# Analysis of an Elasto-Hydrodynamic Seal by Using the Reynolds Equation
## 2 Materials and Methods
### Design Methodology
The general form of the Reynolds equation in the COMSOL handbook is given by [42]:  
\[\frac{\partial}{\partial x}\left(-\frac{\rho h^{3}}{12\mu}\frac{\partial P}{ \partial x}\right)+\frac{\partial}{\partial y}\left(-\frac{\rho h^{3}}{12\mu} \frac{\partial P}{\partial y}+\frac{\rho h}{2}(v_{a}+v_{b})\right)=0 \tag{1}\]  
where \(\rho\) is the density of the fluid, \(h\) is the film thickness (or the height of the clearance), \(\mu\) is the dynamic viscosity, \(P\) is the pressure, \(v_{a}\) is the rotational velocity of the rotor, and \(v_{b}\) is the rotational velocity of the sleeve. In this study, the sleeve was stationary and the effect of  
Figure 2: Working mechanism of the proposed EHD seal: (**a**) initial condition and (**b**) operating condition. the rotational speed of the rotor was neglected. Hence, the governing Reynolds equation was simplified to:  
\[\frac{\mathrm{d}}{\mathrm{d}x}\left(\frac{\rho h^{3}}{12\mu}\ \frac{\mathrm{d}P}{ \mathrm{d}x}\right)=0 \tag{2}\]  
The fluid flowing through the clearance applies pressure on the sleeve and causes structural deformation in the sleeve. The deformation in the structure is governed by the elastodynamics equation as shown below:  
\[\
abla\cdot\mathbf{\sigma}+\mathbf{F}=\rho_{seal}\bar{\mathbf{u}} \tag{3}\]  
where \(\mathbf{u}\) is the deformation occurring in the structure, \(\mathbf{\sigma}\) is the Cauchy stress tensor, \(\mathbf{F}\) is the body force, and \(\rho_{seal}\) is the mass of the seal. \(\
abla\) is the Hamiltonian operator. In this work, the analyses were performed for steady cases, thus the acceleration term was set to zero. The body force was also neglected. Thus, Equation (3) renders into:  
\[\
abla\cdot\mathbf{\sigma}=0 \tag{4}\]  
The Cauchy stress tensor is a function of the deformation, which is governed by the Hooke's law. The simulation geometries for both the sleeve and the flow region, along with the boundary conditions and the meshing, are shown in Figure 3. The back ring, the stator, and the rotor are not modeled, and the deformations of these rigid structures are assumed to be zero. The boundary conditions at the inlet and outlet boundaries for the fluid flow along the clearance are given by:  
\[P_{inlet}=P_{0} \tag{5}\]  
\[P_{outlet}=P_{e} \tag{6}\]  
where \(P_{0}\) is the working pressure acting on the seal and \(P_{e}\) is the pressure at the outlet, which is equal to the atmospheric pressure. Nonetheless, the simulation methodologies presented in this study follow a sound scientific approach and lay the foundation for future model enhancements. The sleeve is fixed on the right end. The inner surface of the sleeve is subjected to pressure due to the fluid flow along the clearance, and the outer surface is subjected to uniformly distributed working pressure, which is equal to the inlet pressure (\(P_{0}\)). The structural deformation of the sleeve reduces the clearance, which in turn affects the flow field, and thus, the resulting pressure distribution along the clearance. The clearance is  
Figure 10.0 (**a**) Sleeve domain and (**b**) flow surface along with boundary conditions and meshing. modeled by the Thin-Film Flow module. In the Thin-Film Flow module, the thickness of the clearance is not geometrically modeled. The flow inside the clearance is modeled along the surface, and the film thickness effects on the flow are included in the flow equations. Figure 3b shows the surface along which the thin film flow is modeled, and this surface is the inner surface of the sleeve shown in Figure 3a. The horizontal lines in Figure 3 are geometrical edges, and they are used to generate the mesh. The bidirectional coupling between the structural deformation in the sleeve and the flow fields is governed by:  
Figure 10.1  
Figure 10.2  
Figure 10.3  
where \(\Delta h\) is the change in the clearance, which is equal to the structural deformation on the inner surface of the sleeve.