $w_{\mathrm{MAX}}=d_{\mathrm{MAX}}$, dashed line: $w_{\mathrm{MAX}}/h=0.55\left(d_{\mathrm{MAX}}/h–1\right)+1$. (b) Micrographs of maximal deformation, from (i) to (iv): dilute segmented flow $\left(d_{\mathrm{MAX}},w_{\mathrm{MAX}}\right)$; from (v) to (vii): packing-induced arrangement of bubble produce deformed oblate shapes. (c) Examples of flows at the corrugation scale.

Figure 6

Bubble velocity variation. (a) Example of evolution of front and rear bubble velocities $V_{\mathrm{F}}$ and $V_{\mathrm{R}}$ in a series of pores. (b) Mean of maximal velocities is identical to initial velocity, $V_{\mathrm{B}}\approx V_0$ for a range of bubble size $d_{\mathrm{Max}}$. (c) Comparison of $V_{\mathrm{B}}$ and $V_{\mathrm{A}}$ for various bubble sizes, solid line: $V_{\mathrm{B}}/V_{\mathrm{A}}=0.5$. (d) Evolution of minimal rear bubble velocity $V_{\mathrm{R},\mathrm{Min}}$ as a function of maximal lateral deformation $w_{\mathrm{Max}}$. (e) Effect of bubble size of front and rear velocity evolution, Ca ranges between $2.3\times {10}^{-1}$ and $3.8\times {10}^{-1}$.