# Analysis of an Elasto-Hydrodynamic Seal by Using the Reynolds Equation
## 2 Materials and Methods
### Design Methodology
The flowchart for the fully coupled simulations is shown in Figure 10.4 The pressure at the inlet of the thin film flow region is applied as a uniform pressure load on the outer surface of the sleeve with the pressure variation along the clearance applied as the pressure load along the inner surface of the sleeve. The pressure field inside the clearance is obtained by solving the Reynolds equation, which is a function of the film thickness (\(h\)). The film thickness is, in turn, affected by the sleeve deformation, which makes the problem bidirectionally coupled. COMSOL Multiphysics is used to solve the bidirectionally coupled physics equations. COMSOL uses Finite Element methods to discretize the elastostatic and the Reynolds equations to form the respective system of equations. The force term in the elastostatic system of equations is a function of the pressure flow field. In the case of the flow field system of equations, the stiffness matrix is a function of (deformation, pressure) and the load vector is a function of pressure. To solve the bidirectionally coupled system of equations,  
Figure 10.5 Flowchart for fully coupled simulations. the fully coupled solver of COMSOL is used with the nonlinear Newton's method. The MUMPS direct solver is used for solving the linear system of equations in the linearized step of the nonlinear Newton's method [42]. In the fully coupled solver, the Reynolds equation is solved to obtain the pressure field inside the clearance, and the pressure field is applied as the boundary load on the inner surface of the sleeve. The elastostatics equations are then solved to determine the structural deformation on the sleeve. The change in the seal clearance height is fed back to the Reynolds equation to calculate the updated pressure. This procedure was performed back and forth until convergence was achieved for the final pressure distribution in the clearance as well as the final deformation of the seal. When carrying out CFD/FEA simulations, as a general rule of thumb, the simulation results need to be validated either with analytical solutions or experimental data. At the time of writing this manuscript, no experimental data was available for the EHD seal design. Thus, we performed an analytical case study to compare its results with the simulation data so that we can prove the fidelity of our simulations. Analytical solutions are usually available for relatively simplified models, while the solutions of the coupled, highly nonlinear partial differential equations require the use of computer simulations. Therefore, we reduced the fully coupled Fluid-Solid-Interaction (FSI) problem to a one-way coupled FSI one. Further, although the one-dimensional Reynolds equation can conveniently be applied to the flow domain, the deformation of the seal is a 2D problem, and solving a 2D elastostatics problem analytically is quite challenging. Therefore, we only compared the theoretical and computational results of the flow domain to validate our simulation methodology. Next, we discuss the analytical modeling of the flow domain. The sleeve was made from structural steel, and carbon dioxide was used as the working fluid. The density and the dynamic viscosity of carbon dioxide are both functions of pressure and temperature. For comparison purposes, the density of the carbon dioxide was modeled by using the ideal gas equation of state, which is given by  
Figure 10.6  
where \(R\) is the gas constant and \(T\) is the working temperature which was considered constant in this study. The viscosity is normally a function of temperature and pressure, but for simplicity, the viscosity was considered to be constant for the analytical comparison. Integration of Equation (2), using the ideal gas equation for density and applying the boundary conditions, yields  
Figure 10.7  
where the pressures are in terms of absolute pressure.