## 3 Results and Discussion  
### CFD Simulation for Validation of Analytical Expressions  
To obtain the correction factor of pressure (\(C_{P}\)) which compensates for the approximation errors that occurred for modeling coflowing lamina streams, computational fluid dynamics (CFD) simulations were conducted using commercial software (CFD-ACE+, Ver. 2019, ESI Group, Paris, France). Using a two-phase module, variations of interfacial locations were estimated by the varying flow rate ratios of two fluids. It was assumed that both fluids behaved as Newtonian fluids. For convenience, both fluids had the same viscosity (\(\mu_{r}=\mu_{t}=1\) cP). The pressure and interface obtained with the CFD simulation were used to obtain the C\({}_{P}\) with respect to the interface. To relocate the interface in the coflowing channel, the flow rate of the test fluid increased from \(Q_{t}=0.1\) mL/h to \(Q_{t}=10\) mL/h. Here, the flow rate of the reference fluid was fixed at \(Q_{r}=1\) mL/h. As shown on the left side panel of Figure 2A, the interfaces (i.e., solid line with red color) were represented with respect to \(Q_{t}\) (i.e., [a] \(\beta=0.095\) for \(Q_{t}=0.1\) mL/h, [b] \(\beta=0.5\) for \(Q_{t}=1\) mL/h, and [c] \(\beta=0.897\) for \(Q_{t}=10\) mL/h). The right-hand panel of Figure 2A showed the distribution of pressure along the coflowing channel with respect to \(Q_{t}\). According to the simulation results, \(\beta\) and the pressure increased considerably at higher flow rates of the test fluid. Figure 2B showed a quantitative comparison of the pressure between the CFD simulation and analytical formula (Equations (1) and (2)). As the pressure was proportional to the channel length (\(L\)), for convenience, the pressure distribution was obtained at \(L_{c}=7\) mm from the ground (i.e., \(P[L_{c}]\)). As shown in Figure 2B(a), variations in pressure across the channel width (\(W\)) were obtained with respect to \(\beta=0.095\), 0.5, 0.743, 0.847, and 0.897. The pressure (\(P[L_{c}]\)) remained unchanged along the channel width. It tended to increase substantially with increasing \(\beta\). Figure 2B(b) showed the variations in pressures obtained with both methods (i.e., CFD simulation and analytical method: Equation (2)) with respect to \(\beta\). In addition, the normalized difference (ND) between the two methods was obtained with respect to \(\beta\). The CFD simulation overestimated the pressure slightly compared with the analytical formula. When \(\beta\) relocated from 0.1 to 0.9, pressure increased significantly from 13.14 kPa to 131.38 kPa. The ND increased from 2% to 13.6%. As shown in Figure 2C(a), variations in \(P_{r}\) and \(P_{t}\) were obtained with respect to \(\beta\). As a principle, both streams should have the same pressure in a straight coflowing channel (i.e., \(P_{r}=P_{t}\)). Here, the pressure of the test fluid stream (\(P_{t}\)) was calculated with respect to \(\beta\) after removing the \(C_{P}\) in Equation (1). However, the difference between \(P_{r}\) and \(P_{t}\) tended to increase when the interface shifted toward the right-side wall. To minimize the difference between \(P_{r}\) and \(P_{t}\) with respect to \(\beta\), it was necessary to include \(C_{P}\) in Equation (1). Based on the viscosity formula derived at the same pressure condition (i.e., Equation (3)), \(C_{P}\) was obtained as \(C_{P}=(Q_{t}/Q_{r})\times(1-\beta)/\beta\) with respect to\(\beta\). Figure 2C(b) showed the variations of \(C_{P}\) with respect to \(\beta\). The \(C_{P}\) tended to increase substantially when the interface moved from the left-side wall to the right-side wall. Linear regression analysis was used to obtain the polynomial expression of \(C_{P}\) as \(C_{P}=2.7378\)\(\beta^{4}-4.3067\)\(\beta^{3}+2.1783\)\(\beta^{2}-0.2644\)\(\beta+0.9592\) (\(R^{2}=0.9918\)). When the interface was positioned between \(\beta=0.3\), and \(\beta=0.7\), both streams had the same pressure within 1% ND, even without including \(C_{P}\).