Abstract
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, and . We build a weak formulation and establish the existence of the solution to the problem.
1. 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 [1โ4]). The experimental results make evidence of the occurrence of two distinct zones, one full of the fluid, namely, the saturated zone ; the other (, where 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 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 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 [1โ3] and rewrite it, here, in a large context, by introducing two positive parameters, namely, and . This formulation of the problem gives as advantages a proportionality relation between and the pressure , and the parameter 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 but also by a saturation variable . This variable characterizes the cavitation phenomena, where in and in . The interface between and constitutes the moving free boundary denoted by . 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 and . 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.
2. Notations and State of the Problem
2.1. Description of the Phenomenon
We consider a global domain with border . The fluid is injected at a given rate over the fixed (internal) boundary (). For each , the experimental results make evidence of the occurrence of two distinct zones: one full of the fluid is the saturated zone , where the pressure ( and the saturation variable , and the other ( is the cavitated zone, where the pressure is constant () and (e.g., the mixture of fluid and air). The free boundary of the region containing fluid is and the region , with border , occupied by the fluid at being given (see Figure 1).
2.2. State of the Problem
The strong formulation of the problem described the phenomenon is written as follows.
For each , find a pair and such that where โ is the thickness of the thin film supposed a regular and given function of the problem. is the speed of the axis supposed being given. is the moving free boundary (with on ). (resp., ) is the normal vector along (resp., ) exterior to (resp., . The saturation variable can be represented by a graph (see, Figure 2).
In (2.1)โ(2.3), there are the diffusion term , the shearing term , and the squeezing term .
2.3. Weak Formulation
Before starting the construction of a weak formulation of the problem (2.1)โ(2.8), we denote by Indeed, multiplying (2.1) by and integrating over , we obtain where is the normal vector along exterior to .
In the same way, we apply to (2.2) By adding (*) and (**) in all , and using (2.3)โ(2.6) then the weak formulation can be written as follows.
Find a pair As implies that .
3. 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 be a real function (see Figure 3) satisfying the following assumptions: Put consider now the problem, given , find such that on , for all vanishing on .
Introducing the operator is as follows: If is a unique solution to the linear problem for every vanishing on .
Lemma 3.1. The operator is continuous from with the weak topology into with the strong topology. Moreover is bounded in .
Proof. Let with for .
Taking in (3.5), we have
Using Cauchy-Schwarz's inequality, we obtain
where is constant depending on , , and . and are two constants depending on , , and .
As
and is Lipschitz continuous function, there exists a constant depending on, , , , and such that
If converge weakly in , then and . Thus , then the continuity of is shown.
Taking in (3.5) and using Cauchy-Schwarz's inequality, we obtain , where is a constant depending on , , , , , , and .
Theorem 3.2. If the function satisfies hypothesis (3.1), then, for every , there exists a solution to the problem (3.3).
Proof. Use Lemma 3.1 and Schauder fixed-point theorem.
Theorem 3.3 (cf. [2, 3]). If the function satisfies hypothesis (3.1), then for every , the solution of the problem (3.3) is unique.
4. 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 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 and (where , ), find such that for every vanishing on .
From Theorems 3.2 and 3.3, we deduce the following theorem.
Theorem 4.1 (cf., [3]). For every and , there exists at least one solution to the problem (4.5). Moreover if and are sufficiently regular, every solution belongs to .
Lemma 4.2. If the function verifies (4.1)-(4.2) and , then one has
Proof. Putting , we have then and .
Using the integration by parts, we obtain
as , then Lemma 4.2 is shown.
The following proposition gives some a priori estimates for pressure .
Proposition 4.3. There exists a constant independent of , , and
Proof. Taking in (4.5), we obtain
By using Lemma 4.2, we have
Applying Poincare's inequality, we have
where is constant depending on .
As, we finally deduce the result.
Proposition 4.4. For every nonnegative and , there exists a constant such that
Proof. From (4.5), we have Multiplying (4.13) by and integrating over , we have Taking then we have Using Proposition 4.3., we obtain where and is constant of Proposition 4.3: We obtain finally
5. An Existence Theorem of the Problem (2.11)-(2.12)
Theorem 5.1. There exists at least one solution to the problem (2.11)-(2.12).
Proof. If and from Proposition 4.3, we can extract a subsequence of , still denoted by, such that
Moreover , for all , then there exists such that
We can now proof that the given in (5.1) and (5.3) is solution of the problem (2.11)-(2.12).
We denote by
Therefore , a.e. in and with a.e. in .
Thus if in (4.5), we find the problem (2.11)-(2.12). In order to prove and , a.e in , we give a brief demonstration (for more details, see [7, pages 1113-1114]).
First, we need to prove . We notice that lim
Now we have to prove that
We define and use (4.13).
Then the couple ( is solution of the problem (2.11)-(2.12).
Theorem 5.2. If on and , then a.e. in .
Proof. Following [7], we construct a sequence solution of the problem
where , and .
From the classical Cauchy-Lipschitz-Picard theorem [2], there exists a unique solution . Multiplying (5.7) by (resp., by ) and integrating over , we obtain
respectively,
Deriving (5.7) with respect to , multiplying by , and integrating over , we obtain
We deduce that there exists and such that
From (5.11) we have
Passing to the limit in (5.7), we deduce that a.e. in .
As is dense in , (2.12) can be rewritten in the following form:
Taking now as test function in (5.14) and passing to the limit over , we deduce
as , therefore a.e. in , that is, a.e. in .
Next work will consist in finding some existence of relationship between the pressure and parameter and in completing numerical analysis study to the problem (2.1)โ(2.8).
Acknowledgment
The author is grateful to Professor Boukrouche M. for a helpful discussion on this subject.