# Characteristics of Air Resistance in Aerostatic Bearings
## 3 Results and Discussion
### Experimental Results and Analysis  
By substituting the experimental air resistance calculated by Equation (9) into Equation (10), the experimental values of the parameter \(k\) can be obtained, as shown in Figure 5.  
From Figure 5, the following observations were derived: (1) For pipelines with different specifications, as the input air pressure increases, the change trend of the parameter \(k\) remains the same; it decreases with a negative exponential trend, and \(k\) finally tends to stabilize. Moreover, under the same air pressure input, regardless of the pipe inner diameter or pipe length, the values of the parameter \(k\) are similar. (2) When the air pressure at the inlet end of the tested pipeline is less than 0.25 MPa, the curve shows a rapid downward trend. When the inlet pressure is greater than 0.25 MPa, the curve tends to be flat. This result is due to the changes in the air flow rate, which leads to a continuous increase in the Reynolds number and continuous thinning of the viscous bottom layer. Therefore, the parameter \(k\) is a function related to the Reynolds number. Figure 3: Mechanical experiment layout. Figure 4: Comparison curve between the simulated and theoretical air resistance values. (**a**) Pipe with 2 mm inner diameter and 0.5 m length. (**b**) Pipe with 2 mm inner diameter and 1.0 m pipe length. (**c**) Pipe with 3 mm inner diameter and 0.5 m pipe length. (**d**) Pipe with 3 mm inner diameter and 1.0 m pipe length. (**e**) Pipe with 4 mm inner diameter and 0.5 m pipe length. (**f**) Pipe with 4 mm inner diameter and 1.0 m pipe length. Figure 5: \(k\) curve. Theoretically, the parameter \(k\) changes with the Reynolds number \(Re\), showing a negative exponential relationship, setting the parameter expression as \(k=a\cdot Re^{-b}\). Using the least-squares method, the parameter \(k\) and Reynolds number \(Re\) were curve-fitted to obtain the best estimated value of the coefficients \(a\) and \(b\) in the expression of the parameter \(k\):  
\[k=14.77 Re^{-0.75} \tag{13}\]  
where \(Re\) is the Reynolds number. To verify the accuracy of the air resistance definition applicable to the capillary tube proposed in this paper, the air resistance calculated using the air resistance definition formula was experimentally compared with the measured air resistance, as shown in Figures 6-9. Figure 6: Comparison curve between the theoretical and experimental values of the gas resistance and input pressure of the tube with an inner diameter of 2 mm and length of 0.5 m.  
Figure 7: Comparison curve between the theoretical and experimental values of the gas resistance and input pressure of the tube with an inner diameter of 2 mm and length of 1.0 m.  
Accordingly, the following observations are obtained from the figures:  
1. An inflection point exists in the theoretical and experimental air resistance values in Figures 6-9, and the change trend of the air resistance value \(R_{g}\) varies from gradually decreasing to gradually increasing. The Mach number at the turning point is approximately 0.3, which is the boundary point between the low-speed flow and high-speed flow. 2. From Equation (10), the air resistance \(R_{g}\) is proportional to \(l\) and inversely proportional to \(d^{5}\). In the capillary tubes of equivalent diameters, the tube lengths experimentally measure 0.5 and 1.0 m. Although the tubes have equivalent diameters, the theoretical air resistance is twice the measured results, and the actual calculation is 1.6-1.8 times the measured results because the density \(\rho\) and parameter \(k\) are actually different. For capillary tubes with the same tube length, the experimentally measured equivalent diameters are 2 and 4 mm. The theoretical air resistance is 0.03 times the measured results, and the actual calculation is 0.8-1 times the measured results, showing a deviation from theory. This is because although the parameter \(k\) is similar for pipelines with different equivalent diameters, the density and flow are completely different, and the variables cannot be controlled. 3.