## 2 Mathematical Models of Pneumatic Flotation
### FSI Numerical Simulations  
FSI analyses were performed to investigate four types of air bar with different hole densities. Based on the typical design of air bar, the design of Type 0 was determined as shown in Figure 1[20]. Type 0 is a commonly applied design of the air holes in a square distribution without holes in the middle. It is known that this design of air holes generates a sufficient flotation height in the middle of the air bar, but the effect of flotation is reduced at the entry and exit positions due to a reduction in the pressure. We therefore suggested alternative designs with additional air holes on both sides of the Type-0 design, which are shown as Types A to C in Figure 1. The additional air holes were distributed in two rows, and only the density of the air holes varied among the alternative designs. The densities in the additional two rows were 0.094 and 0.052 for Types A and B, respectively, while in Type C the two rows had different densities of 0.052 and 0.013. Figure 1: Candidate air-bar designs for the FSI simulations. (**a**) Type 0 is the base design with a 180-degree hole distribution with a density of 0.026. (**b**) Type A is an alternative design based on Type 0, with additional holes at the inlet and outlet with a density of 0.026 in two rows. (**c**) Type B is an alternative based on Type 0, with additional holes at the inlet and outlet with a density of 0.094 in two rows. (**d**) Type C is an alternative based on Type 0, with additional holes at the inlet and outlet with densities of 0.052 and 0.013 in two rows, respectively. The simulation procedure is summarized as follows: The air bar was modeled in three dimensions using CAD software. It had a symmetric shape in the longitudinal direction, and so only half of the real air bar was modeled in order to make the analysis more efficient and rapid (Figure 2a). The pressure at an air hole was determined, as shown in Figure 2b, while the detailed design of the hole is illustrated in Figure 3. The hole diameters at the inlet and outlet were 2 mm and 5 mm, respectively, and the spreading angle of each hole outlet was 90 degrees. A pressure regulator was employed to supply a constant pressure to the air bar, and so the inlet pressure was assumed to be constant, at 0.025 bar in these simulations. The pressure at the hole outlet was calculated as 8.753 Pa under the given geometric parameters (Figure 3). The flow in the nozzle was assumed to be a laminar flow, thus the pressure was uniform and changed at the top of the nozzle. Figure 3: Detail of an air hole, and the pressure distribution from the inlet to the outlet. Figure 2: Summary of the FSI analysis protocol: (**a**) three-dimensional (3D) modeling of the symmetrical air bar, (**b**) determining the pressure at the air hole, (**c**) boundary conditions for the fluid simulations, and (**d**) the pneumatic pressure acting on the substrate induced by the air flow. The boundary conditions for the fluid simulations were assigned as shown in Figure 2c. The input area was selected at the outlet of the air holes where the pneumatic pressure was supplied to produce substrate flotation. Similarly, the output area was selected at the side of the flotation space where the air can spill out. The pneumatic pressure acting on the substrate due to the air flow was analyzed (Figure 2d). Figure 4 summarizes the pressure distribution on the substrate for each design of air bar. It is notable that the direction of the pneumatic pressure reversed at the entry and exit positions (circles in Figure 4). This phenomenon could generate a vortex at both ends of the air bar, resulting in the height of the substrate being reduced compared to that in the middle area. The minimum pressures on each side were \(-3.899\) Pa, \(-2.923\) Pa, \(-1.957\) Pa, and \(-2.912\) Pa for Types 0, A, B, and C, respectively. Table 1 lists the minimum and maximum pressures at the inlet and outlet.