## 2 Materials and Methods
The sizes of the stents were 5, 6, 7, and 8 Fr with inner diameters of 0.67, 1, 1.33, and 1.67 mm, respectively, and outer diameters of 1.67, 2, 2.33, and 2.67 mm, respectively, with multiple side holes in the shaft and coils. The side holes and end holes in the coils were called ports to differentiate them from the side holes in the shaft. Two different types of stents were made, differentiated by the number and position of the side holes in the shaft. The first type was a stent with 22 side holes placed at a 1-cm intervals and at a 90\({}^{\circ}\) angular position. The other was a stent with 45 side holes placed at a 0.5-cm intervals and at a 90\({}^{\circ}\) angular position. The number of ports in each coil was five, including four side holes and one end hole. The diameter of the side holes and ports, except the end holes, was 1 mm, and the diameter of the end holes in the coils depended on the diameter of the stent.The ureter and the stent were combined, and thus the axis of the ureter was shared with that of the stent, meaning that the ureter and the stent had the same axis. The stent was located in the center of the ureter. A concentric stenosis was made in the middle of the ureter, and it was 15 mm long. The degrees of stenosis in the ureter were 0%, 33%, 52%, and 74% in area, as shown in Figure 3.  
Stenosis in the ureter can occur not only in the middle position, but anywhere in the narrow diameter of the ureter [15]. This study assumed the same location and size of stenosis to extend the results of the study to cases in which a previous stenosis occurred, to compare and analyze the effects on the DJS size [16]. The combination of two different curved tubular ureters (4.57 and 5.39 mm), eight different types of stents (5-8 Fr, each with 22 or 45 side holes), and four different degrees of stenosis in the ureter (0%, 33%, 52%, and 74%) resulted in 64 different stented ureter CFD models for numerical analysis. Table 3 shows the details of the eight different models in a certain diameter ureter with a certain size of double-J stent. In two models with a 4.57-mm ureter, 8-Fr stent, and 74% stenosis (Cases 7 and 8), no space was found between the inner ureter wall and the outer stent wall. The mesh generation of the ureter-inserted DJS geometry was conducted using ICEM CFD software. The prism and tetrahedron grid types were used to make the mesh because of the complex shape of the model. The unconstructed grid systems of the ureter-inserted DJS are shown in Figure 4. The total grid elements and node range of the model were \(9.579\times 10^{6}\)-\(13.53\times 10^{6}\) elements and \(3.049\times 106\)-\(19.52\times 106\) nodes, respectively. To check the mesh sensitivity, a mesh independency test  
Figure 3: A stented ureter model and four magnified mid-ureters with different stenoses (0%, 33%, 52%, and 74%). The diameter of the curved tubular ureter is 4.57 mm and the stent is a 6-Fr DJS. The governing equations for fluid flow and the boundary conditions for numerical analysis in a previous study [16] were also used here. The continuity and momentum equations are as follows:  
\[\frac{\partial u_{j}}{\partial x_{j}}=0 \tag{1}\]  
\[\rho(\frac{\partial u_{i}}{\partial t}+u_{j}\frac{\partial u_{i}}{\partial x_{i }})=-\frac{\partial p}{\partial x_{i}}+\eta\frac{\partial}{\partial x_{i}}( \frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}) \tag{2}\]  
where \(\rho\), \(u\), \(\eta\), and \(p\) are the density, velocity vector, viscosity, and pressure, respectively. The governing equations were discretized using the finite volume method with CFX 16.2 software (Ansys Inc.) [17]. The boundary condition and the urine viscosity used in the study were the same as in the previous study [16]. Urine is considered an incompressible and Newtonian fluid. If urine is stagnant in one place for a long time, its viscosity and physical properties will change [18]. However, this study focused on urine flow through the DJS and did not consider the change in the physical properties of urine in stagnant situations.