Solving Reynolds Equation Using Finite Difference Method

The finite difference method is a widely used numerical technique for solving partial differential equations. It discretizes continuous space by replacing differential terms with finite difference approximations, transforming differential equations into algebraic systems. The Reynolds equation is fundamental in fluid lubrication mechanics, describing pressure distribution in applications like bearings and seals.

Solving the Reynolds equation presents challenges due to its nonlinear nature and sensitivity to boundary conditions. The finite difference approach divides the lubrication domain into a grid, employing central/forward/backward differences at each node to approximate derivatives, ultimately forming linear or nonlinear equation systems.

MATLAB implementation typically involves: grid generation using meshgrid(), difference scheme selection (e.g., second-order central differences), boundary condition application through index manipulation, iterative solving using Gauss-Seidel or Newton-Raphson methods, and post-processing with visualization functions like contourf(). For steady-state problems, relaxation factors (ω) are often incorporated in iterative solvers to ensure convergence, while transient formulations require time-marching schemes like implicit Euler method.

Due to nonlinearities, convergence monitoring through residual norms is crucial. Engineering considerations like surface roughness and thermal effects can be integrated through modified source terms or boundary conditions in the discretized equations, implemented via conditional statements or custom function handles in the numerical code.