1 Introduction

In recent years, with the development of microprocessing technology, lab-on-a-chip techniques are increasingly used in bioengineering, chemical engineering and medical testing (Nestorova et al. 2017; Rahman et al. 2015; Shah et al. 2014; Rasponi et al. 2015; Hellé et al. 2015). As a microdevice, lab-on-a-chip compared with traditional experiments has its unique advantages (Jiang et al. 2011; Chen et al. 2013), it allows miniaturization and integration of microdevices into a system for a wide range of applications (Puttaraksa et al. 2016). The cost of design and manufacturing is very low (Humayun et al. 2017), and only little sample is needed to complete the experiment. It is precisely because of these advantages, lab-on-a-chip attracts more and more attention, and received a lot of research results.

As an important part of microfluidic chips, the mixing efficiency of the micromixer plays a decisive role in the ease of application of the microfluidic chip, the accuracy of the analysis and the efficiency of the analysis. Mixing efficiency is a basic phenomenon involved in instruments such as DNA molecular detection devices. Well-designed micromixers are critical to miniaturization of microfluidic chips (Fan et al. 2017). As mentioned by Suh et al., “in microfluidic applications, mixing has been understood as one of the most fundamental and difficult-to-achieve issues” (Suh and Kang 2010). Micromixers are usually divided into two categories, active and passive. Active micromixers can effectively mix different concentrations of samples with external forces such as electric fields, magnetic fields, mechanical agitation, etc. (Beskok 2008; Wu and Yang 2006; Cao et al. 2015; Fu et al. 2010; Chen et al. 2014). It can improve the mixing efficiency of the mixer, and shorten the reaction time. But, with the addition of power source, the structure of the active mixer becomes more and more complex (Zhou et al. 2015). On the contrary, due to the simple manufacture, ease of control, and low cost of the passive mixer is concern (Tran-Minh et al. 2014). In comparison to the active mixer a passive mixer can achieve a better mixing effect by just changing the structure of the microchannel. Hossain and Kim (2016) studied the parametric investigation on flow structure and mixing in a micromixer with two-layer crossing channels which was reported by Xia et al. (2005). A novel micromixer with gaps and baffles is proposed by Xia et al. (2016), the enhanced mass transfer results from the better synergy between the velocity vector and concentration gradient caused by the combination of gaps and baffles. Cortes-Quiroz et al. (2009) applied the optimization of the geometry of the staggered herringbone micromixer (SHM) to maximize mixing index and minimize pressure drop. A planar rhombic micromixer with a converging–diverging element has been systematically investigated by Chung et al. (2008), the planar rhombic micromixer improves the performance at a smaller footprint and lower Reynolds number. Shih et al. studied a high-efficiency planar micromixer with convection and diffusion mixing over a wide Reynolds number range (Shih and Chung 2008), it achieves nearly uniform mixing in this adaptive-design mixer at Reynold (Re) = 0.1 and Re ≥ 20. The combination of patterning the channel bed and splitting and recombining the streams can be used to make controlled lamination in a 2D channel system (Tofteberg et al. 2010). A numerical study of an electrothermal vortex enhanced micromixer has been done by Cao et al. (2008) using vortices induced by the electrothermal effect to enhance the mixing efficiency. By changing the structure of the microchannel, it is possible to increase the contact area and enhance the chaotic convection of the fluid (Du et al. 2010). Numerical simulations have great potential in the design optimization of micromixers. A series of studies on design (Chen et al. 2012; Chen and Shen 2015), simulation (Chen et al. 2016a; Chen and Shen 2017), optimization (Chen and Li 2016, 2017) and fabrication (Chen et al. 2016b, c, 2017) of micromixer have been done in our work.

In this study, we mainly showed the effect of the Koch fractal structure of the microchannel on the mixing performance of micromixers. Compared the effect of two types of microchannels and different Res on the mixing efficiency. The effect on the mixing efficiency is also obvious when the height of the micromixer is different. The efficient mixing performance with different parameters makes it important to study the fractal of microchannels.

2 Theoretical background

2.1 Koch fractal principle

Figure 1 shows the design principle of the Koch fractal microchannel, the length of the primary fractal a = 1.8 mm, the angle of the two sides E, F are 60° and 120°. The angle between the four polylines adjacent to “a” is 60°. The straight lines of the upper and lower borders of the channels are parallel to each other. The side length of secondary fractal b = a/3 = 400 μm is obtained based on the primary fractal. a and b are a unit of the primary and secondary fractal microchannels, respectively. The geometric structure of the Koch fractal micromixer with two inlets and one outlet is designed by the CAD software. Figure 2 shows the whole structure of the Koch fractal microchannel based on the micromixer. The total length of the two inlets is 4.3 mm. The main channel consists of two Koch fractal units of length 5.4 mm. The outlet length and width are 4.2 mm and 0.3 mm respectively. We define the height of the microchannel as H. It is set to 0.1, 0.3 and 0.5 mm for this study.

Fig. 1
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Koch fractal principle

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Fig. 2
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The basic size of the micromixer

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2.2 Governing equations

The following Navier–Stokes (N–S) equations are used for the fluid dynamic analysis (Wong et al. 2004):

$$ \rho \frac{\partial u}{\partial t} + \rho (u \cdot \nabla )u + \nabla p - \eta \nabla^{2} u = 0 $$
(1)
$$ \nabla \cdot u = 0 $$
(2)

where u is the velocity vector, \( \uprho \) is the fluid density, \( \upeta \) is the dynamic viscosity and p is the pressure. Equation (1) is the equation of momentum balance. Equation (2) is the equation of continuity for incompressible fluids.

The following convective diffusion equation is used for concentration analysis:

$$ \frac{\partial {\rm c}}{\partial t} + \nabla \cdot ( -{\rm D}\nabla {\rm c}) = - {\rm u} \cdot \nabla {\rm c} $$
(3)

where c is the concentration, D is diffusion coefficient and u is the velocity vector.

The Re of square microchannel is calculated as follows (Capretto et al. 2011):

$$ D_{d} = \frac{2AB}{A + B} $$
(4)
$$ \text{Re} = \frac{{VD_{d} }}{\nu } $$
(5)

where Re is the Reynolds number, \( D_{d} \) is the equivalent diameter, A is the height, B is the width, V is the velocity and \( \nu \) is the kinematic viscosity. The velocity of the fluid for each simulation has been shown in Table 1. These velocities are calculated from Eqs. 4 and 5.

Table 1 Fluid velocity at different heights and different Re
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Mixing efficiency of the sample can be calculated by the formula as follow.

$$ M = 1 - \sqrt {\frac{1}{N}\sum\limits_{i = 1}^{N} {\left( {\frac{{c_{i} - \bar{c}}}{{\bar{c}}}} \right)^{2} } } $$
(6)

where M is the mixing efficiency, N is the total number of sampling points, \( c_{i} \) and \( \bar{c} \) are normalized concentration and expected normalized concentration, respectively. Mixing efficiency ranges from 0 (0%, not mixing) to 1 (100%, full mixed).

3 Results and discussion

3.1 Mixing efficiency at different heights

In order to studied the effect of different heights on the mixing efficiency, we have defined the height of the microchannels as 0.1, 0.3, 0.5 mm, and study the mixing efficiency of secondary classification with different Re 0.1, 0.5, 1, 5, 10, 50 and 100.

In Fig. 3, we show the positions of section A and section B of the primary and the secondary fractal micromixers and compare the secondary fractal sections. Figure 4 shows the simulation results of different heights and different Res. With Re increasing, laminar flow is gradually broken in each micromixer. It is a meaningful principle that a high Re can encourage the vortical flow, which produces secondary flow to enhance mixing. But, at high Re 100, when the height of the micromixer is increased, the change in the laminar flow state is noticeable. It can be seen clearly from Fig. 6, at different Res, the curves of concentration gradients are nonlinear. As the microchannel height decreases, the mixing efficiency at the same Re is gradually reduced. When Re is 0.1 and 100, the secondary fractal can almost achieve a complete mixing state. When Re is 1 and the channel height is 0.1 mm the mixing effect is not good enough (Fig. 5).

Fig. 3
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The position of section A and B

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Fig. 4
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“At Re 0.1, 1, 10, and 100” the concentration of section A at varying heights (0.1, 0.3 and 0.5 mm) in secondary fractal micromixer

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Fig. 5
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“At Re 0.1, 1, 10, and 100” the concentration of section B at varying heights (0.1, 0.3 and 0.5 mm) in secondary fractal micromixer

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Fig. 6
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Mixing efficiency from simulation of secondary fractal at various Res and varying heights

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3.2 Comparison of the two kinds of Koch fractal micromixers on the mixing efficiency

Based on the above study, we chose the micromixers with a microchannel height of 0.5 mm for a comparative study on the mixing efficiency of two structures of fractal. As shown in Fig. 3, we compared the mixing effect of the two mixers in sections A and B. Two micromixers are simulated and the simulation results are shown in Fig. 7, the mixing index at the exit of the micromixers is shown in Fig. 8.

Fig. 7
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At the Re 0.1, 1, 10 and 100, the concentration of the two fractal micromixer (height 0.5 mm) in section A and B

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Fig. 8
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Mixing efficiency from simulation of both the fractal micromixers at various Res

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In Fig. 7, when Re is less than 10, the laminar flow states of the two fractal microchannels are similar in section A and B. The structure at the turn affects the intensity of vortex and the mixing performance at high Re 100, the laminar flow state disappears. As shown in Fig. 6 that the fractal structure can achieve better mixing efficiency at Re 0.1 and 100. As shown in Fig. 8 the mixing indices of both the fractal micromixers are almost beyond 90%. It is also shown that the secondary fractal has an advantage over the primary fractal micromixer in mixing efficiency at various Re.

3.3 Flow analysis of fractal section

In view of above analysis, at low Re and high Re should draw our attention. Figures 9 and 12 have shown the mixing performance of the micromixers at two Re, namely Re 0.1, and Re 100. That Figs. 9 and 10 present primary fractal micromixers at two Re, and Figs. 11 and 12 present secondary fractal, micromixers at two Re. The color of the streamlines in Figs. 9, 10, 11 and 12 denotes the concentration distribution and the color of arrows in the section A and B represents the velocity magnitude.

Fig. 9
figure 9

Mixing performance of primary fractal at Re 0.1

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Fig. 10
figure 10

Mixing performance of primary fractal at Re 100

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Fig. 11
figure 11

Mixing performance of secondary fractal at Re 0.1

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Fig. 12
figure 12

Mixing performance of secondary fractal at Re 100

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As shown in Figs. 9 and 11, the mixing is limited by molecular diffusion and the chaotic advection is ineffective at Re 0.1. So, the mixing at low Re is dominated by the residence time and depends on the total path of the flow. The secondary fractal channel significantly increases the total path of the micromixer compared to the primary fractal. This undoubtedly contributes to increase in the residence time of the sample within the microchannel, and improves the mixing efficiency of the secondary fractal micromixer. Comparing sections in Figs. 10 and 12, there are obvious chaotic convection phenomena in section A and B. However, the strong chaotic region in Fig. 12 is significantly weaker than Fig. 10. But, due to the impact of the fractal structure, the fluid continuously changes the flow in the secondary fractal microchannels, and a weakly chaotic region of the arrow appears in Fig. 12. These two important factors lead to the result that the mixing efficiency of secondary fractal microchannel is better than the primary fractal microchannel.

4 Conclusions

In this paper, we proposed a novel design for 3D micromixers based on Koch fractal principle. Lots of intensive numerical simulations were conducted to compare primary fractal with secondary fractal micromixer. By comparing the mixing performance of two micromixers, it can be concluded as follows:

  1. 1.

    With increase in the height of the microchannel, the mixing efficiency of micromixers based on Koch fractal principle is getting better. Due to the impact of the fractal structure, the mixing efficiency of three different heights of secondary fractal mixers at each Re is high, beyond 80%. Especially at the height 0.5 mm, the mixing efficiency is beyond 90%.

  2. 2.

    The mixing effect of secondary fractal micromixer is better than that primary fractal micromixer. Both the fractal structures can achieve more than 90% mixing efficiency at a height of 0.5 mm.

  3. 3.

    Fractal structure not only extends the total path of the flow at low Re but also enhances the chaotic advection in fractal microchannel at high Re. It is significant for the mixing performance of passive micromixers. Fractal micromixer will have potential application value in field such as chemical industry and biochemistry due to its outstanding mixing performance.