Numerical Solution of Reynolds Equation for a Compressible Fluid Using Finite Volume Upwind Schemes

When two surfaces interact, friction is inevitable and the lubrication of interacting surfaces becomes a necessity for all sliding systems in order to minimize friction. Modelling and the study of fluid flow in thin gaps is an important aspect of classical lubrication theory. The governing differential equation that describes the pressure distribution in the lubricating film between two surfaces is known as the Reynolds equation. This equation can be solved using various numerical methods to obtain the pressure distribution. In the majority of cases, the solution of the Reynolds equation is obtained by finite difference methods, because of its simplicity in implementation. For an incompressible fluid, this method easily results in a converged solution. However, for a compressible fluid, it is observed that the solution of the Reynolds equation with finite difference method using second-order central difference schemes exhibits instabilities. These instability issues, which is purely numerical in origin, can be addressed by using finite volume discretization and upwind schemes. Hence, the present chapter discusses the use of upwind schemes and th solution of a compressible Reynolds equation using multi-stage Runge-Kutta methods. Further, the use of these solution methods is elaborated with the help of a case study. Steady-state characteristics of externally pressurized porous gas journal bearings are presented. The governing equation that describes the flow of a thin film of lubricant separating two rigid surfaces in relative motion is given by the Reynolds equation, a partial differential equation, which can be derived from the Navier-Stokes equation. The exact close-form solution of the Reynolds equation exists only for a few fortunate cases. However, for a complicated realistic system, the absence of close form solution to the full Reynolds equation renders it unsolvable by known analytical methods. Researchers have therefore resorted to numerical methods to solve the problem. The numerical solution of some form of Reynolds equation is usually required in many fluid-film lubrication analyses or bearing designs. The solution of the hydrodynamic lubrication problem requires obtaining an approximate numerical solution to the Reynolds equation. To discretize the Reynolds equation, finite volume discretization is used as it uses the governing equation in conservative form and ensures the conservation of all properties at control volume.