An arbitrary Lagrangian–Eulerian formulation for the Reynolds equation considering the JFO boundary condition

Abstract In the realm of lubrication theory, the hydrodynamics of the fluid lubricant are commonly elucidated through the Reynolds equation. Its establishment stems from using the Eulerian description method, restricting the fluid domain to a fixed spatial configuration. However, certain scenarios necessitate coupling the Reynolds equation with solid deformation, wherein the fluid boundary undergoes alterations concurrent with the displacement or deformation of the solid. In such cases, the employment of a movable fluid domain becomes imperative. To address this requirement, this contribution introduces the arbitrary Lagrangian–Eulerian method, which effectively transforms the reference system of the Reynolds equation into a fluid domain amenable to arbitrary movement. Subsequently, the weak formulation associated with the movable fluid domain is derived and synergistically combined with a preexisting cavitation formulation. This integration facilitates an efficient solution of the Reynolds equation utilizing the finite element method while duly accounting for the mass‐conserving cavitation effects. Ultimately, a transient hydrodynamic lubrication example is presented to demonstrate the feasibility of the newly developed formulations.