## 3 Governing Equations for Ultra-Thin Gas Film Lubrication  
The schematic diagram of gas-lubricated journal microbearing is shown in Figure 1. For an isothermal, isoviscous, laminar and inertialess ideal gas, the compressible Reynolds equation which governs the gas pressure profiles considering the gaseous rarefaction effects is given below. \[\frac{\partial}{\partial\varphi}\bigg{(}QPH^{3}\frac{\partial P}{\partial\varphi }\bigg{)}+\frac{\partial}{\partial\lambda}\bigg{(}QPH^{3}\frac{\partial P}{ \partial\lambda}\bigg{)}\ =\ \Lambda\frac{\partial(PH)}{\partial\varphi}+2\Lambda\frac{\partial(PH)}{ \partial T} \tag{5}\]  
where \(p=p/p_{a}\) is the dimensionless gas film pressure, \(H=h/C_{b}\) is the dimensionless gas film thickness, \(p_{a}\) and \(C_{b}\) are the ambient pressure and the radius clearance respectively. \(\varphi\) and \(\lambda\) denotes the dimensionless circumferential angle coordinate and axial coordinate of micobearing normalized by the journal radius \(R\). \(\Lambda=6\mu\alpha R^{2}/p_{a}C_{b}{}^{2}\) is the bearing number or compressibility parameter, \(\mu\) is the dynamic viscosity of airflow, \(\alpha\) is the rotational speed of journal. \(Q\) represents the Poiseuille flow rate ratio and \(T\) is the dimensionless time. In the steady state equilibrium condition, the dimensionless film thickness and pressure distribution are time-independent, so the static modified Reynolds equation considering the gaseous rarefaction effect is simplified as follows. \[\frac{\partial}{\partial\varphi}\bigg{(}QPH^{3}\frac{\partial P}{\partial \varphi}\bigg{)}+\frac{\partial}{\partial\lambda}\bigg{(}QPH^{3}\frac{\partial P }{\partial\lambda}\bigg{)}\ =\ \Lambda\frac{\partial(PH)}{\partial\varphi} \tag{6}\]  
The dimensionless pressure boundary condition is depicted as:  
\[\left\{ \begin{array}{l}P(\varphi,\lambda\ =\ \pm\frac{B}{2R})\ =\ 1.0\\ P(\varphi\ =\ 0,\lambda)\ =\ P(\varphi\ =\ 2\pi,\lambda)\\ \frac{\partial P}{\partial\varphi}|_{\varphi=\ 0,\lambda}\ =\ \frac{\partial P}{ \partial\varphi}|_{\varphi=\ 2\pi,\lambda}\end{array} \right. \tag{7}\]  
where \(B\) is the bearing width. Letting \(PH=S\), \((PH)^{2}=S^{2}=\Pi\)[32], the governing Equation (6) can be transformed into the elliptic partial differential equation form -\(\triangledown(c\triangledown u)+au=f\)  
\[ \begin{array}{l}-(\frac{\partial^{2}\Pi}{\partial\varphi^{2}}+\frac{ \partial^{2}\Pi}{\partial\lambda^{2}})+\frac{2\Pi}{\Pi}(\frac{\partial^{2}H}{ \partial\varphi^{2}}+\frac{\partial^{2}H}{\partial\lambda^{2}})+\frac{2\Pi}{ \Pi}(\frac{\partial Q}{\partial\varphi}\frac{\partial H}{\partial\varphi}+ \frac{\partial Q}{\partial\lambda}\frac{\partial H}{\partial\lambda})\\ =-\frac{h}{\Pi}(\frac{\partial H}{\partial\varphi}\frac{\partial\Pi}{\partial \lambda}+\frac{\partial H}{\partial\lambda}\frac{\partial H}{\partial \lambda})+\frac{1}{\lambda}(\frac{\partial Q}{\partial\varphi}\frac{\partial H }{\partial\lambda}+\frac{\partial Q}{\partial\lambda}\frac{\partial H}{\partial \lambda})-\frac{2\Lambda}{\partial Q}\frac{\partial S}{\partial\varphi}-\frac{ \Lambda}{\partial Q}\frac{\partial S}{\partial T}\end{array}  \tag{8}\]  
Figure 1: Schematic diagram and 3D views of gas-lubricated journal microbearings. (**a**) Schematic view; (**b**) 3D graphical representation. The corresponding coefficients in Equation (8) are derived as  
\[\left\{ \begin{array}{l}c\;=\;1\\ a\;=\;\frac{2}{H}(\frac{\partial^{2}H}{\partial\varphi^{2}}+\frac{\partial^{2}H}{ \partial\lambda^{2}})+\frac{2}{QH}(\frac{\partial Q}{\partial\varphi}\frac{ \partial H}{\partial\varphi}+\frac{\partial Q}{\partial\lambda}\frac{\partial H }{\partial\lambda})\\ f\;=\;-\frac{1}{H}(\frac{\partial H}{\partial\varphi}\frac{\partial H}{ \partial\varphi}+\frac{\partial H}{\partial\lambda}\frac{\partial H}{ \partial\lambda})+\frac{1}{Q}(\frac{\partial Q}{\partial\varphi}\frac{ \partial H}{\partial\varphi}+\frac{\partial Q}{\partial\lambda}\frac{ \partial H}{\partial\lambda})-\frac{2\Lambda}{QH}\frac{\partial S}{\partial\varphi} \end{array} \right.