HAL Id: hal-00822388

https://hal.archives-ouvertes.fr/hal-00822388

Submitted on 15 May 2013

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of sci-

entic research documents, whether they are pub-

lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diusion de documents

scientiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

About an inverse problem for a free boundary

compressible problem in hydrodynamic lubrication

K. Ait Hadi, Guy Bayada, M. El Alaoui Talibi

To cite this version:

K. Ait Hadi, Guy Bayada, M. El Alaoui Talibi. About an inverse problem for a free boundary

compressible problem in hydrodynamic lubrication. Journal of Inverse and Ill-posed Problems, De

Gruyter, 2015. �hal-00822388�

Noname manuscript No.

(will be inserted by the editor)

About an inverse problem for a free boundary

compressible problem in hydrodynamic lubrication

K. Ait Hadi · G. Bayada ·

M. El Alaoui Talibi

Received: date / Accepted: date

Abstract In this paper an inverse problem is considered for a non coercive

partial diﬀerential equation, issued from a mass conservation cavitation model

for a slightly compressible ﬂuid. The cavitation phenomenon and compress-

ibility take place and are described by the Elrod model. The existence of an

optimal solution is proven. Optimality conditions are derived and some nu-

merical results are given.

Keywords Inverse problem · Elrod model · optimality system · numerical

results.

Mathematics Subject Classiﬁcation (2000) MCS 35J70 · MCS 35Q35 ·

MCS 49K20 · MCS 65N21 · MCS 65N30.

1 Introduction

The present work comes within the scope of inverse problems in hydrodynamic

lubrication. Numerous works are based upon the computation of the solution

of various forms of the Reynolds partial diﬀerential equation established in

1886. This equation enables one to compute the hydrodynamic pressure (P )

Khalid Ait Hadi

Universit´e Cadi Ayyad, Laboratoire LIBMA de Math´ematiques Appliqu´ees Facult´e des Sci-

ences Semlalia, D´epartement de Math´ematiques, BP 2390, Marrakech Morocco.

E-mail: k.aithadi@ucam.ac.ma

Guy Bayada

Universit´e de Lyon, INSA , Institut Camille Jordan , UMR CNRS 5208 , Bat . Leonard de

Vinci Math 69621 Villeurbanne Cedex France.

E-mail: guy.bayada@insa-lyon.fr

Corresponding author: Mohamed El Alaoui Talibi

Universit´e Cadi Ayyad, Laboratoire LIBMA de Math´ematiques Appliqu´ees Facult´e des Sci-

ences Semlalia, D´epartement de Math´ematiques, BP 2390, Marrakech Morocco.

E-mail: elalaoui@uca.ma

2 K. Ait Hadi et al.

in a lubricated device, from data like the velocities distribution on the surfaces

surrounding the thin ﬁlm ﬂow, the gap between these surfaces(or equivalently

the shape of one of these surfaces) and the rheological characteristics of the

ﬂuids. However, due to severe operational conditions(the gap between the sur-

faces can be of some micrometers only and the relative velocity of the surfaces

some meters/second), some of these data are not really well known.

This is the case for example of the surfaces which are deformed from an initial

known shape by the hydrodynamic pressure inside the ﬂuid. Experimentally,

the knowledge of this pressure (P ) can now be obtained with a good precision.

It becomes then possible to ﬁnd the real shape (h) of the surfaces by solving an

inverse problem with Reynolds equation as state equation. Such information

is important as if the related gap is too small, it means that some contact is

possible between the two surfaces thus inducing wear and possible failure of

the device.

Two other very similar problems in the lubrication ﬁeld can be mentioned.

The ﬁrst one is to ﬁnd the gap (h) such that the pressure (P ) is the greatest

possible. In some speciﬁc situation the solution is the Rayleigh-step bearing

[31] in which the optimum gap h is a discontinuous function. More recently,

generalized Reynolds equation for heterogeneous slip/no slip engineered sur-

faces have been proposed [9]. As a consequence, a no-slip condition for the

velocity is valid on some part of the ﬂuid boundary and slip occurs on the

other part. Due to this non homogeneity, the resulting Reynolds equation con-

tains discontinuous coeﬃcients. Finding the best location of the slip/no-slip

regions is mathematically close to the previous optimization problem.

Another diﬃculty must be considered in these aforementioned identiﬁcation-

optimization problems. It is due to cavitation, the well-known phenomenon in

ﬂuid mechanics when ﬂuid is no longer homogeneous and takes some diphasic

aspect with the appearance of air bubbles. This phenomenon occurs very often

in lubrication and cannot be ignored, especially as the gap (h) is not constant.

Taking cavitation into account implies considering a new non linear operator

as state equation instead of the classical Reynolds equation. Various models

exist in the mechanical literature. The most common one is based upon a vari-

ational inequality for the pressure [27]. However, as it is not a mass preserving

model, it is often replaced by the Jacobsson Floberg Olssen (JFO) model, in

which a new variable θ is introduced. This variable describes the local pro-

portion of ﬂuid (or saturation) considering the presence of air bubbles in the

ﬂow [25]. In a widely referenced paper [20] , it has been pointed out that the

(JFO) model is equivalent to a free boundary complementary problem with

two unknowns, pressure P and saturation θ with

P ≥ 0, 0 ≤ θ ≤ 1, P (1 − θ) = 0

Existence and uniqueness aspects of such models have been mathematically

studied in a lot of papers [2] . However another aspect of the Elrod-Adams pa-

per has been less studied in the mathematical literature: authors try to recover

the somewhat heuristic JFO model by introducing a small compressibility in

the classical (non cavitated) Reynolds equation. They show that the JFO

About an inverse problem for a free boundary 3

model can be obtained as the limit of this compressible model and propose

some numerical procedures to solve it. Although being somewhat heuristic,

this result has been supported by recent rigorous mathematic results [14] for

the asymptotic thin ﬁlm compressible Navier Stokes system. The interest of

the more physical compressible Elrod-Adams model is that there is only one

unknown (the density of lubricant) instead of two (P and θ) in the JFO model.

The price to pay is that the state equation is a non linear degenerate partial

diﬀerential equation. The importance of the choice of a model of cavitation in

the simulation of lubrication devices has often been mentioned [9, 4].

Although these inverse-optimization problems are of permanent interest in the

mechanical literature [3, 12, 15, 17, 19, 21, 29], few mathematical works ex-

ist. One of the ﬁrst attempts [29] is devoted to the optimum Rayleight step

bearing. In [30] the identiﬁcation procedure for a gap with known pressure

is studied. However, in these works cavitation is not considered. In [8], varia-

tional inequality is chosen to model the cavitation and optimality problems are

studied by way of penalization. The Elrod-Adams model is considered in [19]

using some regularizations which enables one to gain optimality conditions.

The present study addresses the identiﬁcation process of the gap for a given

pressure and takes the cavitation into account by way of the slightly compress-

ible model proposed by Elrod-Adams. This model has also the advantage of

preserving the mass ﬂow.

This problem presents many mathematical diﬃculties essentially bound to the

complexity of the criterion considered as a function of the thickness. It is not

even a locally Lipschitz function. We are lead to consider a more general func-

tional framework which is the space of the functions of bounded variation. To

deal with these diﬃculties a double regularization is introduced by approach-

ing the degenerate part of the state equation with a particular sequence of

monotonous and continuously diﬀerentiable functions. Thus a sequence of reg-

ularized control problems can be obtained.

The paper is organized as follows. Section 2 is devoted to the physical

background, the statement of the problem and the proof of the existence of a

control. In section 3, we formulate the regularized control problem for which

an existence result is proved. Optimality conditions are given in section 4,

and some a priori estimates are obtained, so allowing us to pass to the limit.

The optimality system for the initial problem is derived in the one-dimensional

case. In the last section some numerical results show that the optimality system

associated to an inverse algorithm is eﬃcient.

Finally let us mention that from a mathematical aspect, the problem here

studied is very close to the one of the ”dam problem” [13] with unsaturated

porous media or multiple ﬂuid saturated porous media. In this case, the un-

knowns to be determined are the value of the porosity parameters of the soil

[18] and the measured data is the hydraulic pressure.

4 K. Ait Hadi et al.

2 Formulation of the problem and existence of control

2.1 Description of the physical problem (Elrod-Adams model)

Let Ω the square ]0, 1[×]0, 1[ of the (x, y) plane, Γ

0

= {(x, y) ∈ ∂Ω, x = 0},

and Γ = ∂Ω\Γ

0

. The lubricant is assumed to be contained in a three dimen-

sional volume between the lower ﬂat surface Ω with horizontal velocity

−→

U and

an upper one describe by z = h(x, y) at rest. If h is small, the pressure is

known to be independent from z and to obey the following Reynolds equation

deﬁned on Ω [23].

div

ρ

12η

h

3

∇P

= div

h

−→

U ρ

2

!

on Ω, (1)

where P (x, y) is the pressure of the lubricant, h(x, y) is the ﬁlm thickness, ρ

is the density of lubricant, η is the viscosity,

−→

U is the relative velocity of the

surfaces of the mechanism in which lubricant takes place.

For a slightly compressible ﬂuid like water or oil, the law linking P and ρ is

usually described by introducing the bulk number β with: see [25]

P = β log(ρ), ρ > 1, P > 0, (2)

Taking into account the possible existence of a cavitation area where P is zero

and ρ ≤ 1 lead to generalize (2) in

P = β log((ρ − 1)

+

+ 1), ρ ≥ 0, β > 0.

in which f

+

= sup(f, 0). Rewriting (1) in term of ρ, we gain by diﬀerentiating

in the distributional sense

div

β

h

3

12η

∇(ρ − 1)

+

− ρ

h

−→

U

2

!

= 0. (3)

Introducing u = β(ρ − 1) as the primary unknown, equation (3) becomes

div

h

3

12η

∇u

+

−

1 +

1

β

u

h

−→

U

2

!

= 0.

In the sequel, we will consider more precisely the case of a journal bear-

ing (see Figure 1) with a supply line located at x = 0. It is a usual lu-

brication device in which a known quantity of ﬂuid Θ

0

is supplied through

Γ

0

= {(x, y) ∈ ∂Ω, x = 0, 0 < y < 1} while the pressure is assumed to be

known on Γ = ∂Ω\Γ

0

and equal to zero (the atmospheric pressure).

For simplicity sake we assume η =

1

6

,

−→

U ==

1

0

so that the weak formula-

tion of the state problem reads :

(P

h

)

Find u ∈ L

2

(Ω) verifying u

+

∈ V,

1 +

1

β

u

≥ 0

R

Ω

h

3

∇u

+

· ∇φ dxdy −

R

Ω

h

1 +

1

β

u

∂φ

∂x

dxdy =

R

Γ

0

Θ

0

φ dy ∀φ ∈ V

Θ

0

∈ L

∞

(Γ

0

), Θ

0

≥ 0