Lack of contact in a lubricated system

TL;DR

This paper investigates the mathematical modeling of a lubricated contact system, analyzing conditions for the existence and uniqueness of solutions and steady states in a simplified one-degree-of-freedom setup.

Contribution

It provides a rigorous analysis of the coupled Reynolds variational inequality and Newton's law for a lubricated contact problem with a single degree of freedom.

Findings

Proves global existence and uniqueness of solutions under certain geometric conditions.

Establishes conditions for the existence of steady states.

Analyzes the impact of initial conditions and applied load on system behavior.

Abstract

We consider the problem of a rigid surface moving over a flat plane. The surfaces are separated by a small gap filled by a lubricant fluid. The relative position of the surfaces is unknown except for the initial time $t=0$. The total load applied over the upper surface is a know constant for $t>0$. The mathematical model consists in a coupled system formed by Reynolds variational inequality for incompressible fluids and Newton$^{\prime}$s second Law. In this paper we study the global existence and uniqueness of solutions of the evolution problem when the position of the surface presents only one degree of freedom, under extra assumptions on its geometry. The existence of steady states is also studied.