# Research Developments of Aerostatic Thrust Bearings: A Review
## 2 Theoretical Models and Experiment Methods  
### Theoretical Model  
With the developments of ATBs, the theoretical model is continuously improved. In the early stages, the simplified Reynolds equation was solved to predict the performance of the aerostatic bearing. The steady-state pressure governing the Equation of aerostatic bearing in dimensionless form is shown in Equation (1) [10].  
\[\frac{\partial}{\partial\overline{x}}\left(\overline{h}^{3}\frac{\partial \overline{p}^{2}}{\partial\overline{x}}\right)+\frac{\partial}{\partial \overline{z}}\left(\overline{h}^{3}\frac{\partial\overline{p}^{2}}{\partial \overline{z}}\right)+\overline{Q}\delta_{i}=\Lambda_{x}\frac{\partial\left( \overline{p}\overline{h}\right)}{\partial\overline{x}}+\Lambda_{z}\frac{ \partial\left(\overline{p}\overline{h}\right)}{\partial\overline{z}} \tag{1}\]  
where \(\overline{x}\) and \(\overline{z}\) are dimensionless coordinates, \(\overline{h}\) is the dimensionless thickness of the gas film, \(\overline{p}\) is the dimensionless pressure, \(\overline{Q}\) is the dimensionless mass flow factor, \(\delta_{i}\) is a constant coefficient. \(\Lambda_{x}\) and \(\Lambda_{z}\) are dimensionless bearing numbers in the \(x\) and \(z\) directions. The definitions of the above parameters are shown in Equation (2) [10].  
\[\overline{h}=\frac{h}{h_{m}},\overline{x}=\frac{x}{l},\overline{z}=\frac{z}{l},\overline{p}=\frac{p}{p_{s}},\overline{Q}=\frac{24\eta l^{2}p_{a}}{h_{m}^{3} p_{s}^{2}p_{a}}\rho\widetilde{v},\Lambda_{x}=\frac{12\eta ul}{h_{m}^{2}p_{s}}, \Lambda_{z}=\frac{12\eta ul}{h_{m}^{2}p_{s}} \tag{2}\]  
where \(h\) is the thickness of the gas film, \(h_{m}\) is the reference gas film thickness, \(x\) and \(z\) are coordinates, \(l\) is the characteristic length, \(p\) is the gas pressure, \(\rho\) is the gas density, \(\eta\) is the gas dynamic viscosity, \(\widetilde{v}\) is the velocity of orifice gas flow, \(u\) and \(\omega\) are the velocities.  
In the solution of the Reynolds equation, how to accurately calculate the pressure and the flow rate through the feed hole was the focus of the research. Later, it was found that the pressure from feed holes to the outlet of the aerostatic bearings does not monotonically decrease. The pressure increases slightly after a sudden drop around the feed hole. Due to the complexity of three-dimensional gas flow and some assumptions in solving the Reynolds equation, the traditional Reynolds equation was insufficient to capture these phenomena accurately. Gradually, more flow characteristics such as supersonic flow, turbulence, gas vortex, and shock wave were discussed [11; 12; 13; 14]. The flow characteristics and fluctuating pressure of ATB are shown in Figure 3[15]. In recent decades, the demands for the high performance of ATBs are increasing, so it is necessary to study the flow characteristics from a microscopic perspective. To obtain more accurate pressure distribution and its time-dependent characteristics, solving the complete three-dimensional N-S equation has become the most common and acceptable method. The theoretical model is introduced from the following three aspects, gas flow in ATBs, velocity slip effect, and multi-physics coupling.