Tribology Using Homogenization

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2009:027

M A S T E R ' S T H E S I S

Tribology Using Homogenization

Betuel Canhanga Afonso Tsandzana

Luleå University of Technology Master Thesis, Continuation Courses Mathematics

Department of Mathematics

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Tribology Using Homogenization

Betuel Canhanga Afonso Tsandzana and

Department of Mathematics Lule˚a University of Technology

SE-971 87 Lule˚a, Sweden

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Supervisors:

Peter Wall, Ove Edlund and Lars-Erik Persson.

Lule˚a University of Technology, Sweden.

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”As far the laws of mathematics refer to reality, they are not certain, and as far they are certain, they do not refer

to reality.”

Albert Einstein.

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1. INTRODUCTION

Many machine components for example, bearings, pistons rings, gearboxes, breaks consists of parts that to operate must rub against each other causing moving parts which can increase the production costs. To minimize costs of moving parts and to make the machines more reliable it is necessary to lubricate the machine elements.

Lubrication is the action of viscous fluids to diminish friction and wear between solid surfaces. It is fundamental to the operation of all engineer machines and biological process’s. A fluid film can separate two surfaces in relative motion pressed together under and external load, when a fluid film act on this way is called lubricant. Because of the lubricant resistance to motion, the hydrodynamic pressure is built from a lubricant, and this pressure helps to prevent contact between two solid surfaces.

The field of science which deals with practice and technology of lubrication is named Tribology. In the Hydrodynamic lubrication, the flow of fluid through machine elements may be governed by an mathematical model. In 1886 Osborne Reynolds published a theory of lubrication [25]

that is considered the main guide of Tribology.

Consider a liquid flowing through a thin film region separated by two closely spaced moving surfaces, assuming that the fluid pressure is not varying across the film thickness and fluid inertia effects are ignored; from the momentum transport and continuity equation (conservation of mass), the analysis of the fluid movement lead to a elliptic partial differential equation named Reynolds equation, that describe the hydrodynamic pressure used in lubrication theory, this equation is motivated in this master thesis.

Homogenization is a branch within mathematics that involves the study of partial differential equations PDE’s (Reynolds equations for example) with rapidly oscillating coefficients. The main purpose of homogenization of PDE’s, is to approximate PDE’s that have rapidly varying coefficients with equivalent ”homogenized” PDE’s that more easily lend themselves to numerical treatment in a computer.

In this thesis we use the homogenization theory in tribology by homog- enizing the Reynolds type equations. Recent researches in Reynolds type equations and Homogenization theory can be found in [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [14], [16], [17], [18], [19], [21], [22], [23].

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2. DERIVATION OF REYNOLDS EQUATIONS

Let Ω be a domain in R³which consists of a fluid in motion. In the following discussion we assume that there are no source or sinks in the domain, i.e.

no fluid is created or disappears.

2.1. THE EQUATION OF CONTINUITY OF THE FLUID

Consider a fix domain D ⊂ Ω of the fluid bounded by the surface S. The fundamental law of conservation of mass tells that, the rate of change of mass in D equals the flow of mass across S into D. Let ρ = ρ(x₁, x₂, x₃, t) and v = v(x₁, x₂, x₃, t) denote the density and velocity of the fluid in the point x = (x₁, x₂, x₃) at time t. The mass of D at time t is then:

m = Z

D

ρ dx, and since the mass changes at rate

m⁰= Z

D

∂ρ

∂t dx. (2.1)

By considering the flow of mass across the boundary S, we can define the rate of change of mass in D in a different way. Let n be the unit normal at position x on S pointing out of D. The flow of mass across S into D is equal to the rate of change of mass in D and will be:

m⁰= − Z

S

ρv_in_i ds, i = 1, 2, 3; (2.2) from divergence theorem, see [1], applied in (2.2)

m⁰ = − Z

D

∂

∂x_i(ρv_i) dx, (2.3)

comparing (2.1) and (2.3) gives that Z

D

·∂ρ

∂t + ∂

∂x_i(ρv_i)

¸

dx = 0,

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and since D was an arbitrary fixed subset of Ω, the equation above implies

that ∂ρ

∂t + ∂

∂x_i(ρv_i) = 0 (2.4)

must hold in Ω. The equation (2.4) is called the equation of continuity of the fluid, which can be expanded and expressed as

∂ρ

∂t + v_i ∂ρ

∂x_i = 0 when ∂v_i

∂x_i = 0, i.e. when the density of the fluid is constant (incomprehen- sible fluid).

2.2. THE EQUATION OF MOTION

Considering a fix domain D of a fluid bounded by the surface S with the outward directed unit normal n. The motion of the fluid obeys Newton’s second law, that is, the rate of increase of momentum of any part of the fluid is equal to the sum of

1. total body force acting on D, 2. total force acting on S,

3. rate of momentum crossing S into D.

Let the body forces be described by the force density f = f (x, t) (force per unit mass). The total body force acting on D at time t is then given by

Z

D

ρf dx. (2.5)

The total force acting on S at time t is Z

S

σⁿ ds, (2.6)

where σⁿ is the stress vector in the point x = (x₁, x₂, x₃) at time t. The rate at which momentum is flowing across S into D at time t is

− Z

S

v_iρv_jn_j ds. (2.7)

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Summing up the Newton’s second law equations (2.5), (2.6) and (2.7) im- plies that

Z

D

∂

∂t(ρv_i) dx = Z

D

ρf_i dx + Z

S

σⁿ_i ds − Z

S

v_iρv_jn_j ds. (2.8) We will now use this equation to derive the equation of motion, which is the main purpose of this section. The main step in this process is to convert the surface integrals in (2.8) into volume integrals by applying the divergence theorem. The second surface integral in (2.8) can be converted into volume integral directly by using divergence theorem, more precisely

Z

S

v_iρv_jn_j ds = Z

D

∂

∂x_j(v_iρv_j) dx = Z

D

µ ρv_j∂v_i

∂x_j + v_i ∂

∂x_j(ρv_j)

¶ dx.

(2.9) Using (2.9) in (2.8) gives

Z

D

µ∂

∂t(ρv_i) − ρf_i+ ρv_j∂v_i

∂x_j + v_i ∂

∂x_j(ρv_j)

¶ dx =

Z

S

σ_iⁿ ds. (2.10) Making use of the equation of continuity (2.4), (2.10) can be reduced to

Z

D

µ ρ∂v_i

∂t − ρf_i+ ρv_j∂v_i

∂x_j

¶ dx =

Z

S

σ_iⁿ ds. (2.11) Remember that our aim is to convert the surface integrals into volume integral, to do this with the surface integral in (2.11), we need first to use the fact that the stress vector on any surface with normal n can be expressed in terms of the stress vectors corresponding to the surfaces with normal vectors e₁ = (1, 0, 0), e₂ = (0, 1, 0) and e₃ = (0, 0, 1); see more details in [20] pages 510-514. Using this fact

σ_iⁿ= σ_1in₁+ σ_2in₂+ σ_3in₃ = σ_jin_j, where i = 1, 2, 3. (2.12) Substituting (2.12) in the surface integral in (2.11) we will have

Z

D

µ ρ∂v_i

∂x_j

¶ dx =

Z

S

σ_jin_j ds, and applying divergence theorem we produce the equation

Z

D

µ ρ∂v_i

∂x_j

¶ dx =

Z

D

∂σ_ji

∂x_j dx, (2.13)

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and since (2.13) holds for any D, it follows that ρ∂v_i

∂t + ρv_j∂v_i

∂x_j = ∂σ_ji

∂x_j + ρf_i, (2.14)

which is called equation of motion of the fluid holds in Ω.

Up to this point we have four equations; equation of continuity of the fluid (2.4) and three equations of motion of the fluid (2.14) which relates the unknowns functions ρ, the three components v_i and the nine components of σ_ji, i.e. we have thirteen unknowns and four equations. It is obvious that these equations are too few compared to the number of unknowns, but we can reduce the number of unknowns from thirteen to ten by considering the fact that the σ_jimatrix is symmetric, the symmetry of matrix σ_jiis proved in [20], pages 515-519; and we proceed more simplifications by introducing constitutive relations which express σ_ji in terms of v_i and the thermody- namic pressure p; and assuming an equation of the state ρ = ρ(p, T ), where T is the temperature. Let us have this in mind in the proceeding discussion.

2.3. CONSTITUTIVE RELATIONS

Consider an ideal fluid, that is homogeneous, isotropic and not viscous. If p denotes the thermodynamical pressure, then

σⁿ_i = σ_jin_j = −pn_i for any n. Therefore σ_ij = 0 if i 6= j and −p = 1

3σ_kk. Thus

σ_ij = 1

3σ_kkδ_ij = −pδ_ij, here δ_ij is a identity matrix.

We will now generalize this, taking in account the viscous effects. Lets assume that the shear stress is of the form

σ_ij = −pδ_ij+ s_ij, (2.15) where p is the thermodynamic pressure and s_ij the components in the viscous stress tensor. The viscous stress tensor represent the contribution to σ caused by viscous effect. The motion of the fluid may be divided into

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local translation, local rigid rotation and local deformation; where the local deformation is described by strain tensor

e_ij = 1 2

µ∂v_i

∂x_j +∂v_j

∂x_i

¶ .

Lets assume now that it is only the local deformation which contributes to s. We also assume that the constitutive relation is linear and isotropic, i.e, the fluid is Newtonian and satisfy the equation

s_ij = λδ_ije_kk+ 2µe_ij, (2.16) where λ and µ are Lame Viscous coefficient. In general we have that s also depends on the temperature T and the density ρ, that is, λ = λ(T, ρ) and µ = µ(T, ρ).

Using (2.16) in (2.15) we have

σ_ij = −pδ_ij+ λδ_ije_kk+ 2µe_ij. (2.17) Then it follows that

σ_ii= −pδ_ii+ λδ_iie_ii+ 2µe_ii= −3p + 3 µ

λ + 2 3µ

¶ e_ii.

If we define K = λ +2

3µ, then (2.17) can be expressed as σ_ij = −(p − Ke_kk)δ_ij + 2µ

µ e_ij −1

3δ_ije_kk

¶

, (2.18)

the second term in the right hand side has zero trace, and we get that

−1

3σ_ii= p − Ke_kk.

The mechanical pressure P (mean normal stress) is by definition equal to

−1

3σ_ii, thus, we have the relation P = p − Ke_kk between the thermody- namical pressure p and mechanical pressure P. The relation (2.18) with p replaced with respect of P is

σ_ij = −P δ_ij + 2µ µ

e_ij−1 3δ_ije_kk

¶

. (2.19)

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2.4. NAVIER-STOKES EQUATIONS

Inserting constitutive relation (2.19) into (2.14) gives that ρ∂v_i

∂x_j (2.20)

= ρf_i− ∂p

∂x_i + ∂

∂x_j µ

µ∂v_i

∂x_j + µ∂v_j

∂x_i

¶ + ∂

∂x_i µµ

K − 2 3µ

¶∂v_k

∂x_k

¶ . Now, we introduce the equations of state ρ = ρ(p, T ) and derive one equa- tion which correspond to the conservation of energy. We will restrict our discussion to the situation where ρ = ρ(p), K = K(p) and µ = µ(p) are known. With this restrictions in mind the fluid motion is described by equations bellow, equation of continuity of the fluid and the Navier-Stokes equations respectively

∂ρ

∂t + ∂

∂x_i(ρv_i) = 0, ρ∂v_i

∂x_j

= ρf_i− ∂p

∂x_i + ∂

∂x_j µ

µ)∂v_i

∂x_j + µ∂v_j

∂x_i

¶ + ∂

∂x_i µµ

K − 2 3µ

¶∂v_k

∂x_k

¶ .

If the fluid is incompressible, ∂v_i

∂x_i = 0, then the Navier-Stokes equations are reduced to

ρ∂v_i

∂x_j = ρf_i− ∂p

∂x_i + ∂

∂x_j µ

µ∂v_i

∂x_j + µ∂v_j

∂x_i

¶

, (2.21)

in the other hand if µ is constant the equation (2.21) is reduced to ρ∂v_i

∂x_j = ρf_i− ∂p

∂x_i + µ∂²v_i

∂x²_j , (2.22) and if µ = K = 0 (Lame viscous coefficients equals zero) then (2.22) can be simplified, and

ρ∂v_i

∂x_j = ρf_i− ∂p

∂x_i (2.23)

which are called Euler equations.

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Now if we consider a viscous fluid with ρ and µ, constants, then, from (2.22) the governing Navier-Stokes equations are:

∂v_i

∂t + v_j∂v_i

∂x_j = f_i− 1 ρ

∂p

∂x_i +µ ρ

∂²v_i

∂x²_j . The ratio µ

ρ is called kinematic viscosity. Considering two different dimensionless forms of this equation:

1. x^∗_i = x_i

l₀, v_i^∗= v_i

v₀, t^∗ = tv₀

l₀ , p^∗ = p

ρv₀², f_i^∗ = l₀f_i v²₀ ; 2. x^∗_i = x_i

l₀, v_i^∗= v_i

v₀, t^∗ = tµ

l²₀ρ, p^∗= l₀p

µv₀, f_i^∗ = l₀²ρf_i µv₀ , we produce respectively the dimensionless Navier-Stokes equations:

∂v^∗_i

∂t^∗ + v_j^∗∂v_i^∗

∂x^∗_j = f_i^∗−∂p^∗

∂x^∗_i + 1 R

∂²v_i^∗

∂x²_j , (2.24)

∂v_i^∗

∂t^∗ + Rv_j^∗∂v^∗_i

∂x^∗_j = f_i^∗− ∂p^∗

∂x^∗_i +∂²v_i^∗

∂x²_j , (2.25) where R = ρv₀l₀

µ is the Reynolds number. From (2.24) and (2.25) it seems reasonable that for large or small values of the Reynolds number R, the Navier-Stokes equation can be approximated by neglecting the viscous or inertia term. Indeed, for fluid problems where R is large we obtain the governing Euler equations

∂v^∗_i

∂t^∗ + v_j^∗∂v_i^∗

∂x^∗_j = f_i^∗−∂p^∗

∂x^∗_i, for situation where R is small we get Stokes equations

∂v_i^∗

∂t^∗ = f_i^∗− ∂p^∗

∂x^∗_i +∂²v^∗_i

∂x²_j .

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2.5. REYNOLDS EQUATIONS

The derivation of Reynolds equations is based on certain appropriate assumptions, which reduces the Navir-Stokes equations and the law of mass conservation. Note that this assumptions can be motivated by writing the Navier-Stokes equations in dimensionless form and analyzing the order of magnitude of different terms. For simplicity we assume that one of the two surfaces correspond to the domain Ω in the plane x₃ = 0. The distance between the two surfaces is denoted by h = h(x₁, x₂). The upper and lower surfaces are moving with velocity (U_u, V_u) and (U_l, V_l) respectively. Recall the Navier-Stokes equation (2.20), it is reasonable to assume that the iner- tia effects with respect to time at a given point can be neglected, i.e. the left hand said in the equation (2.20). Moreover, also assume that the body force f_i can be neglected and that the only significance velocity gradients

are ∂v₁

∂x₃ and ∂v₂

∂x₃, then equation (2.20) is reduced to:

∂p

∂x₁ = ∂

∂x₃ µ

µ∂v₁

∂x₃

¶

, (2.26)

∂p

∂x₂ = ∂

∂x₃ µ

µ∂v₂

∂x₃

¶

, (2.27)

and ∂p

∂x₃ = 0;

assuming that the pressure is constant in the x₃ direction, i.e. the pressure depends only in x₁ and x₂, p = p(x₁, x₂). It is possible to express the velocity in terms of the pressure. Indeed, integrating (2.26) and (2.27) with respect to x₃ twice gives

v₁(x) = ∂p

∂x₁ x²₃

2µ +A(x₁, x₂)

µ x₃+ B(x₁, x₂); (2.28) v₂(x) = ∂p

∂x₂ x²₃

2µ +A(x₁, x₂)

µ x₃+ B(x₁, x₂).

To find A and B we use the no slip boundary conditions both at the upper and low surfaces, that is

(v₁, v₂) = (U_u, V_u) when x₃= h,

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(v₁, v₂) = (U_l, V_l) when x₃= 0, and we obtain that

∂p

∂x₁ h² 2µ+A

µh + B = U_u; B = U_l. From this it is clear that

A

µ = U_u− U_l h − ∂p

∂x₁ h 2µ(p), and inserting into (2.28) gives that

v₁(x) = ∂p

∂x₁ x²₃

2µ +U_u− U_l

h x₃− ∂p

∂x₁ h

2µ(p)x₃+ U_l (2.29)

= 1 2µ

∂p

∂x₁(x²₃− hx₃) + U_u− U_l

h x₃+ U_l. Similarly we get that

v₂(x) = 1 2µ

∂p

∂x₂(x²₃− hx₃) +V_u− V_l

h x₃+ V_l. (2.30) Since p = p(x₁, x₂), we know that ρ and µ are constants in the x₃- direction (across the fluid film).

Let w be a domain in the lower surface Ω. We denote by D the three dimensional domain obtained by extending Ω between the two surfaces and by S the part of the boundary of D which does not include the surface.

The fundamental law of conservation of mass tells that the rate of change of mass in D is equal to the flow of mass across S into D. The mass m of D at time t is then

m = Z

D

ρdx = Z

w

hρ dx₁dx₂,

and since the mass changes at rate m⁰ =

Z

D

∂ρ

∂tdx = Z

w

∂

∂t(hρ) dx₁dx₂, (2.31) An another expression for m⁰ can be found by considering the flow of mass across the boundary S (there is no flow through the surface). Let n be the

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unit normal at x on S pointing out of D. The flow of mass across S into D is then

m⁰ = − Z

S

(ρv₁n₁+ ρv₂n₂) ds,

if we use the notation ds = dx₃dτ where dτ is the measure on ∂w, and use that ρ does not depend on x₃ we get that

m⁰ = − Z

∂w



ρ Zh

0

v₁ dx₃ n₁+ ρ Zh

0

v₂ dx₃ n₂



 dτ.

Using divergence theorem, see in [1] page 947, we get that

m⁰= − Z

w



 ∂∂x₁



ρ Zh

0

v₁dx₃



 + ∂

∂x₂



ρ Zh

0

v₂dx₃







 dx1 dx₂. (2.32)

comparing (2.31) and (2.32)

∂

∂t(hρ) = − ∂

∂x₁



ρ Zh

0

v₁dx₃



 − ∂

∂x₂



ρ Zh

0

v₂ dx₃



 . (2.33)

Solving (2.33), using (2.29) gives Zh

0

v₁dx₃ = Zh

0

µ 1 2µ

∂p

∂x₁(x²₃− hx₃) + U_u− U_l

h x₃+ U_l

¶ dx₃

= − h³ 12µ

∂p

∂x₁ + (U_u+ U_l)h 2. Similarly using (2.30) gives

Zh

0

v₂dx₃= Zh

0

µ 1 2µ

∂p

∂x₂(x²₃− hx₃) +V_u− V_l

h x₃+ V_l

¶ dx₃

= − h³ 12µ

∂p

∂x₂ + (V_u+ V_l)h 2.

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Inserting into (2.33) gives

∂

∂t(hρ) = ∇ · µρh³

12µ∇p

¶

−U_u+ U_l 2

∂

∂x₁(ρh) −V_u+ V_l 2

∂

∂x₂(ρh).

This is the Reynolds equation which is frequently used to model the pressure distribution in a thin fluid film between two surfaces in relative motion. In many situations the relative motion only take place in one direction, lets say the x₁-direction, which the further reduction

∂

∂t(hρ) = ∇ · µρh³

12µ∇p

¶

−U_u+ U_l 2

∂

∂x₁(ρh). (2.34) For an incompressible fluid (constant density) the equation (2.34) is re- duced to

∂

∂th = ∇ · µ h³

12µ∇p

¶

−U_u+ U_l 2

∂

∂x₁h. (2.35)

Assuming that the film thickness is constant with respect to time we will have

∇ · µ h³

12µ∇p

¶

= U_u+ U_l 2

∂h

∂x₁. (2.36)

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3. HOMOGENIZATION OF REYNOLDS EQUATIONS BY MULTIPLE SCALE

EXPANSION

As follows from the introduction, the Reynolds type equation are used to analyze the influence of the surface geometry and roughness on the hydrodynamic performance of different machine elements when lubricant is flowing into it. The two surfaces through which the lubricant flows between may have different characteristics, but in this study we will only consider the following cases:

1. one surface is smooth and moving, the other is rough and stationary, 2. both surfaces are rough and moving;

that describe the real situation of most of machine elements.

In the case (1), since the film thickness within the machine elements does not change when time changes, i.e. it is constant at some position x, the equation modeling the hydrodynamic performance of machine elements when the lubricant is flowing through it will be time independent.

In Figure 3.1, the surface s₁, is moving with velocity U_l from the left to the right, the rough surface, s₂ is stationary, h is the film thickness, that will be a function independent of time. In the case (2), the film thickness within the machine elements h, is a function depending on time and space because of the movements of rough surfaces. In Figure 3.2, s₁ is moving with velocity U_l, and s₂ with velocity U_u, both from the left to the right. In both cases, h will rapidly change and this changes will cause rapidly oscil- lations of hydrodynamic pressure within the machine elements, therefore a mathematical model describing this phenomena will be composed by differential equations with rapidly oscillating coefficients that can not be easily solved even by numerical methods. To avoid this problem it is reasonable to introduce some well accuracy averaging methods that will transform the problem into another that can be solved (at least numerically) and that the solutions will fit the real problem as much as possible, there is where

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s1 s₂

Uu=0

U_l h(x)

rough stationary surface global film thickness smooth moving surface

Figure 3.1. smooth moving and rough stationary surfaces

s₁ s2

h(x,t)

U_u

U_l

rough stationary surface global film thickness rough moving surface

Figure 3.2. both smooth moving surfaces

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homogenization appears with strong importance. In this chapter we will introduce the homogenization procedure describing the situation (1) and (2).

3.1. ROUGH STATIONARY AND SMOOTH MOVING SURFACES The Reynolds type equation will be of the form

∇ ·

·ρ(p(x))h³(x) 6µ ∇p(x)

¸

= u ∂

∂x₁ [ρ(p(x))h(x)] , (3.1) with u = U_l+ U_u. Because of the roughness on s₂ the film thickness h will depend on the roughness wavelength ε, where ε is a positive sequence that converges to zero and will also affect the oscillation of hydrodynamic pressure, if we assume that the fluid is incompressible, i.e. ρ is a constant, then the equation (3.1) can be expressed as

∇ ·£

h³_ε(x)∇p_ε(x)¤

= λ ∂

∂x₁[h_ε(x)] , λ = 6µu, (3.2) The film thickness will be result of a periodic function h₁ representing the effect of the surface roughness and the global film thickness h₀ representing the geometry of the surfaces,

h_ε(x) = h₀(x) + h₁(y) , y = x ε,

y is the local variable, this means that for small values of ε the function h_ε is rapidly oscillating.

The first step to introduce the homogenization procedure of the equation (3.2) is to assume multiple scale expansion, see in [13] and [24] of the solutions in the following forms:

p_ε(x) = p₀(x, y) + εp₁(x, y) + ε²p₂(x, y) + · · · (3.3) where p_i(x, y) is periodic in y for every x ∈ Ω; Ω ⊂ R² is the domain in which the fluid is flowing and (i = 0, 1, 2, · · · ). Assume that the cells of periodicity is Y = (0, 1) × (0, 1). This means that y is a local variable, describing the behavior of the solution on the unit cell scale and the global behavior is described by the variable x. The homogenized Reynolds equa- tions describes the limiting results when the wavelength of the modelled surface roughness tends to zero.

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Applying the Chain rule on the smooth function f_ε(x) = f (x, y)

the partial derivatives with respect to x_j will be:

∂f_ε

∂x_j(x) = µ∂f

∂x_j + ε⁻¹∂f

∂y_j

¶

(x, y) , j = 1, 2; (3.4) which means that

∇_xf_ε= ∇_xf + ε⁻¹∇_yf (3.5) Using (3.3), (3.4) and (3.5) in (3.2) we get

(∇_x+ ε⁻¹∇_y) · [h³(∇_x+ ε⁻¹∇_y)(p₀+ εp₁+ ε²p₂+ · · · )]

= λ µ ∂

∂x₁ + ε⁻¹ ∂

∂y₁

¶ h, expanding this, we will have

λ µ ∂

∂x₁ + ε⁻¹ ∂

∂y₁

¶

h (3.6)

= {ε⁻²∇_y· (h³∇_y) + ε⁻¹[∇_y· (h³∇_x) + ∇_x· (h³∇_y)] + ∇_x· (h³∇_x)}(p_ε(x)), defining

A₀= ∇_y· (h³(x)∇_y),

A₁ = ∇_y· (h³(x)∇_x) + ∇_x· (h³(x)∇_y), A₂ = ∇_x· (h³(x)∇_x);

(3.6) can be written as:

λ µ ∂h

∂x₁ + ε⁻¹∂h

∂y₁

¶

= (ε⁻²A₀+ ε⁻¹A₁+ A₂)(p₀+ εp₁+ ε²p₂+ · · · ). (3.7) Equating the three lowest power of ε we will have the following system of equations:

A₀p₀= 0, (3.8)

A₀p₁+ A₁p₀ = λ∂h

∂y₁, (3.9)

A₀p₂+ A₁p₁+ A₂p₀ = λ∂h

∂x₁. (3.10)

In order to solve equations (3.8), (3.9) and (3.10) we need the following lemma:

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Lemma 3.1.1. Let F (y) ∈ L²(Ω) be Y -periodic. Consider the boundary value problem

A₀ϕ(y) = F (y)

where ϕ(y) is y -periodic, then, the following holds : 1. There exist a solution ϕ(y) if and only if _{|Y |}¹ R

Y F dy = 0

2. If there exist a solution it is unique up to an additive constant, it means that, if ϕ₀(y) is another solutions of the boundary value pro- blem, then ϕ(y) = ϕ₀(y) + C.

The proof of this lemma can be found in [24]. Now we note that (3.8) has the trivial solution p₀ = 0, since the variable x is just a parameter in (3.8). Lemma (3.1.1) says that, if p₀(x, y) is another solution of (3.8) then p₀(x, y) is a constant with respect to variable y, i.e.

p₀(x, y) = p₀(x),

where p₀(x) is sufficiently differentiable. From equation (3.9) we see that A₀p₁= λ∂h

∂y₁ − A₁p₀, i.e.

∇_y· (h³∇_yp₁) = λ∂h

∂y₁ − ∇_x· (h³∇_yp₀) − ∇_y· (h³∇_xp₀), since ∇_yp₀ = 0, we have

∇_y· (h³∇_yp₁) = λ∂h

∂y₁ − ∇_y· µ

h³∂p₀

∂x₁e₁+ h³∂p₀

∂x₂e₂

¶

, (3.11)

where e₁, e₂are the canonical basis of R². Since the right hand side consists of three linear terms, we expect that p₁(x, y) should be a linear function of three terms. By linearity we will let

p₁(x, y) = ∂p₀

∂x₁v₁(x, y) + ∂p₀

∂x₂v₂(x, y) + v₃(x, y), (3.12) and using (3.12) in (3.11) we will have:

∇_y ·

½ h³

·

∇_y µ∂p₀

∂x₁v₁(x, y) + ∂p₀

∂x₂v₂(x, y) + v₃(x, y)

¶¸¾

(3.13)

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= λ∂h

∂y₁ − ∇_y· µ

h³∂p₀

∂x₁e₁+ h³∂p₀

∂x₂e₂

¶ , that will give us the following cell problems:

∇_y· (h³∇_yv₃− λhe₁) = 0, (3.14)

∇_y· (h³∇_yv₁+ h³e₁) = 0, (3.15)

∇_y· (h³∇_yv₂+ h³e₂) = 0, (3.16) where v_i = v_i(x, y) are their solutions.

Equation (3.10), averaged over the period Y, gives:

Z

Y

µ

A₀p₂+ A₁p₁+ A₂p₀− λ∂h

∂x₁

¶

dy = 0. (3.17) Because of the periodicity of p₂, R

Y A₀p₂dy = 0; using (3.12), expanding (3.17) and the fact that h³∇_xp₁ is periodic, R

Y ∇_y· (h³∇_xp₁)dy = 0, the above integral can be written as:

Z

Y

·

∇_x· µ

h³∇_y µ∂p₀

∂x₁v₁+ ∂p₀

∂x₂v₂

¶¶

+ ∇_x· µ

h³∂p₀

∂x₁e₁+ h³∂p₀

∂x₂e₂

¶¸

dy

= Z

Y

· λ∂h

∂x₁ − ∇_x· (h³∇_yv₃)

¸ dy.

Collecting identical terms we get:

Z

Y

∇_x·

·∂p₀

∂x₁(h³∇_yv₁+ h³e₁) +∂p₀

∂x₂(h³∇_yv₂+ h³e₂)

¸ dy

= Z

Y

· λ∂h

∂x₁ − ∇_x· (h³∇_yv₃)

¸ dy, or

∇_x·

·∂p₀

∂x₁ Z

Y

(h³∇_yv₁+ h³e₁)dy + ∂p₀

∂x₂ Z

Y

(h³∇_yv₂+ h³e₂)dy

¸

(3.18)

= ∇_x· Z

Y

"µ λh

0

¶

−

Ã h^{3 ∂v}_∂y³₁ h^{3 ∂v}_∂y³₂

!#

dy, and expanding (3.18) we produce a matrix

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A(x) =



 R

Y

³

h^{3 ∂v}_∂y¹₁ + h³

´

dy R

Y h^{3 ∂v}_∂y²₂dy R

Y h^{3 ∂v}_∂y¹

2dy R

Y

³ h^{3 ∂v}_∂y²

2 + h³

´ dy



 ,

and the vector

b(x) =

" R

yλh − h^{3 ∂v}_∂y³₁dy

−R

yh^{3 ∂v}_∂y³₂dy

#

; and finally we construct the equation

∇_x· [A(x)∇_xp₀] = ∇_x· b(x). (3.19) Summarizing, we are looking for ∇_xp₀. For that, we need to solve (3.19), but before, we need to have A(x) and b(x); therefore, we solve first the cell problems, ”equations (3.14), (3.15) and (3.16)” that will give v₁, v₂and v₃, then we apply this solutions to find A(x) and b(x). This steps summarize, the homogenization algorithm.

3.2. BOTH ROUGH MOVING SURFACES

If one, or both moving surfaces are rough, the governing Reynolds type equation will then involve time, which means that the hydrodynamic pressure between the two surfaces will be changing according to position and time. Here we also use the formal method of multiple scale expansion.

Let

h(x, y, t, τ ) = h₀(x, t) + h₁(y − τ U_l) + h₂(y − τ U_u), where x = (x₁, x₂), y = (y₁, y₂), y = x

ε, assuming that u_i is constant and τ = t

ε. By using the auxiliary function h define the film thickness h_ε by h_ε(x, t) = h(x, y, t, τ ) and the governing Reynolds equation will have the form

γ∂h_ε

∂t + λ∂h_ε

∂x₁ − ∇ ·¡

h³_ε∇p_ε¢

= 0, (3.20)

γ = 12η and λ = 6ηu, u = U_l+ U_u. Lets perturb a litle our system by assuming the multiple scale expansion of the solution p_ε

p_ε(x, y, t, τ ) = p₀(x, y, t, τ ) + εp₁(x, y, t, τ ) + ε²p₂(x, y, t, τ ) + · · · , (3.21)

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p_i = p_i(x, y, t, τ ). The elements in this equation are defined on the same way as in the preview section and with the same assumptions. The Chain rule then implies that, for

f_ε= f (x, y, t, τ ) the partial derivative with respect to x_j becomes:

∂f_ε

∂x_j = µ∂f

∂x_j + ε⁻¹∂f

∂y_j

¶

(x, y, t, τ ) , (3.22)

and ∂f_ε

∂t = µ∂f

∂t + ε⁻¹∂f

∂τ

¶

(x, y, t, τ ) . (3.23) Using (3.21), (3.22) and (3.23) into (3.20) we get:

γ µ∂

∂t+ ε⁻¹ ∂

∂τ

¶ h + λ

µ ∂

∂x₁ + ε⁻¹ ∂

∂y₁

¶ h

−¡

∇_x+ ε⁻¹∇_y¢

·£ h³¡

∇_x+ ε⁻¹∇_y¢ ¡

p₀+ εp₁+ ε²p₂+ · · ·¢¤

= 0, that can be expanded,

γ µ∂

∂t+ ε⁻¹ ∂

∂τ

¶ h + λ

µ ∂

∂x₁ + ε⁻¹ ∂

∂y₁

¶

h (3.24)

=£

∇_x·¡ h³∇_x¢

+ ε⁻¹∇_x·¡ h³∇_y¢

+ ε⁻¹∇_y·¡ h³∇_x¢

+ ε⁻²∇_y·¡

h³∇_y¢¤

(p_ε), and simplifying by defining

A₀ = ∇_y·¡ h³∇_y¢

, A₁ = ∇_y·¡

h³∇_x¢

+ ∇_x·¡ h³∇_y¢

, A₂= ∇_x·¡

h³∇_x¢

; and applying them into (3.24) we will have:

γ µ∂

∂t+ ε⁻¹ ∂

∂τ

¶ h + λ

µ ∂

∂x₁ + ε⁻¹ ∂

∂y₁

¶ h

=¡

A₂+ ε⁻¹A₁+ ε⁻²A₀¢ ¡

p₀+ εp₁+ ε²p₂+ · · ·¢

= 0.

Tribology Using Homogenization

M A S T E R ' S T H E S I S

Tribology Using Homogenization

Betuel Canhanga Afonso Tsandzana

Tribology Using Homogenization

Betuel Canhanga Afonso Tsandzana and

”As far the laws of mathematics refer to reality, they are not certain, and as far they are certain, they do not refer

to reality.”

Albert Einstein.

Table of Contents

1. INTRODUCTION

2. DERIVATION OF REYNOLDS EQUATIONS

3. HOMOGENIZATION OF REYNOLDS EQUATIONS BY MULTIPLE SCALE

EXPANSION