Existence and Uniqueness of Reynolds equation under natural boundary conditions

TL;DR

This paper establishes the existence and uniqueness of solutions to the Reynolds equation under natural boundary conditions, providing theoretical proofs, an algorithm, and discussing practical implications in engineering.

Contribution

It proves the uniqueness and existence of solutions under natural boundary conditions and introduces a convergent algorithm with theoretical bounds.

Findings

Proved the set of boundaries with positive solutions has a unique greatest element.

Established the zero setting method has only one iterative error solution.

Provided an algorithm reaching the theoretical upper bound for 1D Reynolds equation.

Abstract

For the problem of solving Reynolds equation under natural boundary conditions, the corresponding hypothetical solution can be obtained by assuming the free boundary. If the solution satisfies natural boundary conditions, then the boundary is the boundary we are looking for. Obviously, there is a set S formed by all the boundaries that assume the solution is positive. We prove equivalence between maximum element of S and natural condition. We prove the closeness of set S under addition. Therefore, we prove that the set S must have a unique greatest element. Furthermore, we obtain the uniqueness and existence of solutions of Reynolds equation under natural boundary conditions. In engineering, the zero setting method is often used to find the boundary of Reynolds equation, and we also give the proof that the zero setting method has only one iterative error solution. We give an algorithm for solving Reynolds equation under one-dimensional conditions, which reaches the theoretical upper bound. We discuss the physical meaning of this method at the end.