1 Introduction

The research investigations in the area of micro-scale mixing have received huge momentum in last few decades because of wide applications in clinical diagnostics such as protein folding, drug delivery, biochemical reactions, cells shorting, DNA sequencing, chemical analysis, enzyme reactions etc. (Nguyen and Wu 2005; Rashidi et al. 2018; Lee and Fu 2018; Bayareh et al. 2020; Mehta et al. 2022; Mehta and Pati 2022). The development of microfluidic devices is essential to achieve prompt results with utmost accuracy in medical diagnostics. With the development of microfluidic devices, a clinical laboratory may be integrated into the form of a single-chip (Connolly 1995). One could achieve various advantages such as reduced cost, lesser sample volume, quick result, high precision, automatic measurement, improved analysis, laboratory safety, etc., by reducing the size of clinical systems (Jakeway et al. 2000).

Mixing of two or more fluids homogeneously in an efficient manner within a short period of time is essential in several applications although it is a challenging task (Miranda et al. 2010). The flow in most of the microfluidic devices is laminar in nature. Accordingly, the mixing is guided by molecular diffusion and chaotic advection. It can be noted that the diffusion time (t) is inversely proportional to the diffusion coefficient (D). As the diffusion coefficient is low for most of the fluids, long mixing length and extensive mixing time are required for achieving uniform mixing (Kunti et al. 2017). Several enhanced microfluidic devices have been developed to overcome the difficulties to date in this issue. Based on the mechanism of the mixing, micromixers are classified as active and passive ones. Researchers have found that active micromixer induces turbulent flows either by using moving parts or an external power source such as an electric field (Kunti et al. 2018; Mondal et al. 2021; Vasista et al. 2022), magnetic field (Ballard et al. 2016), acoustics (Ang et al. 2016), pressure, hydrodynamics (Deshmukh et al. 2000), ultrasonic effects (Yang et al. 2001), thermal or mechanical power (Huang and Tsou 2014), etc. In passive micromixers, no external energy source other than the pumping power is used to improve the mixing. To improve the mixing efficiency between Ferro-nanofluid and water in a Y-type semi-active micromixers, a permanent magnet is placed in an appropriate position near the microchannels (Tsai et al. 2009). The prime reasons for wide applications of passive micromixers are easy to fabricate and handle due to less complexity, and inexpensive.

Moreover, to achieve better mixing, passive micromixers have been modified by changing the geometry of the channel (Mondal et al. 2019a, 2020a, b). Ribbed surface and different shape of obstacles have been added in the channel geometry to improve the mixing performance (Borgohain et al. 2018a, b; Wangikar et al. 2018; Mondal et al. 2019b, 2020c). The mixing quality increases with increasing the confluence angle between the inlets and flow rate ratio (Mondal et al. 2018, 2020d). By increasing the asymmetry level of the inlets, mixing quality improves to a significant extent (Calado et al. 2016). Several researchers have proposed passive micromixers with lateral obstructions along the microchannel arranged both symmetric and asymmetric way. Several researchers (Khosravi Parsa et al. 2014; Gambhire et al. 2016; Chen et al. 2016; Sarma and Patowari 2017; Borgohain et al. 2018a; Mondal et al. 2019a) have studied the influence of micromixer geometry on the mixing efficiency and pressure drop using numerical and experimental investigations. Three different geometries: a square-wave mixer, a three-dimensional serpentine mixer and a staggered herringbone mixer are designed and analyzed for improving the mixing efficiency. Better mixing efficiency is obtained in serpentine micro-channel as compared to the straight micro-channel (Das et al. 2017).

A group of researchers (Falk and Commenge 2010; Commenge and Falk 2011; Pinot et al. 2014; Pérez et al. 2020) have carried out extensive research works in the field of micromixing. They showed extensive comparison of mixing efficiency of various types of micromixers. Considering simple relations of mixing in laminar flow, it was observed that to obtain the theoretical mixing time and how to relate it with functioning parameters as the Reynolds number of the flow and the specific power dissipation per mass unit of fluid. They compared the experimental and theoretical mixing times and pointed out that only a few percents of the total mechanical power transmitted to the fluid is effective for mixing. Moreover, to characterize micromixing in conventional stirred tank reactor, they proposed iodide/iodate chemical test reaction named Villermaux–Dushman method.

Several numerical, as well as experimental studies, have been reported in literature analyzing the mixing performance of passive micromixers considering the effect of corrugated shape and obstacles along the channel. However, square wave micromixer with obstacles has not been considered to analyze the performance. In the present work, mixing performance and pressure drop are analyzed both numerically and experimentally in a square wave micromixer with obstacles of three different shape. The results have been parameterized in terms of mixing length, density difference and pressure drop. For experimentation purpose, channels have been fabricated with developed techniques and then experimentally analyzed the mixing of fluid at different flow conditions.

Fig. 1
figure 1

Serpentine square wave micromixer

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2 Problem specification

Figure 1 illustrates a serpentine square wave micromixer without any obstacle. It consists of two inlets which are placed at 90º to each other. The width and the depth of the micromixer are 300 μm and 300 μm, respectively. The side length of the micromixer is 1200 μm. Three different serpentine square wave micromixers with obstruction in channel width are also designed. The geometries of the obstacles are rectangular, triangular and semicircular. Serpentine square wave micromixer with rectangular, triangular and semicircular shaped obstructions are shown in Fig. 2. Water and ethyl alcohol are taken as the input fluid for mixing. A comparison study has been carried out on the mixing performance of the following four variants of micromixer:

  • A Serpentine square wave type micromixer (S0 type).

  • A serpentine square wave with rectangular obstruction type micromixer (S1 type).

  • A serpentine square wave with triangular obstruction type micromixer (S2 type).

  • A serpentine square wave with semicircular obstruction type micromixer (S3 type).

Fig. 2
figure 2

Serpentine square wave micromixers with different shapes of obstacles (a) rectangular, (b) triangular and (c) semicircular

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3 Theoretical formulation

In this work, flow is laminar, incompressible and steady, and both the working fluids are Newtonian. The equations of mass, momentum, and species conservations are as follows:

Mass conservation

$$\vec {\nabla }\cdot \vec {V}=0{\kern 1pt}$$
(1)

Momentum conservation

$$(\vec {V}\cdot \vec {\nabla })\vec {V}= - \frac{1}{\rho }\nabla p+\upsilon {\nabla ^2}\vec {V}$$
(2)

Species conservation

$$(V\cdot \nabla )C=D{\nabla ^2}C$$
(3)

where ρ is the density of the fluid, V is the velocity of the fluid, ν is kinematic viscosity, D the diffusion coefficient and C is the concentration.

The boundary simulations are as follows.

At inlet 1: Constant inlet velocity v0 and concentration of 0 are applied.

At inlet 2: Constant inlet velocity v0 and concentration of 1 are applied.

At the wall: No slip and no penetration conditions are applied.

At outlet: Gauge pressure is considered as zero, and concentration outflow is considered. The mixing index for a given channel configuration is governed by mass Peclet number (Pe = ScRe).

4 Numerical methodology and model validation

The governing equations of conservations of mass, momentum and species transport are solved by a finite volume method-based solver. The second-order upwind scheme is used to discretize both the advection and diffusion terms. The SIMPLE algorithm is used for pressure-velocity coupling. The relative convergence criterion for the residuals is set to 10− 5. To design the 2D Y-shape square wave serpentine passive microchannel CATIA v5 software is used and ANSYS 14.0 FLUENT software is used to study the mixing behaviour of two fluids and pressure drop along the length of the channels.

Prior to numerical simulations, an extensive grid independence test has been carried out to minimize the computational time and resources without sacrificing the accuracy of the results. For each of the micromixer geometry, four different grid systems have been considered. The average density difference at a particular location (location 4) from the junction at the inlet velocity of 0.003 m/s for different grids is presented in Table 1. The simulations are performed with M3, M7, M11 and M15 grids for S0, S1, S2 and S3 type micromixers, respectively. For each geometry under consideration, an error analysis on the prediction of the results has been conducted by selecting the finest grid as a reference for that geometry. It is to be noted that the percentage difference of the results is less than 1% with respect to the reference grid system for the specific geometry.

Table 1 Grid independence test for different geometry (at inlet velocity 0.003 m/s)
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Before presenting the findings of the current investigation, the accuracy of the present numerical solver used for the simulations ascertained. For validation purpose, first the geometry of the channel similar to the work of Miranda et al. (Miranda et al. 2010) is considered. Figure 3 presents the comparison of the mixing efficiency as obtained numerically from the current numerical simulations with that available in the literature (Miranda et al. 2010) for Re = 0.1, Pe = 1450, St = 3.33 × 10− 2. The excellent agreement in the results confirms the accuracy of the solver.

Fig. 3
figure 3

Validation of simulation results against simulation results (Miranda et al. 2010)

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5 Micromixer fabrication and experimental setup

CNC WEDM (Make: RATNAPARKHI (I) PVT. LTD., Nashik, India, Model: Ezeeecut NXG) has been used to fabricate different solid micromixer replica tools from the copper plate. These tools are then used as the pattern for making PDMS (SYLGARD (R) 184 SILICONE ELASTOMER) micromixers by soft lithography technique. The inlet fluids have been injected through the fabricated micromixer by a twine syringe pump (Make: YASHTECH, Model: Mediflow-1000) with different flow rates. The mixing phenomena have been observed by an online camera system (DINO-LITE DIGITAL MICROSCOPE) with a computer. To measure the pressure, pressure sensor devices with a data logger (Make: Ideal Integrated Technology, India) have been connected at the inlet and outlet of the micromixer. The complete experimental setup which includes (a) syringe pump, (b) online monitoring system, (c) pressure sensor and data logger is shown in Fig. 4. Initially, fluid with three different flow rates is made to flow through simple serpentine square wave micromixer. Then the results are compared with that of mixers with obstructions.

Fig. 4
figure 4

Experimental setup for mixing

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6 Results and discussion

The computational analyses have been carried out for serpentine square wave micromixer without obstruction as well as with obstruction of three different shapes at three different inlet velocities (0.003 m/s 0.015 m/s and 0.030 m/s). At zero distance from the junction, the density difference between the two fluids is 211 kg/m3. With increasing the channel length, the density difference decreases. When the density difference is equal to zero, the two liquids are assumed to mix perfectly.

6.1 Influence of channel geometry on mixing length

One can explain the effect of channel geometry on the mixing performance and pressure drop by analyzing the streamlines. Figure 5 represents the streamlines for all four types of micromixers at an inlet velocity of 0.003 m/s. It is seen from Fig. 5(a) that the streamlines are straight and parallel to each other for flow through micromixer without obstacles (S0 type). The streamlines become curved for flow through micromixers with obstacles, and the waviness of streamlines depends on the geometry of the obstacles (Falk and Commenge 2010; Commenge and Falk 2011). With the increase in the curvature of streamlines, fluids get mixed even though the flow is laminar. It is observed that the density difference between the two fluids gradually decreases in the downstream from the junction point, and finally, the density difference becomes zero. This is owing to the fact that the time of contact is more as the flow progresses. It is observed that for micromixers with obstacles of different shapes, density difference becomes zero for a shorter distance measured from the junction. This can be explained as follows. The aspect ratio of the mixer decreases in case of micromixer with obstacles (Song et al. 2012; Mondal et al. 2019b), as a result, fluids get mixed well, and it leads to better diffusion. This implies that the presence of obstacles in the mixer imparts quick mixing as compared to that of micromixer without obstacle(Jiang et al. 2004; Clark et al. 2018; Gidde et al. 2022). Though literally, obstruction means the higher flow resistance, here the presence of obstructions is advantageous as far as better mixing is concerned(Gidde et al. 2018a,b). Comparing the mixing length of micromixers with various shape of obstacles, it is seen that the micromixer with triangular shape obstruction gives quick mixing as compared to that of other two as can be seen from Fig. 6.

Fig. 5
figure 5

Streamlines of serpentine square wave micromixers (a) without obstacle, with (b) rectangular, (c) triangular and (d) semicircular at velocity 0.003 m/s

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Fig. 6
figure 6

Density differences in different type microchannels while inlet velocity 0.003 m/s

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6.2 Influence of inlet velocity on mixing length

A comparative study of mixing performance has been carried out by varying the inlet velocity for a particular microchannel geometry. The analysis is performed for S0 type micromixer with different inlet velocities of 0.003 m/s, 0.015 m/s and 0.030 m/s. The respective considered Reynolds numbers are 1, 5 and 10. It is observed that there is a significant difference in density at a particular location in S0 type micromixer while velocity of the fluid at the inlet is varied. Figure 7 shows the density difference for S0 type micromixer at different locations for different inlet velocities. It is seen that with the increase in inlet fluid velocity, mixing length decreases because of the fact that the strength of the recirculation gets enhanced with the inlet velocity.

Fig. 7
figure 7

Density differences in S0 type microchannels (inlet velocity (a) 0.003 m/s (b) 0.015 m/s (c) 0.03 m/s)

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Fig. 8
figure 8

Variation of pressure drop (Pa) obtained from computational study for different micromixers at different inlet velocities

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6.3 Influence of channel geometry on pressure drop

The pressure drop is another important parameter that needs to be analyzed for efficient design of micromixers. The micromixer should operate in such a way that not only the mixing length is shorter but also have a relatively lower pressure drop. Intuitively, pressure drop depends on the flow velocity, and it increases with the increase in the velocity. The pressure drops in all four designed micromixers, i.e., S0, S1, S2, and S3 types have been calculated by varying the inlet velocity and is presented in Fig. 8. It is observed from Fig. 8 that the pressure drop in the micromixers with obstacles of different shape is higher as compared to that of micromixer without obstacle (S0 type). This is due to the fact that flow resistance due to the geometry of the flow passage increases due to the presence of the obstructions. It is seen that S1 type micromixer experiences the maximum pressure drop as compared to S2 and S3 type micromixers and the least pressure drop occurs in S2 type. Thus, S2 type of micromixer is preferable as far as minimum pressure drop criteria are concerned.

Fig. 9
figure 9

Images of (a) S0, (b) S1, (c) S2 and (d) S3 type micromixer during experimentation

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7 Experimental validation

In the experimental analysis, a little amount of red and blue inks/colours has been added to the fluids for better visualization of the characteristics. It is observed that at the junction point and at some distance from the junction, there is a clear separation between the two fluids. Here the mixing point is measured offline from the images as captured by an online camera. The observations have been done from the inlet side. A measurement line is drawn from the junction point to that location which indicates the mixing length. The mixing length at a flow rate of 0.003 m/s for different types of micromixers is shown in Fig. 9. A comparison of the mixing length obtained from experimental analyses is shown in Table 2.

Table 2 Density difference at different location from the junction while inlet velocity 0.003 m/s
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From the experimental analysis, it is also seen that the pressure drop in serpentine square wave micromixer with obstacles is higher as compared to that of micromixer without obstacle. Comparing pressure drop among the micromixers having serpentine square wave with different obstacles such as S1, S2, S3 type, it is observed that in S2 type micromixer experiences less pressure drop (Fig. 10). With respect to pressure drop in different types of micromixer with an obstacle, S2 type of micromixer is the most preferable due to less pressure drop in it.

Table 3 Density difference at different location from the junction while inlet velocity 0.030 m/s
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Table 4 Density differences at different location from the junction while inlet velocity 0.015 m/s
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Table 5 Mixing length for different type of micromixers obtained from experiments
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Fig. 10
figure 10

Variation of pressure drop (Pa)/ velocity (m/s) obtained from experiments for different micromixers at different inlet velocities

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The obtained results from the experimental analysis are not exactly same when compared to those of computational analysis due to some system limitations. There are some differences due to various experimental conditions such as micromixer material, dimensions, edge smoothness, working temperature, low magnifying imaging system, accuracy of the syringe pump, higher least count of the available data logger and monitoring device, etc. The deviation values of mixing length are varied from 0.5 to 7.2%, and the mean deviation is 2.61%. The comparison between the experimental and computational values of pressure drop (Pa) for different design micromixer at different inlet velocity is shown in Fig. 11. For pressure drop calculation by comparing all the cases, it is observed that the mean deviation is 9.4%. It is also observed that the higher deviations are found in case lower values of pressure drop and the higher least count readability is the main reason for this.

Fig. 11
figure 11

Comparison of values of the pressure drop (Pa) for different micromixers at different inlet velocities obtained from the experimental and computational investigations

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8 Conclusions

In this work, numerical and experimental investigations have been performed in serpentine square wave micromixer with and without obstructions along the width to analyze the mixing length and pressure drop. Three different shapes of the obstacles, namely, rectangular, triangular and semicircular, are considered. The main findings from the present work are summarized as follows:

  • Placing obstacles in the micromixer creates disordered advection effect, which is a novel method to shorten the mixing length in passive micromixers.

  • Micromixer with any shape of obstruction shows rapid mixing as compared to that of without obstruction.

  • The mixing length decreases by 29.57%, 31.29% and 29.46% in micromixer with rectangular, triangular, and semicircular shape obstacle respectively as compared to micromixer without obstacle.

  • The pressure drop increases in case of obstacles in the micromixer compared to the case without obstacle.

  • The micromixer with triangular shape obstruction gives the minimum mixing length, while the corresponding pressure drop is also the lowest among all the micromixers with obstacles.