1 Introduction

Micromixers are widely used for different applications such as lab-on-a-chip and micro-total analysis systems [1]. For the laminar flow in smooth-walled microchannels, the Reynolds number (Re= Vl/ν, where V is the average velocity, l is characteristic length and ν is the kinematic viscosity of the fluid) is low (Re < 100) due to small dimensions of channels and low fluid flow velocity. Hence, the micromixing process is limited to molecular diffusion [2]. However, in the turbulent flow, in which the flow velocity is very high, the mixing quality is affected by convection and turbulent diffusion [3]. Generally, micromixers are divided into active and passive mixers. Active micromixers require external energy sources such as thermal field [4], acoustic field [5], electric field [6, 7], pressure field [8] and magnetic field [9] to enhance the mixing index. In contrast to active micromixers, passive ones do not use any external actuator to drive the fluids. Values of the Peclet number (Pe = Vl/D, where D is the molecular diffusivity) are relatively high (Pe > 100) for passive micromixers. Since the Peclet number is the ratio of mass transfer due to convection and that due to diffusion, the mixing length is in order of centimeter [10]. Thus, one of the most important advantages of lab-on-a-chip devices, i.e. miniaturization, cannot be employed. Therefore, many researchers have performed numerical and experimental investigations to overcome this limitation. They usually used complex geometries to generate chaotic advection and enhance the contact surface between the fluids to increase molecular diffusion [11]. Thus, lamination and chaotic-advection patterns have been proposed to promote the mixing efficiency. In addition to complicated fabrication of lamination based micromixers, it was found that a high pressure drop occurs along the micromixers. Quantitatively, the ratio of mixing efficiency and pressure drop can be used as a criterion to evaluate the performance of passive micromixers [12]. Hessel et al. [13] reported that the mixing length Lm = Pe × w for laminar flow, where w is the channel width. In addition, multi-lamination designs have been used by the researchers to decrease the mixing length. In multi lamination patterns, the fluids enter the mixer with different arrangements compared to Y-type or T-type patterns. They include multiple flows [14], hydrodynamic focusing [15], interdigitated mixing [16, 17] and cyclone arrangements [18]. Hong et al. [19] introduced a two-dimensional modified Tesla configuration fabricated from cyclic olefin copolymer (COC) using thermal bonding. The increase in the mixing quality of this micromixer was due to chaotic advection due to Coanda effect. A relatively high Reynolds number (Re > 25) was required to create stretching and folding of main streams in chaotic-advection based micromixers [20]. Planar microchannels are simpler strategies to create chaotic advection compared to non-planar ones. Since the present micromixer is a planar one, some of planar patterns are described in more details. Alam et al. [21] studied a baffle based planar micromixer with curved microchannel for a Reynolds number range of 0.1–60. They compared the effect of obstructions on the mixing efficiency and demonstrated that micromixers with circular and hexagonal baffles exhibit the same mixing efficiency at different Reynolds numbers. They proposed a mixer with circular obstructions and achieved the mixing efficiency of 88%. They also showed that the pressure drop is an increasing function of the number of obstructions. Secondary vortices were generated for Re ~ 60 due to channel curvature. Chen et al. [22] investigated a novel Koch fractal micromixer experimentally and numerically. The mixing index of 90% was achieved at four different Reynolds numbers. They concluded that the mixing quality decreases slightly and the pressure drop decreases considerably after rounding corners. For example, the pressure drop was reduced from 21,295 to 12,789 Pa at Re = 100. Hama et al. [23] used reverse-staggered herringbone in a micromixer to promote its mixing performance. They demonstrated that the mixing efficiency of 100% can be obtained after three mixing cycles under different operating conditions. It was revealed that the Reynolds number has a very weak impact on the mixing quality within the first three cycles. Khosravi Parsa and Hormozi [24] investigated the mixing enhancement in a convergent–divergent sinusoidal micromixer and showed that the mixing index decreases with the Reynolds number for 0.2 < Re < 20 due to that the centrifugal forces and the expansion vorices are not generated completely. They found that the mixing performance increases with the Reynolds number for 20 < Re < 75 due to the generation of Dean and expansion vortices when the phase shift is located in the range of π/2 and 3π/4. In convergent–divergent micromixers, the expansion vortices are generated when the cross-sectional area increases abruptly, leading to an increase in the contact surfaces between the fluids. Mondal et al. [25] compared the mixing performance of micromixers with two configurations, called raccoon and serpentine. Their results revealed that the mixing index increases with the wavelength, however, the mixing performance of raccoon one is more appropriate than the serpentine one for given Reynolds numbers and wavelengths. Recently, a comprehensive review on passive and active micromixers was presented by Bayareh et al. [26] in which the effect of different parameters on mixing index has been discussed in details.

In the present work, a relatively simple novel micromixer with convergent–divergent channels is proposed (Fig. 1) and its mixing performance is evaluated numerically and experimentally. The aim is to evaluate the influence of designing parameters, inlet mass flow rate and number of cycles on the mixing quality.

Fig. 1
figure 1

Schematic of the proposed micromixer: L = 100 μm, W = 100 μm, W1 = 1000 μm, W2 = 1200 μm, R = 250 μm, d = 100 μm, L′ = 3200 μm and α = 60o

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2 Numerical simulations

The governing equations for an incompressible, viscous and laminar flow are continuity and Navier–Stokes equations [12]:

$$ \nabla \cdot \varvec{u} = 0 $$
(1)
$$ \rho \left( {\frac{{\partial \varvec{u}}}{\partial t} + \varvec{u} \cdot \nabla \varvec{u}} \right) = - \nabla P + \mu \nabla^{2} \varvec{u} $$
(2)

where \( \varvec{u} \) indicates the velocity vector, p the pressure, ρ the density, and μ the dynamic viscosity.

Also, convection–diffusion equation for species is expressed in the following:

$$ \frac{\partial C}{\partial t} + \varvec{u} \cdot \nabla C = D\nabla C $$
(3)

where C (mol/m3) is the species concentration and D = 10−11 m2/s is molecular diffusivity.

The SIMPLEC algorithm is used to couple the velocity and pressure fields. A second-order upwind scheme is employed to reduce the numerical diffusion. Boundary conditions of uniform velocity and constant pressure (Pg = 0) are used at inlets and outlet of the mixer. No-slip boundary condition is imposed on all walls of the microchannel. The commercial software COMSOL Multiphysics 5.3a is employed to solve the mathematical models. The computational domain is discretized by unstructured triangular meshes (Fig. 2a). The mixing efficiency results at the outlet of the mixer are compared for different grid resolutions. The grid study is performed by varying the discretization level from predefined fine element size (with 317,209 nodes) to extra fine element size (with 8,859,351 nodes). It is found that the further refinement of predefined extra fine element size (with 1,861,862 nodes) showing very little variation in the results (maximum difference is less than 2%) (Fig. 2b). This figure shows the mixing efficiency along the micromixer when the inlet flow rate is 0.2 ml/min (Re = 37). It should be pointed out that the characteristic length to calculate the Reynolds number is hydraulic diameter, 4A/Ƥ, where A and Ƥ are area and perimeter, respectively. Thus, the grid resolution of 1,861,862 is employed for further simulations. The grid resolution is more refined on corner boundaries and in the channel entrance.

Fig. 2
figure 2

a Three-dimensional grid used for numerical simulations, and b mixing efficiency calculated along the micromixer for Re = 37 and different grid resolutions

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Mixing efficiency, ME, is calculated to quantify the performance of the micromixer [7]:

$$ ME = 1 - \frac{{\left[ {\mathop \sum \nolimits_{i = 1}^{N} \frac{{\left( {C_{i} - \bar{C}} \right)^{2} }}{N}} \right]^{1/2} }}{{\bar{C}}} \approx 1 - \frac{{\sigma_{C} }}{{\bar{C}}} $$
(4)

where \( \sigma_{C} \) and \( \bar{C} = \mathop \sum \limits_{i = 1}^{N} C_{i} /N \) are the standard deviation and the average concentration, respectively. N denotes the number of nodes in the considered area. Therefore, ME = 100% indicates a perfect mixing.

3 Fabrication and experimental setup

The proposed micromixer consists of a set of semicircular microchannels (Fig. 1). The radius of the circles is R = 250 μm. The inlet of the micromixer is Y-shaped with the dimensions of 100 μm × 100 μm × 1000 μm. The blockage ratio is defined as ξ = y/d, where y is the vertical distance between the upper semicircles and the end points of lower ones. In the present simulations, y varies between d/8 to d. To fabricate the microchip, polydimethylsioxane (PDMS) is employed using soft lithography technique (for more details see references [6] and [26]). Methyl red with 20 mM concentration and water with blue pigment are injected by a syringe pump from inlets A and B, respectively. The experiments are performed for four flow rates 0.05 (Re = 9), 0.1 (Re = 19), 0.2 (Re = 37) and 0.4 ml/min (Re = 75). The temperature of the laboratory is maintained at 25 °C that is employed to measure its properties for relevant simulations. The mixing process is recorded by a Jiusion 1000× USB digital microscope with 1000× magnification and 2 Megapixel lens. Figure 3 illustrates a picture of the experimental system (For more details see Ref. [7]).

Fig. 3
figure 3

A picture of the experimental system

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4 Results

The mixing length (or mixing time) in passive micromixers is much longer than that in active ones to achieve reasonable mixing quality. In the case of convergent–divergent mixers, several cycles are required to obtain high mixing efficiency. For example, Parsa and Hormozi [24] obtained the mixing efficiency of 60% to about 100% for three to nine mixing cycles of a divergent–convergent sinusoidal micromixer at Re = 0.2–75. Here, a novel micromixer is proposed to reach higher mixing index using less mixing cycles.

4.1 Validation

Since the present investigation involves numerical results, they are compared with the present experimental results qualitatively and the numerical ones of Parsa and Hormozi [24] who considered the mixing process in a convergent–divergent sinusoidal micromixer quantitatively. Figure 4 presents the mixing efficiency at different cross-sections of the micromixer. The figure demonstrates that the present results are in excellent agreement with those reported by Parsa and Hormozi [24].

Fig. 4
figure 4

Mixing efficiency as a function of Reynolds number for a sinusoidal microchannel with phase shift of π and wave length to amplitude ratio of 3.3

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4.2 Effect of inlet velocity

Inlet velocity is a significant parameter to describe the flow characteristics. Concentration distribution along the micromixer is presented in Fig. 5 for different Reynolds numbers. This figure shows that as the Reynolds number increases, the mixing index improves qualitatively.

Fig. 5
figure 5

Distribution of concentration along the micromixer at different Reynolds numbers

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Figure 6 shows the mixing efficiency along the microchannel for different Reynolds numbers. The mixing efficiency is calculated for the micromixer with four cycles. It can be seen that the mixing quality increases by increasing the Reynolds number (or injecting velocity). In other words, there is not an optimum value for injection velocity. The maximum value of mixing efficiency (ME = 98%) occurs when the volume flow rate is 0.4 ml/min (Re = 75). Present numerical simulations are performed for two cases of time dependent and stationary (steady state). Time dependent results are used for verification of the experimental data. It is worth noting that most of the numerical simulations presented by previous researchers have been assumed to be steady. Hence, in their simulations, the value of mixing index is related to the situation in which the calculations do not change. Thus, their reported mixing efficiency is higher compared to time dependent investigations. It is confirmed in the present simulations. For example, at Re = 37, mixing efficiency reaches about 78% for the mixing time of 480 s after four mixing cycles, while mixing efficiency is about 90% for the same conditions in stationary simulations. Generally, due to the designed geometry of the present micromixer, irregularities cause the fluids to collide in each cycle and the diffusion increases with the number of cycles. The molecular permeation phenomenon due to layered mixing and irregular displacement leads to well mixing at low Reynolds numbers. The basic idea in designing this micromixer is the variation of microchannel cross section, folding, stretching, bending and breaking the fluid flows. In the throttling areas and the inlets, two streams experience successive displacements in the sequence of convergences and divergences. These longitudinal layer displacements allow for high mixing of two fluids. The two fluids move in layers, and as the concentration distribution shows, the mixing rate increases with each curvature of the microchannel. Another factor in this mixer is the irregular displacements, leading to well performance of micromixer at low Reynolds numbers. This factor results in an increase in the fluid retention in the microchannel at low Reynolds numbers due to an enhancement in the contact surface of the two fluids. The streamlines follow the convergence and divergence of the microchannel. The successive variation of cross section along the microchannel, especially in the curved portions, put the two fluid layers in contact.

Fig. 6
figure 6

Mixing efficiency versus time for different volume flow rates calculated after the second mixing cycle

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Generation of expansion and Dean vortices is now evaluated by considering the streamlines along the micromixer (Fig. 7) and concentration contours perpendicular to the cross sections (Fig. 8). Most of the researchers, e.g. Parsa and Hormozi [24], used the same approach to determine the role of expansion and Dean vortices in the sinusoidal microchannels with different phase shifts. Figure 7 shows that, for Re = 9, expansion vortices are not generated due to very low inflow velocity. It is observed that weak expansion vortices are formed after the throat area for Re = 19. It is revealed that as the Reynolds number increases, the expansion flow becomes stronger and the subsequent vortices are greater in size, leading to higher mixing quality. Thus, the generation of expansion vortices is an effective parameter to enhance the mixing quality in the present micromixer. Concentration contours are presented in Fig. 8 in some vertical planes on the traverse cross sections along the micromixer for two Reynolds numbers, Re = 9 and 75. The figure demonstrates that the Dean vortices are not generated due to the periodical changes in the channel direction.

Fig. 7
figure 7

Streamlines on horizontal mid-plane for different Reynolds numbers

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Fig. 8
figure 8

Concentration contours perpendicular to the cross section of microchannel for two Reynolds numbers

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On the other hand, due to the proximity of the upper and lower arcs (small aspect ratio), Dean vortices do not generated in this micromixer. The centrifugal forces near the upper and lower walls of the microchannel are much less than that in its center. Since the aspect ratio is low, the upper and lower walls of the microchannel are close together, preventing the formation of Dean vortices, which is why this micromixer is suitable for low-Reynolds-number flows (Re < 75). Figure 9 compares the distribution of concentration in a horizontal plane obtained from time-dependent numerical simulations and experiments for Re = 9. It is observed that there is a reasonable agreement between the numerical and experimental results. Experimental images confirm that the mixing quality increases as the number of mixing cycles increase. In addition, it enhances with the time.

Fig. 9
figure 9

Comparison between the images obtained from numerical simulation (left) and the experiments (right) for inlet Reynolds number of 9 and the time of: a 240 s, b 300 s, c 360 s and d 434 s

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4.3 Effect of number of cycles

Figure 10 shows the concentration distribution for four micromixers with three to six mixing cycles. Due to the simplicity of the design of this micromixer, the mixing rate can be increased to an acceptable magnitude at low Reynolds numbers by increasing the mixing cycles. As expected, the mixing rate increases as the number of cycles enhances. In other words, the curves and irregularities in the channel lead to an increase in the mixing quality. Figure 11 presents the mixing efficiency for different number of cycles. The results show that the mixing efficiency for the micromixer with three and six mixing cycle reaches 77% and 98%, respectively. Going through the irregular path successively allows the fluids to get high mixing due to molecular diffusivity. It was demonstrated in Sect. 4.2 that the formation of secondary flow is the main cause of mixing for Re > 19. Hence, the mixing index is relatively high even for three mixing cycle. Motion of two fluids through the curved surfaces reduces the pressure on the walls and then in the center of the microchannel, resulting in the fluid being drawn to these areas and creating curved streamlines. Micromixers with this structure will usually work well at Reynolds numbers from 5 to 75. As mentioned in the introduction, the ratio of mixing efficiency and pressure drop can be used as a criterion to evaluate the performance of passive micromixers [12]. We expect that the pressure drop increases with the number of mixing cycles. Figure 12 shows the pressure drop as a function of number of cycles. This figure demonstrates that the pressure drop increases considerably with the cycles. To evaluate the performance of micromixer considering the number of mixing cycles, the ratio of mixing efficiency to the pressure drop is compared. The value of ME/ΔP (Pa−1) for the micromixer with three, four, five and six mixing cycles is 6.36, 2.23, 0.74, and 0.5, respectively. Hence, ME/ΔP decreases with the number of mixing cycles. Therefore, even though the mixing quality improves with the number of cycles, the pressure drop affects the performance of the micromixer. Cheri et al. [12] reported that ME/ΔP is in order of 0.01 for a baffled based micromixer. Parsa and Hormozi [24] obtained the maximum value of ME/ΔP at different Reynolds numbers for convergence–divergence sinusoidal micromixer. For example, their results showed that ME/ΔP = 0.002 at Re = 5. They showed that the pressure drop is an increasing function of the Reynolds number. It can be concluded that the proposed micromixer is much more efficient compared to those proposed by Cheri et al. [12] and Parsa and Hormozi [24] who reported the values of pressure drop and mixing efficiency. It should be pointed out that the mixing efficiency of the present micromixer is as high as those proposed by Cheri et al. [12] and Parsa and Hormozi [24].

Fig. 10
figure 10

Concentration distribution along the micromixer for different number of cycles for inlet Reynolds number of 37

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Fig. 11
figure 11

Mixing efficiency along the micromixer for different numbers of mixing cycles and inlet Reynolds number of 37

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Fig. 12
figure 12

Pressure drop for different numbers of mixing cycles and inlet Reynolds number of 37

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4.4 Effect of blockage ratio

The geometry of the proposed micromixer is such that the cross-section of the throats can be changed. It is expected that as the throat cross section decreases, the mixing quality increases significantly. However, it should be pointed out that the amount of pressure drop will also increase. Hence, it is necessary to evaluate the magnitude of ME/ΔP for different blockage ratios. Distribution of concentration along the micromixer is plotted in Fig. 13 for blockage ratios of 0.125 and 1. This figure qualitatively shows that as the cross-section of the throat decreases, the mixing performance of the fluid enhances. This is due to that the impact of molecular diffusion increases as the layer of fluids become closer to each other. In addition, stronger expansion vortices are formed for lower cross-section of the throat (Fig. 14). Figure 14 also demonstrates that the Dean vortices are not formed for the lowest blockage ratio. Sequence of convergences and divergences leads to that the penetration of fluids increases when the cross-sectional area of the throats decreases more sharply. Figure 15 confirms that the mixing efficiency increases by about 100% when the blockage ratio is 0.25. It is worth noting that the mixing efficiency reaches 100% for both cases of ξ = 0.125 and 0.25. The pressure drop is plotted in Fig. 16 for different blockage ratios. Considering Figs. 15 and 16, ME/ΔP can be calculated. It is found that it is 0.044, 0.252, 2.34, and 202 Pa−1 for ξ = 0.125, 0.25, 0.5, and 1, respectively. Since, the efficiency of the micromixer with the blockage ratio of 0.25 is about 100%, it is an excellent mixer for practical applications.

Fig. 13
figure 13

Concentration distribution along the micromixer for different blockage ratios for inlet Reynolds number of 37

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Fig. 14
figure 14

Concentration contours perpendicular to the cross section of microchannel for two blockage ratios and Re = 37

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Fig. 15
figure 15

Mixing efficiency along the micromixer for different blockage ratios

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Fig. 16
figure 16

Pressure drop along the micromixer for different blockage ratios

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5 Conclusions

The present paper studied numerically and experimentally a convergence–divergence based micromixer to evaluate its mixing characteristics. The influence of inlet velocity, number of mixing cycles and blockage ratio on the mixing performance of the micromixer was investigated. Dean vortices were not formed due to low aspect ratio of the proposed micromixer. However, generation of expansion vortices was the main factor of mixing for Re > 19. This micromixer is suitable for low-Reynolds-number flows (Re < 75). It was demonstrated that there is not an optimum value for injection velocity of fluids. As the Reynolds number increases, the mixing index increases. The results showed that the mixing rate increases as the number of cycles enhances. ME/ΔP (Pa−1) was calculated to evaluate the performance of micromixer. The value of ME/ΔP for the micromixer with three, four, five and six mixing cycles was 6.36, 2.23, 0.74, and 0.5, respectively. In addition, the effect of blockage ratio on mixing index was investigated. It was revealed that as the blockage ratio increases, the expansion vortices become stronger, leading to an increase in the mixing quality. The results showed that the efficiency of the micromixer with the blockage ratio of 0.25 is about 100%. Since its ME/ΔP value is 0.252, it is an excellent mixer for practical applications.