Lubrication is the process whereby a fluid (the lubricant) fills the gap between two surfaces in close proximity and relative motion. Lubrication is employed to reduce friction and avoid contac

Ciuperca, S.

Jay, M.

Tello del Castillo, Jose Ignacio

English

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Inverse Problems in Lubrication Theory

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International Conference on PDEs. For the 60th birthday of Michael Chipot
An inverse problem in Lubrication Theory
S. Ciuperca1, M. Jay2 and J. I. Tello3
1.- Universite´ de Lyon 1, Lyon, France
2.- INSA Lyon, Lyon, France
3.- Universidad Polite´cnica de Madrid, Spain
Poitiers, February 18-20 , 2010
Program:
1.- Introduction
2.- Modelling Reynolds equation
3.- An inverse problem in Journal Bearing systems with cavitation
1. Introduction
Lubrication is the process whereby a fluid (the lubricant) fills the gap
between two surfaces in close proximity and relative motion. Lubrication is
employed to reduce friction and avoid contact
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We assume 0 < h << L, 0 < h << B
• Lubrication is known from the ancient civilizations (Egyptians, Greeks...)
• During the middle age, animal fat is used to lubricate wagon wheels, ships
(rudder) etc..)
• Leonardo Da Vinci did experiments on friction and obtain experimental
data on friction coefficient.
• Industrial revolution brought machineries and the necessity to understand
the process of lubrication to improve the efficiency of the machines.
• XIX century, Petrov (student of Ostrogradsky) Frictional forces and the
relevance of the viscosity coefficient of the lubricant.
• Introduction of residues from the oil refining industry as lubricants to
substitute the animal and vegetable oils
• Reynolds introduce in 1886 Reynolds Equation
• Sommerfield 1904. Explicit solutions for the 1-dimensional journal bearing.
• Rayleigh 1908. Optimization in the shape of lubricated surfaces to maxi-
mize the load supported by the engine.
• A.G.M. Michel 1929, study the journal bearing system with axial oil flow.
• Elrod 1960, Elrod-Adams 1975 modeled the phenomenon of Cavitation:
Physical phenomenon whereby null pressure regions appear
• Cameron 1966.
• M. Chipot and M. Luskin 1986, studied the compressible Reynolds eq.
• Bushman, A. Friedman, Hu, Rodrigues, Litman,
• G.Bayada, M.Chambat, S.Ciuperca, I.Hafidi, M.Jay, S.Martin etc.. (Lyon)
• C.Va´zquez, J.Durany, I.Arregui, G.Garc´ıa, B.Cid etc.. (Galicia-Spain)
• JI Dı´az, J.Carrillo, S.A´lvarez, R.Oujja etc. (Madrid - Spain)
Lubrication is today studied in different processes
- Journal Bearing systems,
- Row ball bearing,
- Magnetic recording,
- Hard disk device,
- Earth Physics etc..
2.Modelling Reynolds equation
Starting from the usual conservation principles
mass conservation:
∂ρ
∂t
+ div (uρ) = 0,
momentum conservation: (e.g. incompressible case)
∂(ρu)
∂t
+ (u · ∇)(ρu) = −∇P + µ
2
div(∇u + (∇u)T ).
Dimensional analysis:
∂ρ
∂t +
∂(ρu1)
∂x +
∂(ρu2)
∂y +
∂(ρu3)
∂z = 0
↓ ↓ ↓ ↓
ρ0/T0 ρ0U1c/L ρ0U2c/B ρ0U3c/H
U3c
H
' U1c
L
;
U3c
H
' U2c
B
.
Then
U3c
U1c
<< 1,
U3c
U2c
<< 1.
X-component of N-S
∂ρu1
∂t + u1
∂ρu1
∂x + u2
∂ρu1
∂y + u3
∂ρu1
∂z = −∂P∂x + µ∆u1
↓ ↓ ↓ ↓ ↓ ↓
ρ0U1c
T0
ρ0U
2
1c
L
U2cρ0U1c
B
U3cρ0U1c
H δxP µ
U1c
L2
µU1c
B2
µU1c
H2
Re =
UcL
µ
; − ∂P
∂x
+ µ
∂2u1
∂z2
= 0.
X -Component − ∂P
∂x
+ µ
∂2u1
∂z2
= 0,
Y -Component − ∂P
∂y
+ µ
∂2u2
∂z2
= 0,
Z -Component: − ∂P
∂z
= 0.
B.C. u1 = u2 = 0, u3 =
∂h
∂t
, on z = h(x, y),
u1 − U1 = u2 − U2 = u3 = 0, on z = 0.
Then
u1 =
1
2µ
∂P
∂x
z(z − h) + U1(1− z
h
),
u2 =
1
2µ
∂P
∂y
z(z − h) + U2(1− z
h
).
We define
qx :=
∫ h
0 u1dz =
U1h
2
− h
3
12µ
∂P
∂x
,
qy :=
∫ h
0 u2dz =
U2h
2
− h
3
12µ
∂P
∂y
.
Integrating (1)
∂(hρ)
∂t
+
∂
∂x
(ρqx) +
∂
∂y
(ρqy) = 0.
Reynolds Equations (1886)
Incompressible fluids
∂
∂x[U1h− h
3
6µ
∂P
∂x ] +
∂
∂y [U2h− h
3
6µ
∂P
∂y ] = − ∂∂th, in Ω,
P − P0 = 0, on ∂Ω.
−div(h3∇P ) = −6µU∇h− ∂h
∂t
in Ω, +BC
Cavitation has been modeled in different ways:
• Considering the positive part of the solution to Reynolds equation (p)+
• Variational inequalities

−div
[
h3
6µ∇p
]
≥ −U ∂∂xh− ∂∂th, p ≥ 0
(
−div
[
h3
6µ∇p
]
+ U ∂∂xh +
∂
∂th
)
p = 0 on Ω
p = p0 on ∂Ω.
• Elrod-Adams Model

−div
[
h3
6µ∇p
]
= −U ∂∂x(θh)− ∂∂t(θh), in Ω,
θ ∈ H(p),H is the heaviside function
p = p0, on ∂Ω.
3.- Journal Bearing system with cavitation
• Unknown position of the inner
cylinder which support a given load
• We only allow to the inner cylinder
to displace parallel to the outer one
• 2-parameters inverse problem
OOs
h1
h2
w
Scheme of the journal bearing
• Os the center of the shaft,
• G. Bayada 1972, J. Durany, J. Pereira, F. Varas. Numerical sol. 2008
• Cartesian coordinates of Os, (η1, η2) Polar coordinates of Os, (η, α)
• η1 = η cosα, η2 = η sinα.
• h(θ, η, α) = ρ(θ)− η cos(θ − α) = ρ(θ)− η1 cos θ − η2 sin θ.
• 0 ≤ x ≤ 1
• Ω := S1 × (0, 1)
Unknowns:
• the pressure p : (θ, x) ∈ Ω→ IR
• the position of the inner cylinder (given by Os := (η, α))
Known Data:
• Load supported by the shaft (F = (F1, F2)),
• viscosity µ, (assumed constant)
• velocity (assumed constant) and geometry of the surfaces (given by ρ).
To find:
p ∈ K :=
{
ϕ ∈ H10(Ω) : ϕ ≥ 0
}
and
(η1, η2) ∈ A := set of admissible positions
such that ∫
Ω
h3∇p · ∇(ϕ− p) ≥
∫
Ω
h
∂
∂θ
(ϕ− p) ∀ϕ ∈ K
∫
Ω
p cos θdθdx = F1
∫
Ω
p sin θdθdx = F2
where h(θ, η, α) = ρ(θ)− η cos(θ − α) = ρ(θ)− η1 cos θ − η2 sin θ
Assumptions:
ρ ∈ C3(IR), ρ, is 2pi − periodic.
ρ′′(θ) + ρ(θ) > 0, ∀ θ ∈ [0, 2pi) (technical assumption)
min
0≤θ≤2pi ρ(θ) = 1.
Notation:
a(α) := min
α−pi2<θ<α+pi2

ρ(θ)
cos(θ − α)
 .
A := {(η, α); 0 ≤ η < a(α)}
Let G = (G1, G2) : A→ IR2,
G1(η1, η2) :=
∫
Ω
p cos θdθdx− F1 (1)
G2(η1, η2) :=
∫
Ω
p sin θdθdx− F2 (2)
for p defined as the unique solution to the variational inequality for fixed η1, η2.
Thus the problem is equivalent to
Find (η1, η2) ∈ A such that G(η1, η2) = (0, 0)
Theorem 1 For any F ∈ IR2 there exists, at least, a solution to G = 0.
For fixed (η1, η2) there exists a unique solution to the variational problem p (see Kinderlehrer-
Stampacchia 1980) and G is well defined.
Some basic results in Degree Theory in IRn
Definition.
Let S be a bounded open subset of IRn and f : S → IRn, f ∈ C1(S). Let y0 ∈ IRn such
that y0 6∈ f (∂S), suppose that Df (x) is invertible for all x ∈ f−1(y0). We define the degree
of f in S at y0 by:
d(f, S, y0) :=
n∑
i=1
sign(det(Df (xi))),
for all xi ∈ f−1(y0).
Theorem 2 Let S be a bounded open set in IRn. Let f0 and f1 be two continuous
functions from S to IRn. We assume moreover that S is a star domain with respect to
the point y0 ∈ S and that
[f0(x)− y0]t · [f1(x)− y0] > 0, for any x ∈ ∂S.
Then
d(f0, S, y0) = d(f1, S, y0).
Properties of a(α)
•
a(α) := min
α−pi2<θ<α+pi2

ρ(θ)
cos(θ − α)
 .
• a(α) represents the maximum displacement of Os in the α-direction.
• Since
lim
θ→α±pi2
ρ(θ)
cos(θ − α) =∞
we guarantee the existence of at least one minimum.
• Since ρ′′ + ρ > 0 the minimum is unique.
• θα represents the contact point, it is unique and C1.
•
1 = min
0≤θ≤2pi ρ ≤ a(α) ≤ max0≤θ≤2pi ρ = ρM .
• There exists s < pi2 such that |θ˜α − α| < s
• We introduce
A = {(η1, η2) : 0 ≤ η ≤ a(α)− },
• We fix (η, α) ∈ ∂A
• We take ϕ = 0 and ϕ = 2p as test functions in the variational inequality to obtain
∫
Ω
h3 |∇p|2 = −
∫
Ω
p
∂h
∂θ
• Notice that
−
∫
Ω
p
∂h
∂θ
∼ (
∫
Ω
pcosθ,
∫
Ω
psinθ) · (−η2, η1)t
• We introduce
h˜(θ, α) := h(θα, a(α)− , α) + [ρ′′(θα) + ρ(θα)](1− cos(θ − θα)).
•
−
∫
Ω
p
∂h˜
∂θ
= c1(
∫
Ω
pcosθ,
∫
Ω
psinθ)t · (η sin θα,−η cos θα) in (η, θα) ∈ ∂A
For (η, θα) ∈ ∂A we have
(
∫
Ω
pcosθ,
∫
Ω
psinθ)t · (η sin θα,−η cos θα) =
∫
Ω
p(h− h˜)′ +
∫
Ω
h3 |∇p|2
In order to take limits when → 0 we need some estimations
•
inf
0≤α≤2pi
∫
Ω
h3(θ, η, α)|∇p|2dθdx > c−12 for  << 1.
• Since,
∣∣∣∣∫Ω p(h− h˜)′
∣∣∣∣ =
∣∣∣∣∣∣
∫
Ω
∂p
∂θ
(h− h˜)
∣∣∣∣∣∣ ≤
1
2
∫
Ω
h3|∇p|2 + 1
2
∫
Ω
∣∣∣∣h− h˜∣∣∣∣2
h3
and ∫
Ω
∣∣∣∣h− h˜∣∣∣∣2
h3
≤ c2.
We compare the fields G with (−η2, η1) in ∂A by using the vector field W
W : (η, α) ∈ A→ (η sin θα,−η cos θα) ∈ IR2
to obtain
G(η1, η2) ·W (η1, η2) ≥ η(c(−1/2 − 1)− ‖F‖), in ∂A.
Then for  << 1
deg(G,A, 0) = deg(W,A, 0) = deg((−η2, η1), A, 0) = 1
by continuity of G we have the existence of at least a solution.
Open Questions:
Uniqueness of solutions
Existence and uniqueness for Elrod-Adams model


Inverse Problems in Lubrication Theory

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