Existence of Weak Solution for a Free Boundary Lubrication Problem

This paper is interested in a free boundary problem modelling a phenomenon of cavitation in hydrodynamic lubrication. We reformulate the problem (see Boukrouche, (1993)) in a large context by introducing two positive parameters, namely, 𝑁0 and 𝑎. We build a weak formulation and establish the existence of the solution to the problem.


Introduction
The lubrication fields have many applications; one example is the study of the rotary mechanisms such as the bearing, joints. The study is concerned in looking for a moving free boundary problem related to the cavitation modelling in lubrication see 1-4 . The experimental results make evidence of the occurrence of two distinct zones, one full of the fluid, namely, the saturated zone Ω t ; the other Ω c Ω c Ω \ Ω t , where Ω is the global domain , is the cavitated zone e.g., the mixture of fluid and air . Two approaches have been used to cope with phenomena. One of them 5 homogenizes the phenomena and considers it as a 2D phenomena, so introducing θ the saturation variable lubricant concentration ; the other one 6 takes full account of the three-dimensional character of this phenomena, with appearance of air bubbles and introduces in Ω c the relative height as supplementary unknown for more details, see 1, 3 . We use here the first approach but both approaches lead to the same mathematical problem. In this paper, we take the problem studied in 1-3 and rewrite it, here, in a large context, by introducing two positive parameters, namely, N 0 and a. This formulation of the problem gives as advantages a proportionality relation 2 ISRN Mathematical Analysis between N 0 and the pressure p, and the parameter a which allows the control of a squeezing effects. The mathematical modelling is made according to the model of Jakobsson-Floberg see 1, 3 , where the lubricant is not defined only by a pressure p but also by a saturation variable θ. This variable θ characterizes the cavitation phenomena, where θ ≡ 1 in Ω t and 0 ≤ θ < 1 in Ω c . The interface between Ω t and Ω c constitutes the moving free boundary denoted by Γ t . The problem is a convection-diffusion problem type, and the Reynolds equation is elliptic in the saturated zone and hyperbolic in the other one. We note that the study of existence and uniqueness point of views to the problem has been established in 3 in the particular case of the N 0 0 and a 1. For this, the author proved the existence of solution for this kind of problem by way of an approximation by an elliptic problem. In our work, we followed, exactly the same way, and the aim of this study is to construct a weak formulation and establish the existence of solution to the problem.
The plan of the paper is as follows. Section 2 proposes a state of the problem and a weak formulation. Section 3 introduces an elliptic nonlinear problem and gives the existence and uniqueness of the solution to this problem. Section 4 proposes an approximation of the elliptic nonlinear problem by a family of linear problems and proves a priori estimate. Last section gives a theorem of existence of the solution to the problem.

Description of the Phenomenon
We consider a global domain Ω with border ∂Ω. The fluid is injected at a given rate w over the fixed internal boundary Γ I Γ ex ∂Ω \ Γ I . For each t ∈ 0; T , the experimental results make evidence of the occurrence of two distinct zones: one full of the fluid is the saturated zone Ω t , where the pressure p p > 0 and the saturation variable θ θ ≡ 1 , and the other Ω c Ω c Ω \ Ω t is the cavitated zone, where the pressure is constant p 0 and 0 ≤ θ < 1 e.g., the mixture of fluid and air . The free boundary of the region Ω t containing fluid is Γ t and the region Ω 0 , with border Γ I , occupied by the fluid at t 0 being given see Figure 1 .

State of the Problem
The strong formulation of the problem described the phenomenon is written as follows. For each t ≥ 0, find a pair p, θ ∈ L 2 0, T;

2.9
h t, x is the thickness of the thin film supposed a regular and given function of the problem. V is the speed of the axis supposed being given. υ is the moving free boundary with υ 0 on Γ I . n resp., n is the normal vector along Γ t resp., Γ I exterior to Ω t resp., Ω 0 . The saturation variable θ can be represented by a graph see, Figure 2 . In 2.1 -2.3 , there are the diffusion term ∇ K h, t, N 0 ∇p , the shearing term ∇ hV , and the squeezing term ∂h/∂t.

Weak Formulation
Before starting the construction of a weak formulation of the problem 2.1 -2.8 , we denote by

2.10
Indeed, multiplying 2.1 by ϕ ∈ E and integrating over 0, T × Ω t , we obtain where n ex is the normal vector along ∂Ω exterior to Ω.
In the same way, we apply to 2.2 By adding * and * * in all Ω Ω t ∪ Ω c , and using 2.3 -2.6 then the weak formulation can be written as follows. Find a pair p, θ ∈ L 2 0, T;

An Elliptic Nonlinear Problem
To solve the problem 2.11 -2.12 , we will approximate it by an elliptic problem in the same way as Boukrouche 3 and Gilardi 7 . Let β be a real function see Figure 3 satisfying the following assumptions: consider now the problem, given ε > 0, find p ∈ H 1 Q such that Introducing the operator τ is as follows:

3.4
If p ∈ H 1 Q , τ p is a unique solution q to the linear problem

3.6
Using Cauchy-Schwarz's inequality, we obtain where α is constant depending on h, N 0 , and ε. C 1 and C 2 are two constants depending on h, V , and a. As and β is Lipschitz continuous function, there exists a constant C depending on h, V , N 0 , a, and ε such that If p ε → p converge weakly in H 1 Q , then p ε → p, L 2 Q and p ε T → p T , L 2 Ω T . Thus τ p ε → τ p , then the continuity of τ is shown. Taking ϕ q in 3.5 and using Cauchy-Schwarz's inequality, we obtain q H 1 Q ≤ k 1 , where k 1 is a constant depending on h, V , N 0 , β, T , a, and ε.

Approximating Problems
In order to solve the problem 2.11 -2.12 , we consider a new family of problems of type 3.3 in which the function β is an approximation of the Heaviside function see Figure 4 . Therefore we consider a family of functions Consider now the following approximating problem.
For fixed ε and a where ε, a ∈ 0; 1 , find p ε such that Applying Poincare's inequality, we have where α 1 is constant depending on Ω.
As p ε L 2 Σ I ≤ α 2 ∇p ε 2 L 2 Σ I , we finally deduce the result. Passing to the limit in 5.7 , we deduce that p p − a.e. in Q.
As E H 1 0 0, T; V is dense in L 2 0, T; H 1 Ω , 2.12 can be rewritten in the following form: wϕ, ∀ϕ V, a.e. in 0, T . 5.14 Taking now p ε as test function in 5.14 and passing to the limit over ε, we deduce as w ≥ 0, therefore p − 0 a.e. in Q, that is, p ≥ 0 a.e. in Q.
Next work will consist in finding some existence of relationship between the pressure p and parameter N 0 and in completing numerical analysis study to the problem 2.1 -2.8 .