Figure 1

Schematic diagram of plugs moving in a meandering microchannel. (a) Plugs moving in a meandering microchannel. (b) Flow patterns in plugs at different stages of the meandering microchannel. Vortices of different sizes are formed near the wall of the microchannel. The vortex near the inner wall (with radius $R_i$) is smaller, while the vortex near the outer wall (with radius $R_o$) is larger. (c) Simplified rectangular-shaped plugs with sliding walls in a frame of reference moving with the plugs. In stage I, as shown on the left, the sliding velocities on the top and bottom walls are, respectively, $-\omega R_i$ and $-\omega R_o$. When the plug arrives at a location with a different curvature (stage II), the sliding velocities on the top and bottom walls exchange. (d) A plug train refers to a series of $N$ consecutive different plugs which repeat periodically.Reuse & Permissions

Figure 2

Cartesian coordinate systems for a plug train moving in a microchannel.Reuse & Permissions

Figure 3

Flow field in an ethanol plug of an ethanol/air plug train moving in a microchannel with $\widehat{L}$ $=$1.57: (a) ${\widehat{u}}_x$ from the 2D model. (b) $u_x$ from $\mu$PIV [52]. (Reprinted with permission from [52]. Copyright 2005 American Chemical Society.) (c) ${\widehat{u}}_y$ from the 2D model. (d) $u_y$ from $\mu$PIV [52]. (Reprinted with permission from [52]. Copyright 2005 American Chemical Society.) (e) Dimensionless velocity vectors from the 2D model. (f) Velocity vectors from $\mu$PIV [52]. (Reprinted with permission from [52]. Copyright 2005 American Chemical Society.) (g) Contours of dimensionless stream function $\widehat{\varphi }$ from the 2D model with the increment between two successive streamlines $\Delta \widehat{\varphi }=0.05$. The crosses and the corresponding values are the locations and the magnitudes of the maximum/minimum of the dimensionless stream function.Reuse & Permissions

Figure 4

Mechanisms to enhance mixing in a plug moving in meandering microchannels: stretching of liquid filaments by vortices, folding of liquid filaments by changing channel curvature, and diffusion. The dimensionless plug length is $\widehat{L}=2$, the radius ratio of the microchannel is $\widehat{R}=0.7$, and the viscosity ratio is ${\mu }_1/{\mu }_2=1$. (a) Stretching of liquid filaments in a liquid plug as it is moving in a microchannel. (b) Folding of liquid filaments by alternation of the direction of the channel curvature. The scattering of tracing particles at different times ($\widehat{t}=0$, 17.8, 35.6, 53.4) illustrates the folding effect. The corresponding locations of the plug in the meandering microchannel are, respectively, 0, 0.5, 1, 1.5, and 2 periods. (c) Poincaré section for a plug flowing in a meandering microchannel (for 1000 periods with 20 particles). This Poincaré section was generated by releasing tracer particles into the flow and recording their positions after each period of the meandering microchannel. (d) Scattering of tracer particles by the diffusion effect, which accelerates the mixing process. The Peclet number $\mathrm{Pe}={10}^4$.Reuse & Permissions

Figure 5

Mixing process in a typical liquid plug under the effects of stretching, folding, and diffusion. The dashed horizontal line ${\eta }_{\mathrm{asymp}}=0.00707$ indicates the homogeneous state with randomly distributed particles. The radius ratio of the microchannel is $\widehat{R}=0.7$, the Peclet number is $\mathrm{Pe}={10}^4$, the viscosity ratio is ${\mu }_1/{\mu }_2=1$, and the dimensionless plug length is $\widehat{L}=2$.Reuse & Permissions

Figure 6

Effect of curvature on the mixing process in plugs. The Peclet number is $\mathrm{Pe}={10}^4$, the viscosity ratio is ${\mu }_1/{\mu }_2=1$, and the dimensionless plug length is $\widehat{L}=2$. (a) Variation of mixing indices with time for microchannels of different curvatures. The dashed horizontal line ${\eta }_{\mathrm{asymp}}=0.00707$ indicates the homogeneous state with randomly distributed particles. (b) Flow pattern in plugs moving in meandering microchannels with different curvatures. Solid (black) contours indicate streamlines at stage I, while dashed (red) contours indicate streamlines at stage II. (c) Visualization of mixing patterns at $\widehat{t}=100$ for microchannels with different curvatures.Reuse & Permissions

Figure 7

Effect of the Peclet number on mixing process in plugs. The radius ratio of the meandering microchannel is $\widehat{R}=0.7$, the viscosity ratio is ${\mu }_1/{\mu }_2=1$, and the dimensionless plug length is $\widehat{L}=2$. (a) Variation of mixing indices with time for different Peclet numbers. The dashed horizontal line ${\eta }_{\mathrm{asymp}}=0.00707$ indicates the homogeneous state with randomly distributed particles. (b) Visualization of mixing patterns at $\widehat{t}=100$ for different Peclet numbers.Reuse & Permissions

Figure 8

Effect of the viscosity ratio on the mixing process in plugs. The radius ratio of the meandering microchannel is $\widehat{R}=0.7$, the Peclet number is $\mathrm{Pe}={10}^4$, and the dimensionless plug length is $\widehat{L}=2$. (a) Variation of mixing indices with time in plugs of different viscosities. The dashed horizontal line ${\eta }_{\mathrm{asymp}}=0.00707$ indicates the homogeneous state with randomly distributed particles. Inset: Variation of mixing indices with viscosity ratio at $\widehat{t}=100$. (b) Flow pattern in plugs for fluids of different viscosities; secondary vortices are formed when the viscosity ratio ${\mu }_1/{\mu }_2$ is low. (c) Visualization of mixing patterns at $\widehat{t}=100$ for fluids of different viscosities.Reuse & Permissions

Figure 9

Flow diagrams of plug trains with different viscosity ratios: (a) distortion of vortex pattern by shear stress at the interface ${\tau }_{xy,2}$; (b) formation of secondary vortices when the shear stress is sufficiently high due to the low viscosity ratio ${\mu }_1/{\mu }_2$.Reuse & Permissions

Figure 10

Effect of dimensionless plug length on mixing process in plugs. The radius ratio of the meandering microchannel is $\widehat{R}=0.7$, the Peclet number is $Pe={10}^4$, and the viscosity ratio is ${\mu }_1/{\mu }_2=1$. (a) Variation of mixing indices with time in plugs with different $\widehat{L}$. The dashed horizontal line ${\eta }_{\mathrm{asymp}}=0.00707$ indicates the homogeneous state with randomly distributed particles. Inset: The variation of mixing indices with $\widehat{L}$ at $\widehat{t}=100$. (b) Visualization of mixing patterns at $\widehat{t}=100$ for plugs with different $\widehat{L}$. (c) Flow pattern in plugs of different $\widehat{L}$. (d) The flow in a plug can be divided into three regions, the front region, the middle region, and the rear region. The velocity profiles at three typical cross sections, $\widehat{x}=0$, $\widehat{x}=1$, and $\widehat{x}=1.5$, are plotted. In the middle region (such as $\widehat{x}=0$), the velocity profile is parabolic, while in the front/rear regions (such as $\widehat{x}=1.5$), the velocity profile is flat in a large part of the transverse direction ($-0.5<\widehat{y}<0.5$).Reuse & Permissions

Figure 11

Matching of the velocity components ($u_y$) and the shear stresses (${\tau }_{xy}$) at the interface of two consecutive plugs [plug $i$ and plug ($i$+1), where $1\le i<N$].Reuse & Permissions

Figure 12

Schematic diagram of a 3D liquid plug moving in a 3D microchannel. The width of the microchannel is $w$. The depth of the microchannel is $d$. The plug length is $L$. The middle depth plane $z=0$ is indicated.Reuse & Permissions

Figure 13

Effect of the channel depth on the flow pattern in plugs. The width of the microchannel is $w$, the depth of the channel $d$ is varied from $w$ to $8w$, and the length of the plug is $L=2w$. The Reynolds number $\mathrm{Re}=\frac{\rho Vw}{\mu }={10}^{-2}$. (a) Contours of velocity component $u_x/V$ in the middle depth plane $z=0$ as indicated in Fig. 12. (b) Contours of velocity component $u_y/V$ in the middle depth plane $z=0$. (c) Profiles of $u_x/V$ along X-X indicated in Fig. 12.Reuse & Permissions