This study is composed of four experiments with a meandering square microchannel. The first experiment was intended to measure the liquid velocity within the area of interest in . The area was scanned by the two-axis scanner shown in . The y-component of the velocity field vy(X, Y, Z) is shown in .
Each horizontal row in has a different flow rate shown in . The Reynolds number Re is defined by Re = VW/v for each row was 11, 22, and 33, respectively, where V is the cross-sectional average of the primary velocity, W is the channel width, and v is the kinematic viscosity. Top views (the XZ-plane) from the second to the fourth column in were reconstructed from volume images. The second column  shows a plane at 25% of the channel depth apart from the top, and the third , 50% (symmetric plane), and the fourth , 75%. The locations for five cross-sectional images (the XY-plane) in the first column  were indicated by the lines in the second column. Each inset in the third column is a side view (the YZ-plane) of the location indicated by the line right below where the effect of the primary flow caused by −1.5° tilting with respect to the X-axis vanishes (see the Methods section). The curvature ratio W/R varied periodically with large amplitude (from −35 to 35%) along the streamwise direction where R is the radius of curvature. The maximum Dean number for each row is 6.53, 13.07, and 19.6, respectively. The Dean number is defined by Dn = Re√(W/R).
As shown in the insets in , the measured velocity field presented a pair of counter-rotating vortices (Dean cells), one in the upper half and the other in the lower half, as indicated above the inset in . The inset in is divided into four areas which are separated by the black color, i.e. vy
= 0. Each area has circular color patterns. The outermost color in an area indicates the direction of vy
across the area. For example, the outermost color in the upper left area is blue which means that the direction of vy
is upward (negative vy
). The red color at the center of the upper left area does not mean positive vy
but more negative vy
because there is an abrupt color change from blue to red. A strong negative velocity caused an aliasing effect or a phase wrapping. Finally, we did the same analysis with changing the relative orientation between the incident light and the microchannel to verify the structure of the Dean cells. The centrifugal force exerted at each fluid element generated the vortices. Hence, the secondary flow moved from the inside to the outside side wall along the horizontal symmetric line. Since the curvature was alternating, the rotational direction of the vortices was also alternating, as shown by comparing to (m). The first order of Dean’s solution12,13
for secondary flow velocity in a circular tube states that the maximum velocity component normal to the interface of the Dean cells is (0.0096Dn2
< 24. The measured values in the present study showed good agreement with it. It is shown in that the secondary flow was developing because of alternating curvature. The length required for full development increased with increasing flow rate, as indicated by the three lines in the middle of , (n), and (v). As mentioned before, depict the y
-component of the velocity field in the symmetric plane. In this symmetric plane, the effect of the primary flow dominates because the secondary flow is almost normal to the incident light [at the positions, for instance, marked by
in , the secondary flow is exactly normal to the incident light] and its magnitude is much smaller than the primary flow. The primary velocity profile deviated from the Poiseuille flow and the location, where maximum velocity is, moved from the center toward the outside side wall.
The second experiment was conducted to visualize the mixing pattern, as shown in . The same image-producing method as the first experiment was used. The Reynolds number for each row in Figs. (a–h) and (i–p) is 11 and 33, the Dean number is 6.53 and 19.6, and the Péclet number Pe = VW/D is 1.26 × 107 and 3.79 × 107, respectively, where D is the molecular diffusivity. The Pe´clet number is the ratio of convection to molecular diffusion. Convection dominates mixing phenomena in this condition. The mixing patterns in each row were imaged at the location labeled by the marks in . The mixing patterns were aligned to have the same direction of the primary flow [the label locations in correspond to the left side of the mixing patterns in ]. The different colors of the figure frames denote different curvature.
It is clearly shown that vertical interfaces  at the confluence were distorted and evolved by the counter-rotating vortices with alternating rotation. For instance, the interface intersecting with the symmetric plane moved back and forth. When comparing with (f), the interface located at the center of the channel proceeded to the outer channel wall [indicated by the arrow in ] by 25% of the channel width. The proceeded distance calculated with the first order of Dean’s solution is 20.8%. A movie clip for the volume-rendered mixing pattern is shown in Video 1 in the ESI†
= 11). Even though the secondary flow is exactly reversed when the curvature is reversed, the interface intersecting with the symmetric plane does not oscillate around the channel center. It is supposed that the biased oscillation is caused by the initial condition imposed before the first turn. When one sees the mixing from the top over the channel, it looks as if the two liquids are completely mixed. Alternately curving a channel, however, was not enough to make a complete mixing even at the end of the channel.
The third and fourth experiments were designed to enhance two-liquid mixing and to visualize the enhancement mechanism. The hypothesis was that injecting air into two liquids enhances mixing. By injecting air, a bubble-train flow30–32
was sustained in a quasi-steady-state. Through the third experiment, the velocity field, which governs a mixing phenomenon, was imaged. The velocity field of bubble-train flow in the meandering microchannel is characterized by the Dean number Dnj
based on the overall superficial velocity j
] and the capillary number Ca
. Those are defined as Dnj
) and Ca
/σ, respectively, where Ql
are the liquid and gas flow rates, respectively, A is the cross-sectional area, μ is liquid viscosity, σ is interfacial tension, and Ub
is bubble velocity. The bubble velocity Ub
and, therefore, Ca
can be estimated from the overall superficial velocity.31
The Dean number and the capillary number were about 5.4 and 2.7 × 10−4
, respectively, under the present experimental conditions. Since the flow has rapid, transient phenomena and the camera frame rate (1/T
) is limited, it was impossible to acquire an instantaneous volume image. Therefore, the probe beam of SDDOCT was fixed at the locations indicated by the blue and red dots in without B
- and C-scans. Continuous camera exposure constitutes depth versus
time images like in . The three blue dots, which are indicated by the numbers 1–3 in , evenly divide the channel width into quarters. were taken at the lower, center, and upper blue dots, respectively. The red dot indicated by the number 4 is located 20 µm away from the outside side wall where was taken.
Fig. 5 Depth versus time images showing liquid velocities projected on the incident light vy(y, t). (a) Schematic diagram illustrating the three-dimensional flow field which gives rise to mixing enhancement. (b–d) Liquid slug in bubble-train that flows (more ...)
In the liquid region far from the bubbles in , the secondary flow caused by the curvature was also observed. The maximum primary liquid velocity around the center of the cross-section far from the bubbles is about 1.8 times higher31,32
than the bubble velocity with a capillary number of 2.7 × 10−4
. Therefore, a toroidal vortex per liquid slug was generated before and after the bubbles and made a longitudinal recirculation flow when this flow was seen in a coordinate system moving with the bubbles. The longitudinal recirculation adds a complexity to the transversal recirculation by the secondary flow to be the key mechanism for mixing enhancement. is a schematic diagram for the three-dimensional flow field which gives rise to the mixing enhancement. Strong vertical (along the y
-axis) liquid motions resulting from the toroidal vortices were detected as shown in . On the other hand, there is a bypass liquid flow at the corners of the square channel from the front liquid slug to the back because the cross-sectional average of primary liquid velocity is about 0.8 of bubble velocity.31,32
The bypass flow is imaged in with the longitudinal recirculation flow and can be observed only at the corners in this capillary number regime.
In order to visualize the enhanced mixing, the unmixed and vertically arranged two liquids were pumped into inlet 1 using the configuration A shown in . At the same blue dots, depth versus
time images were recorded with the same gas and liquid flow rates and are shown in . The Péclet number Pej
based on the overall superficial velocity was 1.04 × 107
in this condition. At a liquid slug far from the bubbles, the primary velocity profile made fluid elements disperse parabolically and this axial dispersion was perturbed by the alternating secondary dispersion shown in . In addition, longitudinal recirculation enhanced radial mixing near the bubbles. This transient mixing was also imaged at the 16 cross-sections along the lines of interest in . The transient mixing patterns at the 5th, 6th, 11th, 12th, 15th, and 16th cross-sections are presented in and Videos 2–4 in the ESI†
to compare with . Mixing performance was greatly enhanced. It was supposed that a complete mixing was done by the 15th cross-section with the current resolution limit (6.3 µm). As a matter of fact, the thickness of liquid layers (approx. initial thickness × 0.5logPe
, where log Pe
is the number of folding for a complete mixing) can be as small as a micron right before a complete mixing occurs when the Péclet number is high. If we have a ultra broadband light source with a shorter center wavelength, a submicron imaging is possible with SDDOCT. It is nevertheless difficult to measure concentration variation over a cross-section in order to determine a mixing uniformity.
Depth versus time images showing an enhanced mixing pattern of water and polystyrene suspension within a liquid slug of bubble-train flow. Each image was taken at a different point of interest shown in .
Fig. 7 Transient mixing patterns enhanced by the bubble-train flow were imaged at the 5th, 6th, 11th, 12th, 15th, and 16th cross-sections (see Videos 2–4 in the ESI†). It was supposed that a complete mixing was done by the 15th cross-section (more ...)
The characteristic time for the longitudinal recirculation is defined as the time for the liquid to move from one end of the liquid slug to the other end. With the current capillary number, the longitudinal recirculation time is 3lls
is the length of the liquid slug.32
The characteristic time for the alternating secondary flow is defined as the time for the liquid slug to go through one period of the meandering microchannel, lp
is length of one period of the microchannel. Therefore, 3lls
means how many perturbations one can expect during the time for the liquid to move from one end of the liquid slug to the other end. In other words, lp
means how many foldings by longitudinal recirculation one can expect during the time for the liquid slug to turn a period of the meandering microchannel. In the case of , the number lp
was 1.18 and the total number of folding for the entire channel was about 8 which is almost equal to the number of folding for a complete mixing, log Pej
The performance of two-liquid mixing is sensitive to arrangements of two liquids at the inlet. If the interface between the two liquids is oriented horizontally at inlet 1 using the configuration B shown in , the mixing will not be efficient because the perturbation caused by the alternating Dean cells will be of no use.