## 3 Results
### Effect of Microchannel Size  
We next used our FEM simulation to explore the effect of the mixer micro-channel dimensions on the mixing efficiency. Figure 7 shows the mixing efficiencies as a function of the serpentine mixer length for three different height-to-width ratios (r = H/W): 0.9, 1.8, and 3.6 at a constant flow rate of 100 \(\upmu\)L/min. Figure 7a shows the mixing efficiencies for H = 45 \(\upmu\)m and W = 12.5, 25, and 50 \(\upmu\)m, respectively. Figure 7b shows the mixing efficiencies for H = 22.5, 45, and 90 \(\upmu\)m for W = 25 \(\upmu\)m.  
The results in Figure 7a indicate that for a fixed height of H = 45 \(\upmu\)m and a channel width of W = 12.5 \(\upmu\)m (r = 3.6), the micro-mixer can achieve mixing efficiencies of over 90%, faster than for W = 25 and 50 \(\upmu\)m, and that to achieve the same mixing efficiency with a wider channel, a longer channel length is required. In addition, wider channels also require more of the sample to maintain the same flow rate and, hence, sample consumption can be an issue. However, whilst the narrower microchannels appear to be more efficient at mixing, these also have the disadvantage that when mixing and jetting biological samples, the channels can potentially become blocked. When examining the effect of height in Figure 7b for a fixed channel width of W = 25 \(\upmu\)m, we find that the most efficient mixing  
Figure 6: Numerical results: Mixing efficiency as a function of the serpentine mixer length for flow rates in the range of 80–200 \(\upmu\)L/min. Within this flow rate range, our 1200 \(\upmu\)m long serpentine micromixers can achieve homogenous (>90 %) mixing whilst producing a stable liquid jet for sample delivery at the outlet. occurs for H = 22.5 \(\mu\)m (r = 0.9). The largest height (H = 90 \(\mu\)m) gives the least efficient mixing, whilst a channel height of (H = 45 \(\mu\)m) provides an intermediate mixing efficiency. The average Reynolds number for the devices that was used in the experimental studies (r = 1.8) is Re = 2.13, whilst Re = 4.27 for r = 3.6, and Re = 1.07 for r = 0.9. The results from Figure 7a,b indicate that a smaller microchannel cross-section results in uniform mixing that is faster and has higher efficiency due to an increase in the average velocity of the solutions and consequently has an enhanced chaotic mixing effect. The fabrication approach using SU8 developed for our microfluidic devices enables highly flexible geometries to be achieved without requiring any alteration of the photolithography mask design. This means that all of the ratios in terms of height and width simulated here can be readily reproduced experimentally and, according to our previous experimental work [40; 41], will also result in stable jetting. Ratios are r = 0.9, 1.8, and 3.6. The minimum achievable mixing times for the serpentine micromiker as a function of the flow rate are shown in Figure 8; the smaller the serpentine, the faster the mixing. The minimum achievable mixing time is defined here as the amount of time required for the reagents to travel in the serpentine micromiker before at least 95% mixing efficiency is achieved. Consistent with the results in Figure 7, Figure 8a shows that for a fixed channel height (H = 45 \(\mu\)m), as the channel width increases, the minimum achievable mixing time also increases (from 0.30 ms for W = 12.5 \(\mu\)m to 1.28 ms for W = 50 \(\mu\)m) and that even increasing the flow rate to 300 \(\mu\)L/min cannot make up for the difference in mixing times compared to the 12.5 and 25 \(\mu\)m width channels. By decreasing the channel width to 12.5 \(\mu\)m, we can maintain sub-ms mixing (0.85 ms), even with flow rates as low as 120 \(\mu\)L/min whilst still maintaining a stable jet. Figure 8b shows that for a fixed channel width (W = 45 \(\mu\)m) as the height of the mixing channel decreases, shorter and shorter mixing times can be achieved for a given flow rate. With the thinnest mixer (H = 22.5 \(\mu\)m) sub-ms mixing times can be achieved with flow rates as low as 80 \(\mu\)L/min.