# A New Approach for Estimating the Friction in Thin Film Lubrication

Authors: Dag Lukkassen; Annette Meidell; Peter Wall
Journal: Mathematical Problems in Engineering
 (2010)

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## Abstract

An important problem in the theory of lubrication is to model and analyze
the effect of surface roughness on, for example, the friction and load carrying
capacity. A direct numerical computation is often impossible since an
extremely fine mesh is required to resolve the surface roughness. This
suggests that one applies some averaging technique. The branch in mathematics
which deals with this type of questions is known as homogenization. In this
paper we present a completely new method for computing the friction. The
main idea is that we study the variational problem corresponding to the
Reynolds equation. We prove that the homogenized variational problem is
closely related to the homogenized friction. Finally we use bounds on the
homogenized Lagrangian to derive bounds for the friction. That these bounds
can be used to efficiently compute the friction is demonstrated in a typical
example.


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## Body

## 1. Introduction

A fundamental problem in lubrication theory is to describe the flow behavior between two adjacent surfaces which are in relative motion. For example, this type of flow takes place in different kinds of bearings, hip joints, and gearboxes. The main unknown is often the pressure in the fluid. After computing the pressure it is possible to compute other fundamental quantities as the friction on the surfaces and the load carrying capacity. In this paper we develop a completely new technique to find the friction in problems, where the effects of surface roughness are taken into account.Let the bearing domain,Ω, be an open bounded subset of ℝ2, and points in Ω are denoted by x=(x1,x2). Let us assume that the lower surface is smooth and moving while the upper surface is rough and stationary, see Figure 1. The velocity of the upper surface is V=(V1,V2). To express the film thickness we introduce the following auxiliary function:(1.1)h(x,y)=h0(x)+hr(y),
where h0 and hr are smooth functions. Moreover, hr is Y-periodic, and we can without loss of generality assume that the cell of periodicity is the unit cube in ℝ2, that is, Y=(0,1)×(0,1). By using the auxiliary function h we can model the film thickness hε by
(1.2)hɛ(x)=h(x,xɛ),ɛ>0.
This means that h0 represents the global film thickness, the periodic function hr describes the roughness on the upper surface, and ɛ>0 is a parameter which describes the wavelength of the roughness.Figure 1
A smooth lower surface in motion and a rough stationary upper surface.If the pressure is zero on the boundary, the fluid is Newtonian and has viscosityμ, then the pressure due to the relative motions of the surfaces is modeled by the Reynolds equation, see, for example, [1] or [2]: find pɛ∈W01,2(Ω) such that(1.3)div⁡(hɛ3∇pɛ)=6μdiv⁡(hɛV).
On the surface x3=0 the friction force, Fɛ, is given by(1.4)Fɛ=∫ΩμhɛV+hɛ2∇pɛdx,
see, for example, [1].In many applications the surfaces are rotating around the same axis. For example this is the situation in thrust pad bearings. Equation (1.3) in polar coordinates reads(1.5)∂∂x1(hε3x2∂pε∂x1)+∂∂x2(x2hε3∂pε∂x2)=6μωx2div⁡(hεe1),
where ω is the angular speed, e1=(1,0), x1 the angular coordinate, and x2 is the radial coordinate. The friction torque is then(1.6)Tε=∫Ωμωx23hεdx+∫Ωhεx22∂pε∂x1dx,
see, for example, [1].We note that both (1.3) and (1.5) for the pressure pɛ∈W01,2(Ω) are of the form(1.7)div⁡(Aɛ∇pε)=div⁡(bε),
where(1.8)Aɛ(x)=(a1(x,xɛ)00a2(x,xɛ)),bɛ(x)=(b1(x,xɛ)b2(x,xɛ)).
For (1.3) we have ai(x,x/ɛ)=hɛ3(x) and bi(x,x/ɛ)=6μVihɛ(x) and for (1.5) a1(x,x/ɛ)=hɛ3(x)/x2, a2(x,x/ɛ)=x2hɛ3(x), and bɛ(x)=6μωx2hɛ(x)e1.A well-known fact from the calculus of variation is that the solutionpɛ of (1.7) also is the solution of the variational problem(1.9)min⁡p∈W01,2(Ω)⁡Iɛ(∇p),
where (1.10)Iɛ(∇p)=∫Ω[12Aɛ∇p⋅∇p-bɛ⋅∇p]dx.
That is, Iɛ(∇pɛ)=minpIɛ(∇p).In order to computeFɛ and 𝒯ɛ we have to find the partial derivatives of pɛ. However, this is a delicate problem. The main reason is that for small values of ɛ (i.e., the roughness scale is much smaller than the global scale) the distance between the surfaces, hɛ, is rapidly oscillating. Thus a direct numerical treatment of (1.7) or (1.9) will require an extremely fine mesh to resolve the surface roughness. In many situations it is impossible even in practice. One approach is then to do some type of averaging. The field of mathematics which handles this type of averaging is known as homogenization, see, for example, [3] or [4]. Lately, homogenization has been used with success by many authors to study different types of the Reynolds equations. In particular, homogenization was applied to find the homogenized friction and friction torque in [5]. It should be mentioned that even though homogenization is used the numerical analysis is expensive since the homogenization procedure includes the solution of a number of local problems.In this work we develop a new technique for estimating the homogenized friction. The main idea is that we first prove that the homogenized friction is closely related to the homogenized (averaged) variational problem corresponding to (1.9); secondly we apply our lower and upper bounds on the homogenized Lagrangian for (1.9) to obtain lower and upper bounds on the friction. As the bounds are close the mean value of the lower and upper bound will give a very accurate estimate of the friction. The benefit of the proposed method for estimating the homogenized friction, instead of computing it directly, is that no local partial differential equations have to be solved. This means that the new method requires a less computation time. We also illustrate the efficiency of the new method by applying it in a typical numerical example.It should be mentioned that in the present paper the surface roughness is modelled as a periodic extension of a representative part of the roughness. This is crucial for the proposed method for computing the homogenized (averaged) friction. However, by using stochastic homogenization theory it is also possible to analyze related problems where the surface roughness is described as a realization of a stationary random field, see, for example, the book [4]. The first development of the stochastic theory, in the context of lubrication, was made in [6]. This work was limited to two-dimensional transverse and longitudinal roughness. Patir and Cheng were the first to propose a model for general roughness patterns, see [7, 8]. Their derivation was heuristic, and it is now well known that it does not properly model situations where the roughness anisotropy directions are not identical to the Cartesian coordinate axes, see, for example, [9]. From a practical point of view the stochastic approach roughly leads to that one in the averaging process has to consider a number of realizations on a part of the roughness. The result from this averaging can also be obtained by the periodic approach if the representative part is chosen sufficiently large. In applications it is therefore sufficient to consider the surface as a periodic extension of a measured representative part. The results in this paper are based on bounds related to a known periodic roughness. In future works it would though be interesting to derive bounds for the homogenized Lagrangian in the case when only the distribution of the roughness is known and investigate if there is a connection with the friction.


## 2. Homogenization

The main idea in homogenization is to prove that there exists ap0 such that pɛ→p0 as ɛ→0 and that p0 solves a corresponding homogenized (averaged) problem, which does not involve any rapid oscillations. This means that p0 may be used as an approximation of pɛ for small values of ɛ. In this section we recall the main homogenization results concerning (1.7) and (1.9), for (1.7) see [10–14] and for (1.9) see [15].LetWper1,2(Y) be the closure of smooth Y-periodic functions with respect to the usual norm in W1,2(Y). Now we introduce the three local problems: find w1, w2, and v in L2(Ω;Wper1,2(Y)) such that(2.1)div⁡y[A(x,y)(ei+∇ywi(x,y))]=0,onY,i=1,2,div⁡y[b(x,y)-A(x,y)∇v(x,y)]=0,onY.
Here e1=(1,0) and e2=(0,1). Note that the domain in the local problems (2.1) is Y and not Ω. When the local solutions w1, w2, and v are known they are used to define the homogenized matrix A0(x), the homogenized vector b0(x), and the homogenized scalar c0(x) in the following way:(2.2)A0ei=∫YA(ei+∇ywi)dy,i=1,2,b0=∫Y(b-A∇yv)dy,c0=∫Y12A∇v⋅∇vdy.
It is now possible to define the homogenized integral functional I0 as(2.3)I0(∇p)=∫Ωf0(x,∇p)dx,
where the homogenized Lagrangian f0 is given by(2.4)f0(x,ξ)=12A0(x)ξ⋅ξ-b0(x)⋅ξ-c0(x).
The solutions pɛ of (1.9) converge weakly in W01,2(Ω) to the solution p0 of the homogenized variational problem(2.5)min⁡p∈W01,2(Ω)⁡I0(∇p),
that is, I0(∇p0)=minpI0(∇p). Moreover, we have the following convergence:(2.6)lim⁡ε→0⁡Iε(∇pε)=I0(∇p0)=∫Ω(12A0(x)∇p0⋅∇p0-b0(x)⋅∇p0-c0(x))dx.
We remark that p0 also solves the Euler equation corresponding to (2.5), that is, (2.7)div⁡(A0∇p0)=div⁡b0inΩ.


## 3. Bounds

As mentioned in the introduction the new technique for computing the friction, which is developed in the present work, is based on our previous results concerning lower and upper bounds on the homogenized Lagrangianf0 given in (2.4). The bounds were presented in [16], but for the reader's convenience we review the main results here.Let us start by introducing some functions which will appear in the formulation of the bounds. If either(i,j)=(1,2) or (i,j)=(2,1), then(3.1)ai+(x)=(∫01(∫01aidyj)-1dyi)-1,ai-(x)=∫01(∫011aidyi)-1dyj,bi+(x)=ai+(x)∫01∫01bidyj∫01aidyjdyi,bi-=∫01[(∫011aidyi)-1∫01biaidyi]dyj,ci+(x)=12ai+(x)(∫01∫01bidyj∫01aidyjdyi)2-12∫01(∫01bidyj)2∫01aidyjdyi,ci-(x)=12∫01[(∫011aidyi)-1(∫01biaidyi)2]dyj-12∫01∫01bi2aidyidyj.These functions are now used to definef- and f+ as(3.2)f-(x,ξ)=12A-ξ⋅ξ-b-⋅ξ+c1-+c2-,f+(x,ξ)=12A+ξ⋅ξ-b+⋅ξ+c1++c2+,
where ξ∈ℝ2,(3.3)A±=(a1±00a2±),b±=(b1±b2±).
The main result in [16] is that we have the following bounds on the homogenized Lagrangian f0: (3.4)f-(x,ξ)≤f0(x,ξ)≤f+(x,ξ).
From this it is obvious that (3.5)min⁡⁡I-(∇p)≤min⁡⁡I0(∇p)≤min⁡⁡I+(∇p),
where the minimum is taken over all p in W01,2(Ω) and I- and I+ are defined as(3.6)I-(∇p)=∫Ωf-(x,∇p)dx,I+(∇p)=∫Ωf+(x,∇p)dx.
If we use the notation p- for the minimizer in the left hand side of (3.5) and p+ the minimizer in the right hand side of (3.5), that is,  (3.7)I-(∇p-)=min⁡p⁡I-(∇p),I+(∇p+)minp⁡I+(∇p),
then (3.5) can be rewritten as(3.8)I-(∇p-)≤I0(∇p0)≤I+(∇p+).
This means that I0(∇p0) can be estimated with high accuracy if the lower and upper bounds in (3.8) are close. This fact will be crucial later on.The pressuresp- and p+ which are minimizers in the variational problems given in (3.7) can be found by solving the corresponding Euler equations. Indeed,(3.9)div⁡(A±∇p±)=div⁡(b±)inΩ.
We remark that the bounds are optimal in the sense that there are surface roughnesses for which the bounds f- and f+ coincide with f0.In [17, 18] it was shown by many numerical examples that the difference of the bounds solutions p- and p+ is very small. Moreover, it was seen that the homogenized solution p0 of (2.7) is between p+ and p-. This together means that (p++p-)/2 can be used as a very good approximation of p0. From a computational point of view this is useful since it is much easier to find p+ and p- than p0. The reason for this is that to be able to compute p0 one first has to find A0 and b0 which involves the solutions of many local problems (parameterized in x), while one only has to integrate to find A± and b±, which are needed for the computation of p±. Thus it is clear that ∫Ω(p++p-)/2dx may be used to compute the load carrying capacity. It is not obvious how p+ and p- can be used to calculate the friction in different applications (if possible). The main result in this paper is that we prove how this can be done.


## 4. Bounds for the Friction

The physical interpretation ofI0(∇p0) has not been known. Hence the physical meaning of the estimates (3.8), that is, I-(∇p-)≤I0(∇p0)≤I+(∇p+), is also unclear. However, in this section we will prove that I0(∇p0) is closely related to the homogenized friction, which is induced by the relative motion. Moreover, we will show that we can obtain bounds for the friction via the bounds (3.8).Let us consider the case with the Reynolds equation given in Cartesian coordinates, that is, whenai(x,x/ɛ)=hɛ3(x) and bi(x,x/ɛ)=6μVihɛ(x) (Vi is the constant speed in the xi-direction). As mentioned in the introduction, the friction force, Fɛ, on the surface x3=0 is given by(4.1)Fɛ=∫ΩμhɛV+hɛ2∇pɛdx.
The component of the friction force in the direction V of the motion is(4.2)Fɛ=Fɛ⋅1|V|V=∫Ωμ|V|hɛ+hɛ2|V|V⋅∇pɛdx.Let us now consider the generalized formulation of (1.7): find pɛ∈W01,2(Ω) such that(4.3)∫ΩAɛ∇pɛ⋅∇ϕdx=∫Ωbɛ⋅∇ϕdx,
for any ϕ∈W01,2(Ω). Formally this equation is obtained by first multiplying (1.7) by ϕ and thereafter using partial integration. In particular, for ϕ=pɛ, it holds that(4.4)∫ΩAɛ∇pɛ⋅∇pɛdx=∫Ωbɛ⋅∇pɛdx.
This implies that(4.5)Iɛ(∇pɛ)=∫Ω12Aɛ∇pɛ⋅∇pɛ-bɛ⋅∇pɛdx=-∫Ω12bɛ⋅∇pɛdx=-3μ∫ΩhɛV⋅∇pɛdx.
Thus(4.6)∫ΩhɛV⋅∇pɛdx=-13μIɛ(∇pɛ).
From (4.2) and (4.6) it follows that(4.7)Fɛ=μ|V|∫Ω1hɛdx-16μ|V|Iɛ(∇pɛ).
By using (2.6) it is possible to pass to the limit in this equality. We get(4.8)Fɛ⟶K-16μ|V|I0(∇p0)≔F0,
where the constant K is(4.9)K=μ|V|∫Ω∫Y1h(x,y)dydx.
By (3.8) and (4.8) we get bounds on the homogenized (averaged) friction in the direction of the motion, ℱ0,(4.10)F0-≤F0≤F0+,
where(4.11)F0-=K-16μ|V|I+(∇p+),F0+=K-16μ|V|I-(∇p-).If the difference betweenI+(∇p+) and I-(∇p-) is small, then the average of ℱ0-+ℱ0+ is a good approximation of the friction ℱ0. The benefit of this approximation is that it is easier to compute ℱ0- and ℱ0+ than ℱ0. The reason for this is that in order to compute ℱ0 we have to solve three local problems (parameterized in x), see (2.1), but computation ℱ0- and ℱ0+ does not involve solution of any local problems.In order to calculateℱ0- and ℱ0+ we have to find I+(∇p+) and I-(∇p-). Recall that(4.12)I+(∇p+)=∫Ω12A+∇p+⋅∇p+-b+⋅∇p++c1++c2+dx,(4.13)I+(∇p-)=∫Ω12A-∇p-⋅∇p--b-⋅∇p-+c1-+c2-dx.
From practical reasons it is sometimes a good idea to rewrite (4.12) and (4.13). Indeed, by choosing p+ as the test function in the generalized formulation of (3.9) we get that(4.14)∫ΩA+∇p+⋅∇p+dx=∫Ωb+⋅∇p+dx.
This implies that the expression (4.12) for I+(∇p+) can be rewritten as(4.15)I+(∇p+)=∫Ωc1++c2+-12b+⋅∇p+dx.
In the same we get that(4.16)I-(∇p-)=∫Ωc1-+c2--12b-⋅∇p-dx.It should be mentioned that it is possible to pass to the limit in (4.2) directly. Indeed, it was proved in [14] that ∇pɛ two scale converges to ∇p0+∇yp1, where p1 is of the form(4.17)p1(x,y)=w1(x,y)∂p0∂x1+w2(x,y)∂p0∂x2+v(x,y).
For the friction we get(4.18)Fɛ⟶∫Ω∫Yμ|V|h+h2|V|V⋅[∇p0+∇yp1]dydx.
We remark that the result obtained in this way agrees with [19], where the asymptotic behavior of pressure and stresses in a thin film flow with a rough boundary was analyzed by introducing two parameters corresponding to the film thickness and wavelength of the roughness. Also note that a direct asymptotic analysis of (4.2) leads to that p0, w1, w2, and v have to be found.


## 5. Bounds for the Friction Torque

In many applications the surfaces are rotating around the same axis. For example, this is the situation in thrust pad bearings. The governing equation for the pressure is then (1.5), that is, a1(x,x/ɛ)=hɛ3(x)/x2, a2(x,x/ɛ)=x2hɛ3(x) and bɛ(x)=6μωx2hɛ(x)e1. The friction torque is then given by (1.6), that is,(5.1)Tɛ=∫Ωμωx23hɛdx+∫Ωhɛx22∂pɛ∂x1dx,
see, for example, [1]. In the same way as for the friction force we get that(5.2)Iɛ(∇pɛ)=-∫Ω12bɛ⋅∇pɛdx=-6μω∫Ωhɛx22∂pɛ∂x1dx.
Combining (1.6) and (5.2) gives(5.3)Tɛ=∫Ωμωx23hɛdx-16μωIɛ(∇pɛ).
In the limit(5.4)Tɛ⟶C-16μωI0(∇p0)≔T0,
where the constant C is(5.5)C=μω∫Ω∫Yx23h(x,y)dydx.
In a similar way, as for the friction in the previous section, we can obtain bounds on the friction torque from (3.8) and (5.4). Indeed,(5.6)T0-≤T0≤T0+,
where(5.7)T0-=C-16μωI+(∇p+),T0+=C-16μωI-(∇p-).
As for the friction this motivates that the average of 𝒯0- and 𝒯0+ can be used to approximately find the homogenized friction torque 𝒯0.


## 6. A Numerical Example

In this section we apply the new method for computing the friction in a typical application. Indeed, we will estimate the friction force in a step bearing with bicosinusoidal roughness. More precisely, we consider the case whereh0 and hr are of the form(6.1)h0(x)={h1,0<x1≤L12,0<x2<L2,h2,L12<x1<L1,0<x2<L2,hr(y)=Acos⁡(2πy1)cos⁡(2πy2).
The data that is used is presented in Table 1. In the numerical analysis we used the software COMSOL and MATLAB.Table 1
Table of input data for the numerical example.
ParameterDescriptionValueUnitL1Pad length (x1)0.1mL2Pad length (x2)0.2mμFluid viscosity0.2Pa sARoughness amplitude0.4·10-5mh1Maximum ofh02.0·10-5mh2Minimum ofh01.0·10-5mV=(V1,V2)Velocity of the upper surface(1,0)m/sIt is natural to ask if the effect of the surface roughness is negligible or not. In order to give some answer to this question we consider the two extreme cases, where the distance between the surfaces is assumed to behmin=h0-A and hmax=h0+A (A is the amplitude of the surface roughness). Denote the friction forces corresponding to hmin and hmax by ℱmin respective ℱmax. Since hmin≤hmax it is obvious that the homogenized friction force, ℱ0, satisfies the estimates ℱmax≤ℱ0≤ℱmin. In our example we get that ℱmin=559.85 N and ℱmax=252.61 N. We also have that the friction force, ℱ00, corresponding to that the distance between the surfaces is just h0 (i.e., no roughness) is ℱ00=348.64 N (observe that the choice of roughness implies that h0 equals h0 plus the average of the roughness hr). This indicates that the influence of the surface roughness on the friction is significant and thus has to be taken into account.We will now illustrate that the new proposed method, based on bounds, gives very good estimates of the friction. Let us first note that the upper surface only moves in thex1-direction, that is, V2=0. From this it follows that b2(x,x/ɛ)=0, which in turn implies that b2±=0 and c2±=0. Due to symmetry it also holds that a1±=a2±. In Table 2 the coefficients for the bounds (3.2) are presented.Table 2
Table of input data for the numerical example.
h01.0000·10-52.0000·10-5a1+(h0)1.1136·10-158.2365·10-15a1-(h0)7.9000·10-167.5359·10-15b1+(h0)1.2000·10-52.4000·10-5b1-(h0)1.0641·10-52.3291·10-5c1+(h0)00c1-(h0)-2.8983·103-3.6014·102Moreover, the constantK in the bounds in (4.10) for ℱ0- and ℱ0+ is 309.83. The bounds become(6.2)F0-=K+16μ|V|∫Ω12b+⋅∇p+-c1+dx=356.65N,F0+=K+16μ|V|∫Ω12b-⋅∇p--c1-dx=395.41N.
The new method presented in this paper is that we use the average of ℱ0-+ℱ0+ as an approximation of the friction. We get that(6.3)F0-+F0+2=376.03N.
It should be observed that if we approximate the distance between the surfaces with h0 plus the mean value of hr and thereafter compute the friction, then the friction (denoted by ℱ00) is underestimated.Let us also compare the average ofℱ0-+ℱ0+ with the homogenized friction. Indeed, first we solve the local problems (2.1). Secondly, we compute the homogenized matrix A0(x), the homogenized vector b0(x), and the homogenized scalar c0(x) given by (2.2). We obtain that(6.4)c0(x)={1.8141⋅102,0<x1≤L12,0<x2≤L2,1.4912⋅103,L12<x1≤L1,0<x2≤L2,A0(x)=(a0(x)00a0(x)),b0(x)=(b01(x)0),
where(6.5)a0(x)={7.8836⋅10-15,0<x1≤L12,0<x2≤L2,9.4686⋅10-16,L12<x1≤L1,0<x2≤L2,b01(x)={2.3643⋅10-5,0<x1≤L12,0<x2≤L2,1.1301⋅10-5,L12<x1≤L1,0<x2≤L2.
After computing p0 given by (2.7) we find that(6.6)I0(∇p0)=∫Ω(12A0(x)∇p0⋅∇p0-b0(x)⋅∇p0-c0(x))dx=-∫Ω12b0(x)⋅∇p0+c0(x)dx=-79.653,
which implies that(6.7)F0=K-16μ|V|I0(∇p0)=376.21N.As mentioned before, homogenization can be used to find good approximate answers to various problems for small values ofɛ. This is due to various convergence results as ɛ→0. However, it is natural to ask how small ɛ has to be in order to have a good approximation. It has been shown in many examples in previous works that homogenization applies for rather large values of ɛ, much larger than in most typical applications. In the present example we have that ℱ0.01=376.08 N, ℱ0.005=376.15 N, and ℱ0.004=376.17 N.


## 7. Concluding Remarks

We have proposed a new method, which takes surface roughness into account, for computing the friction in, for example, different bearings. The main idea is that we use the variational formulation of the corresponding Reynolds equation. First we prove that the homogenized friction is closely related to the homogenized variational problem. Thereafter we use our previous results concerning bounds for the homogenized Lagrangian to obtain bounds for the homogenized friction. That the method is applicable is shown in a typical example.In order not to hide the main ideas we have considered the case when only one surface is rough. If both surfaces are assumed to be rough, then the coefficients (which are related to the distance between the surfaces) in the variational problem will involve rapid oscillations not only in space but also in time. The homogenization in this case has been studied in, for example, [15, 20, 21]. Bounds for the homogenized Lagrangian follow in this case in the same way, see [17]. Taking this into account it is obvious that the ideas in the present work also apply in order to compute the friction when both surfaces are rough. In [18] less sharp bounds (Reuss-Voigt type) were used to compute the corresponding p±. These bounds are in general wider. However, they are easier to compute, but can still give sufficient information. Clearly, the results in this work imply that the Reuss-Voigt type bounds can be used to generate bounds for the homogenized friction.



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*Source: 963983-2011-01-02.xml*