# Numerical Analysis via Mixed Inverse Hydrodynamic
## 2 Mathematical Model
### Fluid Mechanics  
#### 2.2.1 Inverse Hydrodynamic Lubrication Method, IHL  
The behavior of a fluid in a thin film follows the Reynolds equation. Assuming that the working fluid is an incompressible and Newtonian fluid, the Reynolds equation can be expressed as:  
\[\frac{\partial}{\partial x}\left(\frac{h^{3}}{\mu}\frac{\partial p}{\partial x }\right)+\frac{\partial}{\partial y}\left(\frac{h^{3}}{\mu}\frac{\partial p}{ \partial y}\right)=6\left(u\frac{\partial h}{\partial x}+2\frac{\partial h}{ \partial t}\right) \tag{2}\]  
where \(p\) is the pressure at the contact, \(h\) is the thickness of the oil film, and \(\mu\) is the fluid viscosity. In this study, the Reynolds boundary condition is applied to hydrodynamic lubrication analysis, so it is not calculated below the reference pressure (0 MPa). That is, even if a rapid pressure change occurs during the instroke/outstroke, it is replaced by 0 below the reference pressure. The approach provides an efficiency solution to the oil film pressure, as the Reynolds boundary condition can be easily realized in the numerical solution of the Reynolds equation. Figure 1: Finite element model: (**a**) seal assembly and (**b**) simulation. Figure 2: Static contact pressure distribution, outstroke. If the seal width is short, a steady state can be assumed and, because it is axisymmetric, Equation (2) can be simplified to Equation (3) [5]. \[\frac{\partial}{\partial x}\left(\frac{h^{3}}{\mu}\frac{\partial p}{\partial x} \right)=6u\frac{\partial h}{\partial x} \tag{3}\]  
Integrating Equation (3) gives Equation (4). \[\frac{dp}{dx}=6\mu u\frac{h-h_{0}}{h^{3}} \tag{4}\]  
where \(h_{0}\) is the oil film thickness at the position where \(\partial p/\partial x=0\), that is, the position where the pressure is the greatest. Substituting Equation (4) into Equation (3) gives:  
\[h^{3}\frac{d^{2}p}{dx^{2}}+3\frac{dh}{dx}\left(h^{2}\frac{dp}{dx}-2\mu u \right)=0 \tag{5}\]  
The film thickness at the inflection point \(I\) is calculated from Equation (6). \[h_{I}=\sqrt{2\mu u/\left(\frac{dp}{dx}\right)_{max}} \tag{6}\]  
By substituting Equation (6) into Equation (4), the relationship between \(h_{0}\) and \(h_{1}\) can be obtained. \[h_{0}=\frac{2}{3}h_{I}=\frac{1}{3}\sqrt{8\mu u/\left(\frac{dp}{dx}\right)_{ max}} \tag{7}\]  
Using Equation (4) with nondimensional variables \(\lambda\) and \(H(x)\), Equation (9) can be obtained to calculate the distribution of the oil film:  
\[\lambda=\frac{h_{0}{}^{2}}{6\mu u}\frac{dp}{dx},\ \ H(x)=\frac{h(x)}{h_{0}} \tag{8}\]  
\[\lambda H(x)^{3}-H(x)+1=0 \tag{9}\]  
As shown in Equation (9), the Reynolds equation was developed as a cubic algebraic equation for film thickness. The imaginary root must be correctly identified and solved when calculating the root of a cubic polynomial. Otherwise, convergence to the correct solution may not be possible owing to numerical instability. The type set of roots of the cubic polynomial is governed by \(\lambda\) in Equation (8), i.e., it depends on the value of the pressure gradient at a specific position \(x\). The set of roots can be classified into three types based on the pressure gradient. If the pressure gradient has a positive \(x\) value and is smaller than \(\left(\partial p/\partial x\right)_{max}\), the equation has one negative real root and two positive real roots; therefore, a value that makes the oil film distribution continuous among the two positive real roots. In the section where the pressure gradient has a positive value and is larger than \(\left(\partial p/\partial x\right)_{max}\), an effective film thickness cannot be defined because the equation has a pair of complex conjugate roots and a negative real root. Therefore, in IHL theory, the oil film thickness in this section is assumed to be the oil film thickness at the inflection point, which corresponds to the horizontal straight section in Figure 3 and the oil film distribution.