## 3 Results and Discussion  
### Quantification of the Correction Factor of Co-Flowing Method for Accurate Viscosity Measurement  
In this study, the co-flowing method was used to sequentially measure the viscosities of both the blood and plasma. According to the previous method, the fluid viscosity can be obtained by detecting the interface in the co-flowing channel. However, the performance of the co-flowing method varies depending on the interfacial location [14]. A correction factor was suggested to minimize the measurement error. A previous study reported that the correction factor depends on the depth/width and interface [28]; that is, when a microfluidic channel is newly designed, a correction factor should be obtained via experiment or numerical simulation. Figure 2(A-i) shows the mathematical representation of the co-flowing channel in terms of two fluidic resistance elements. As the co-flowing channel was filled with blood (i.e., subscript '\(\prime\)') and reference fluid (i.e., subscript '\(\prime\)'), frictional loss of each fluid was represented with fluidic resistance elements (i.e., \(R_{t}\), \(R_{r}\)). At the outlet ('\(\
abla\)'), the pressure was set to zero (i.e., \(P\) = 0). \(Q_{t}\) and \(Q_{r}\) denote the flow rates of the blood and reference fluid, respectively. For a simple mathematical representation, the two fluid streams are represented as \(\mathrm{R_{t}}\) and \(\mathrm{R_{r}}\). Both the fluidic resistance elements were connected in parallel. Based on the Poiseuille flow equation (i.e., \(\Delta P\) = \(R\) \(\times\)\(Q\)), the pressure of each fluid stream can be expressed as  
\[P_{r}\ =\ \frac{12\ \mu_{r}L}{(1-\alpha_{t})wh^{3}}\times Q_{r} \tag{1}\]  
\[P_{t}\ =\ \frac{12\ \mu_{t}L}{CF\ \alpha_{t}wh^{3}}\times Q_{t} \tag{2}\]  
In Equation (2), the correction factor (\(CF\)), which depends on the interface (i.e., \(CF\) = \(CF\) [\(\alpha_{t}\)]), was added to compensate for the mathematical modeling error resulting from the approximation of a real physical problem [28]. As the two fluid streams were confined in straight and rectangular channels, both the streams had the same pressure at a distance (\(L\)) from the outlet. Based on the same pressure condition (i.e., \(P_{t}\) = \(P_{r}\)), the viscosity of the test fluid was expressed as  
\[\mu_{t}\ =\ \mu_{r}\times\left(\frac{Q_{r}}{Q_{t}}\right)\times\left(\frac{ \alpha_{t}}{1-\alpha_{t}}\right)\times CF\ (\alpha_{t}) \tag{3}\]  
Here, the test fluid and reference fluid were selected as the plasma and 1 \(\times\) PBS, respectively. To simulate variations in \(\alpha_{t}\) by means of a CFD Solver (CFD-ACE+, ESI Group, Paris, France), the viscosity ratio of the two fluids was given as \(\mu_{t}/\mu_{r}\) = 1.674. As shown in Figure 2(B-i), microscopic images representing \(\alpha_{t}\) were summarized with respect to \(Q_{t}/Q_{t}=1/0.2\), \(1/0.5\), \(1/1\), \(0.5/1\), and \(0.2/1\). Additionally, based on the numerical simulation results, as shown in Figure 2(B-ii), the variations in \(\alpha_{t}\) were clearly visualized with a binary color representation with respect to \(Q_{t}/Q_{r}\). Consequently, \(\alpha_{t}\) showed a tendency to increase at higher value of \(Q_{t}/Q_{r}\). Both the methods provide consistent variations in \(\alpha_{t}\) with respect to \(Q_{r}/Q_{t}\). To obtain the variations in \(CF\) with respect to \(\alpha_{t}\), the analytical formula of the correction factor was derived as  
\[CF\ (\alpha_{t})\ =\ \left(\frac{\mu_{t}}{\mu_{r}}\right)\times\left(\frac{1- \alpha_{t}}{\alpha_{t}}\right)\times\left(\frac{Q_{t}}{Q_{r}}\right) \tag{4}\]  
Based on Equation (4), as shown in Figure 2C, variations in the correction factor were obtained as a function of the interface. The \(CF\) obtained from the numerical simulation was higher than that obtained from the experimental results. In this study, according to the linear regression analysis of the experimental results, the \(CF\) was best fitted as \(CF\) = 0.7004 \(\alpha_{t}\) + 0.6216 (R\({}^{2}\) = 0.9469).