# Performance of the Compliant Foil Gas Seal with Surface Micro-Textured Top Foil+
## 3 Theoretical Model  
### Reynolds Equation and Gas Film Thickness Control Equation  
The compliant foil gas seal problem can be categorized as a micro-scale flow field problem. Therefore, the flow field can be considered as a laminar flow in this study. Moreover, the following assumptions are made: there is no slip at the fluid-solid interface, the sealing medium is an ideal gas, and the physical parameters remain unchanged. The pressure solution model can be obtained as follows:  
\[\frac{\partial}{\partial\theta}\left(PH^{3},\frac{\partial P}{\partial\theta} \right)+\left(\frac{R}{L}\right)^{2}\frac{\partial}{\partial Z}\left(PH^{3}, \frac{\partial P}{\partial Z}\right)=\Lambda\frac{\partial(PH)}{\partial \theta}+2\gamma\Lambda\frac{\partial(PH)}{\partial T} \tag{1}\]  
Considering the Coulomb friction between the bump foil and the sealing cavity and ignoring the concavity or convexity of the flat foil in the radial direction, [PERSON]'s [28] stiffness model with one end fixed and one end free is used to characterize the deformation of the bump foil. Thus, the static film thickness formula can be obtained as follows:  
\[H= \begin{cases}1+\epsilon\cos\theta+H_{\text{c}}+\frac{(P-1)}{K_{\text{b}}}& (\theta,Z)\in\text{Textured region}\\ 1+\epsilon\cos\theta+\frac{(P-1)}{K_{\text{b}}}&(\theta,Z)\in\text{Non- textured region}\end{cases}  \tag{2}\]  
The dimensionless parameters are defined as follows:  
\[Z=\frac{z}{L},\;H=\frac{h}{h_{0}},\;P=\frac{p}{p_{\text{a}}},\;T=\tau t,\; \Lambda=\frac{6\omega\mu r^{2}}{p_{\text{a}}h_{0}^{2}},\;\gamma=\frac{\tau}{ \omega},\;H_{\text{c}}=\frac{h_{\text{c}}}{h_{0}},\;K_{\text{b}}=\frac{k_{ \text{b}}h_{0}}{p_{\text{a}}} \tag{3}\]  
where \(z\) is the axial coordinate, \(t\) is the motion time, \(k_{\text{b}}\) is the bump foil stiffness, \(h_{0}\) is the average gas film thickness, \(p_{\text{a}}\) is the ambient pressure, \(\tau\) is the angular velocity of the vortex, \(\mu\) is the viscosity of the lubricant gas, \(\gamma\) is the perturbation ratio, \(\Lambda\) is the compressibility coefficient, \(\varepsilon\) is the eccentricity, and \(P\), \(Z\), \(H\), \(T\), \(H_{\text{c}}\), and \(K_{\text{b}}\) are the corresponding dimensionless parameters. After the displacement perturbation and velocity perturbation at the equilibrium position, the dimensionless perturbation is obtained as follows:  
\[\Delta X=\frac{\Delta x}{h_{0}}=|\Delta X|e^{iT},\;\Delta Y=\frac{\Delta y}{h_ {0}}=|\Delta Y|e^{iT} \tag{4}\]  
The Taylor series expansions for the gas film pressure, gas film thickness, and foil deformation with the perturbation parameters are expressed as follows:  
\[\left\{ \begin{array}{l}P_{t}=P+P_{x}\Delta X+P_{x}{}^{\prime}\Delta\dot{X}+ P_{y}\Delta Y+P_{y}{}^{\prime}\Delta\dot{Y}\\ H_{t}=H+H_{x}\Delta X+H_{x}{}^{\prime}\Delta\dot{X}+H_{y}\Delta Y+H_{y}{}^{ \prime}\Delta\dot{Y}\\ U_{t}=U+U_{x}\Delta X+U_{x}{}^{\prime}\Delta\dot{X}+U_{y}\Delta Y+U_{y}{}^{ \prime}\Delta\dot{Y}\end{array} \right. \tag{5}\]  
After considering the time term, the gas film thickness deviates from the static gas film thickness \(H\), and the gas film thickness under perturbation is obtained as follows:  
\[H_{t}=H+(\Delta X\sin\theta-\Delta Y\cos\theta)+\Delta U_{t} \tag{6}\]  
Combining Equations (5) and (6), the following can be deduced:  
\[\left\{ \begin{array}{l}H_{x}=U_{x}+\sin\theta,H_{x}{}^{\prime}=U_{x}{}^{ \prime}\\ H_{y}=U_{y}-\cos\theta,H_{y}{}^{\prime}=U_{y}{}^{\prime}\end{array} \right. \tag{7}\]  
According to the relationship between the gas film pressure and the deformation of the bump foil, the following can be obtained:\[P=K_{\rm b}U+C_{\rm b}\gamma\frac{\partial U}{\partial T}+1 \tag{8}\]  
Substituting Equations (5) and (7) into Equation (8) gives the following equation, where \(C_{\rm b}\) is the dimensionless damping of the foil:  
\[\left\{ \begin{array}{l}P_{x}=K_{\rm b}H_{x}-C_{\rm b}{H_{x}}^{\prime}-K_{\rm b }\sin\theta\\ P_{x}^{\prime}=K_{\rm b}{H_{x}}^{\prime}+C_{\rm b}{h_{x}}-C_{\rm b}\sin\theta \end{array} \right.\left\{