A New Approach for Estimating the Friction in Thin Film Lubrication

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.

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:ℎ𝑥,𝑦=ℎ₀(𝑥)+ℎ_𝑟𝑦, (1.1) where ℎ₀ and ℎ_𝑟 are smooth functions. Moreover, ℎ_𝑟 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 ℎ_𝜀 by ℎ_𝜀(𝑥)𝑥=ℎ𝑥,𝜀,𝜀>0. (1.2) This means that ℎ₀ represents the global film thickness, the periodic function ℎ_𝑟 describes the roughness on the upper surface, and 𝜀>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 viscosity 𝜇, then the pressure due to the relative motions of the surfaces is modeled by the Reynolds equation, see, for example, [1] or [2]: find 𝑝_𝜀∈𝑊₀^1,2(Ω) such thatdivℎ³_𝜀∇𝑝_𝜀=6𝜇divℎ_𝜀𝑉. (1.3) On the surface 𝑥₃=0 the friction force, 𝐹_𝜀, is given by𝐹_𝜀=_Ω𝜇ℎ_𝜀ℎ𝑉+^𝜀2∇𝑝_𝜀𝑑𝑥, (1.4) 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𝜕𝜕𝑥₁ℎ³_𝜀𝑥₂𝜕𝑝_𝜀𝜕𝑥₁+𝜕𝜕𝑥₂𝑥₂ℎ³_𝜀𝜕𝑝_𝜀𝜕𝑥₂=6𝜇𝜔𝑥₂divℎ_𝜀𝑒₁, (1.5) where 𝜔 is the angular speed, 𝑒₁=(1,0), 𝑥₁ the angular coordinate, and 𝑥₂ is the radial coordinate. The friction torque is then𝒯_𝜀=_Ω𝜇𝜔𝑥³₂ℎ_𝜀𝑑𝑥+_Ωℎ_𝜀𝑥₂2𝜕𝑝_𝜀𝜕𝑥₁𝑑𝑥, (1.6) see, for example, [1].

We note that both (1.3) and (1.5) for the pressure 𝑝_𝜀∈𝑊₀^1,2(Ω) are of the formdiv𝐴_𝜀∇𝑝_𝜀=div𝑏_𝜀, (1.7) where𝐴_𝜀(𝑥)=𝑎₁𝑥𝑥,𝜀00𝑎₂𝑥𝑥,𝜀,𝑏_𝜀(𝑥)=𝑏₁𝑥𝑥,𝜀𝑏₂𝑥𝑥,𝜀. (1.8) For (1.3) we have 𝑎_𝑖(𝑥,𝑥/𝜀)=ℎ³_𝜀(𝑥) and 𝑏_𝑖(𝑥,𝑥/𝜀)=6𝜇𝑉_𝑖ℎ_𝜀(𝑥) and for (1.5) 𝑎₁(𝑥,𝑥/𝜀)=ℎ³_𝜀(𝑥)/𝑥₂, 𝑎₂(𝑥,𝑥/𝜀)=𝑥₂ℎ³_𝜀(𝑥), and 𝑏_𝜀(𝑥)=6𝜇𝜔𝑥₂ℎ_𝜀(𝑥)𝑒₁.

A well-known fact from the calculus of variation is that the solution 𝑝_𝜀 of (1.7) also is the solution of the variational problemmin_{𝑝∈𝑊01,2(Ω)}𝐼_𝜀∇𝑝, (1.9) where 𝐼_𝜀∇𝑝=_Ω12𝐴_𝜀∇𝑝⋅∇𝑝−𝑏_𝜀⋅∇𝑝𝑑𝑥. (1.10) That is, 𝐼_𝜀(∇𝑝_𝜀)=min_𝑝𝐼_𝜀(∇𝑝).

In order to compute 𝐹_𝜀 and 𝒯_𝜀 we have to find the partial derivatives of 𝑝_𝜀. However, this is a delicate problem. The main reason is that for small values of 𝜀 (i.e., the roughness scale is much smaller than the global scale) the distance between the surfaces, ℎ_𝜀, 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 a 𝑝₀ such that 𝑝_𝜀→𝑝₀ as 𝜀→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 𝑝_𝜀 for small values of 𝜀. 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].

Let 𝑊^1,2_per(𝑌) be the closure of smooth 𝑌-periodic functions with respect to the usual norm in 𝑊^1,2(𝑌). Now we introduce the three local problems: find 𝑤₁, 𝑤₂, and 𝑣 in 𝐿²(Ω;𝑊^1,2_per(𝑌)) such thatdiv_𝑦𝐴𝑥,𝑦𝑒_𝑖+∇_𝑦𝑤_𝑖𝑥,𝑦=0,on𝑌,𝑖=1,2,div_𝑦𝑏𝑥,𝑦−𝐴𝑥,𝑦∇𝑣𝑥,𝑦=0,on𝑌. (2.1) 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:𝐴₀𝑒_𝑖=𝑌𝐴𝑒_𝑖+∇_𝑦𝑤_𝑖𝑏𝑑𝑦,𝑖=1,2,₀=_𝑌𝑏−𝐴∇_𝑦𝑣𝑐𝑑𝑦,₀=_𝑌12𝐴∇𝑣⋅∇𝑣𝑑𝑦. (2.2) It is now possible to define the homogenized integral functional 𝐼₀ as𝐼₀∇𝑝=_Ω𝑓₀𝑥,∇𝑝𝑑𝑥, (2.3) where the homogenized Lagrangian 𝑓₀ is given by𝑓₀()=1𝑥,𝜉2𝐴₀(𝑥)𝜉⋅𝜉−𝑏₀(𝑥)⋅𝜉−𝑐₀(𝑥). (2.4) The solutions 𝑝_𝜀 of (1.9) converge weakly in 𝑊₀^1,2(Ω) to the solution 𝑝₀ of the homogenized variational problemmin_{𝑝∈𝑊01,2(Ω)}𝐼₀∇𝑝, (2.5) that is, 𝐼₀(∇𝑝₀)=min_𝑝𝐼₀(∇𝑝). Moreover, we have the following convergence:lim_𝜀→0𝐼_𝜀∇𝑝_𝜀=𝐼₀∇𝑝₀=_Ω12𝐴₀(𝑥)∇𝑝₀⋅∇𝑝₀−𝑏₀(𝑥)⋅∇𝑝₀−𝑐₀(𝑥)𝑑𝑥. (2.6) We remark that 𝑝₀ also solves the Euler equation corresponding to (2.5), that is, div𝐴₀∇𝑝₀=div𝑏₀inΩ. (2.7)

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 Lagrangian 𝑓₀ 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), then𝑎⁺_𝑖(𝑥)=¹₀¹₀𝑎_𝑖𝑑𝑦_𝑗⁻¹𝑑𝑦_𝑖⁻¹,𝑎⁻_𝑖(𝑥)=¹₀¹₀1𝑎_𝑖𝑑𝑦_𝑖⁻¹𝑑𝑦_𝑗,𝑏⁺_𝑖(𝑥)=𝑎⁺_𝑖(𝑥)¹₀∫¹₀𝑏_𝑖𝑑𝑦_𝑗∫¹₀𝑎_𝑖𝑑𝑦_𝑗𝑑𝑦_𝑖,𝑏⁻_𝑖=¹₀¹₀1𝑎_𝑖𝑑𝑦_𝑖⁻¹¹₀𝑏_𝑖𝑎_𝑖𝑑𝑦_𝑖𝑑𝑦_𝑗,𝑐⁺_𝑖(𝑥)=12𝑎⁺_𝑖(𝑥)¹₀∫¹₀𝑏_𝑖𝑑𝑦_𝑗∫¹₀𝑎_𝑖𝑑𝑦_𝑗𝑑𝑦_𝑖²−12¹₀∫¹₀𝑏_𝑖𝑑𝑦_𝑗²∫¹₀𝑎_𝑖𝑑𝑦_𝑗𝑑𝑦_𝑖,𝑐⁻_𝑖(𝑥)=12¹₀¹₀1𝑎_𝑖𝑑𝑦_𝑖⁻¹¹₀𝑏_𝑖𝑎_𝑖𝑑𝑦_𝑖²𝑑𝑦_𝑗−12¹₀¹₀𝑏²_𝑖𝑎_𝑖𝑑𝑦_𝑖𝑑𝑦_𝑗. (3.1)

These functions are now used to define 𝑓⁻ and 𝑓⁺ as𝑓⁻()=1𝑥,𝜉2𝐴⁻𝜉⋅𝜉−𝑏⁻⋅𝜉+𝑐⁻₁+𝑐⁻₂,𝑓⁺()=1𝑥,𝜉2𝐴⁺𝜉⋅𝜉−𝑏⁺⋅𝜉+𝑐⁺₁+𝑐⁺₂, (3.2) where 𝜉∈ℝ²,𝐴^±=𝑎^±₁00𝑎^±₂,𝑏^±=𝑏^±₁𝑏^±₂. (3.3) The main result in [16] is that we have the following bounds on the homogenized Lagrangian 𝑓₀: 𝑓⁻(𝑥,𝜉)≤𝑓₀(𝑥,𝜉)≤𝑓⁺(𝑥,𝜉). (3.4) From this it is obvious that min𝐼⁻∇𝑝≤min𝐼₀∇𝑝≤min𝐼⁺∇𝑝, (3.5) where the minimum is taken over all 𝑝 in 𝑊₀^1,2(Ω) and 𝐼⁻ and 𝐼⁺ are defined as𝐼⁻∇𝑝=_Ω𝑓⁻𝐼𝑥,∇𝑝𝑑𝑥,₊∇𝑝=_Ω𝑓⁺𝑥,∇𝑝𝑑𝑥. (3.6) 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, 𝐼⁻∇𝑝⁻=min_𝑝𝐼⁻∇𝑝,𝐼⁺∇𝑝⁺min_𝑝𝐼⁺∇𝑝, (3.7) then (3.5) can be rewritten as𝐼⁻∇𝑝⁻≤𝐼₀∇𝑝₀≤𝐼⁺∇𝑝⁺. (3.8) 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 pressures 𝑝⁻ and 𝑝⁺ which are minimizers in the variational problems given in (3.7) can be found by solving the corresponding Euler equations. Indeed,div𝐴^±∇𝑝^±=div𝑏^±inΩ. (3.9) 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 of 𝐼₀(∇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, when 𝑎_𝑖(𝑥,𝑥/𝜀)=ℎ³_𝜀(𝑥) and 𝑏_𝑖(𝑥,𝑥/𝜀)=6𝜇𝑉_𝑖ℎ_𝜀(𝑥) (𝑉_𝑖 is the constant speed in the 𝑥_𝑖-direction). As mentioned in the introduction, the friction force, 𝐹_𝜀, on the surface 𝑥₃=0 is given by𝐹_𝜀=_Ω𝜇ℎ_𝜀ℎ𝑉+^𝜀2∇𝑝_𝜀𝑑𝑥. (4.1) The component of the friction force in the direction 𝑉 of the motion isℱ_𝜀=𝐹_𝜀⋅1|𝑉|𝑉=_Ω𝜇|𝑉|ℎ_𝜀+ℎ_𝜀2|𝑉|𝑉⋅∇𝑝_𝜀𝑑𝑥. (4.2)

Let us now consider the generalized formulation of (1.7): find 𝑝_𝜀∈𝑊₀^1,2(Ω) such that_Ω𝐴_𝜀∇𝑝_𝜀⋅∇𝜙𝑑𝑥=_Ω𝑏_𝜀⋅∇𝜙𝑑𝑥, (4.3) for any 𝜙∈𝑊₀^1,2(Ω). Formally this equation is obtained by first multiplying (1.7) by 𝜙 and thereafter using partial integration. In particular, for 𝜙=𝑝_𝜀, it holds that_Ω𝐴_𝜀∇𝑝_𝜀⋅∇𝑝_𝜀𝑑𝑥=_Ω𝑏_𝜀⋅∇𝑝_𝜀𝑑𝑥. (4.4) This implies that𝐼_𝜀∇𝑝_𝜀=_Ω12𝐴_𝜀∇𝑝_𝜀⋅∇𝑝_𝜀−𝑏_𝜀⋅∇𝑝_𝜀𝑑𝑥=−_Ω12𝑏_𝜀⋅∇𝑝_𝜀𝑑𝑥=−3𝜇_Ωℎ_𝜀𝑉⋅∇𝑝_𝜀𝑑𝑥. (4.5) Thus_Ωℎ_𝜀𝑉⋅∇𝑝_𝜀1𝑑𝑥=−𝐼3𝜇^𝜀∇𝑝_𝜀. (4.6) From (4.2) and (4.6) it follows thatℱ_𝜀=𝜇|𝑉|_Ω1ℎ_𝜀1𝑑𝑥−|𝑉|𝐼6𝜇^𝜀∇𝑝_𝜀. (4.7) By using (2.6) it is possible to pass to the limit in this equality. We getℱ_𝜀1⟶𝐾−|𝑉|𝐼6𝜇⁰∇𝑝₀∶=ℱ₀, (4.8) where the constant 𝐾 is𝐾=𝜇|𝑉|_Ω𝑌1ℎ𝑥,𝑦𝑑𝑦𝑑𝑥. (4.9) By (3.8) and (4.8) we get bounds on the homogenized (averaged) friction in the direction of the motion, ℱ₀,ℱ⁻₀≤ℱ₀≤ℱ⁺₀, (4.10) whereℱ⁻₀1=𝐾−|𝑉|𝐼6𝜇⁺∇𝑝⁺,ℱ⁺₀1=𝐾−|𝑉|𝐼6𝜇⁻∇𝑝⁻. (4.11)

If the difference between 𝐼⁺(∇𝑝⁺) and 𝐼⁻(∇𝑝⁻) is small, then the average of ℱ⁻₀+ℱ⁺₀ is a good approximation of the friction ℱ₀. The benefit of this approximation is that it is easier to compute ℱ⁻₀ and ℱ⁺₀ 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 ℱ⁻₀ and ℱ⁺₀ does not involve solution of any local problems.

In order to calculate ℱ⁻₀ and ℱ⁺₀ we have to find 𝐼⁺(∇𝑝⁺) and 𝐼⁻(∇𝑝⁻). Recall that𝐼⁺∇𝑝⁺=_Ω12𝐴⁺∇𝑝⁺⋅∇𝑝⁺−𝑏⁺⋅∇𝑝⁺+𝑐⁺₁+𝑐⁺₂𝐼𝑑𝑥,₊∇𝑝⁻=_Ω12𝐴⁻∇𝑝⁻⋅∇𝑝⁻−𝑏⁻⋅∇𝑝⁻+𝑐⁻₁+𝑐⁻₂𝑑𝑥. (4.12)(4.13) 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_Ω𝐴⁺∇𝑝⁺⋅∇𝑝⁺𝑑𝑥=_Ω𝑏⁺⋅∇𝑝⁺𝑑𝑥. (4.14) This implies that the expression (4.12) for 𝐼⁺(∇𝑝⁺) can be rewritten as𝐼⁺∇𝑝⁺=_Ω𝑐⁺₁+𝑐⁺₂−12𝑏⁺⋅∇𝑝⁺𝑑𝑥. (4.15) In the same we get that𝐼⁻∇𝑝⁻=_Ω𝑐⁻₁+𝑐⁻₂−12𝑏⁻⋅∇𝑝⁻𝑑𝑥. (4.16)

It should be mentioned that it is possible to pass to the limit in (4.2) directly. Indeed, it was proved in [14] that ∇𝑝_𝜀 two scale converges to ∇𝑝₀+∇_𝑦𝑝₁, where 𝑝₁ is of the form𝑝₁𝑥,𝑦=𝑤₁𝑥,𝑦𝜕𝑝₀𝜕𝑥₁+𝑤₂𝑥,𝑦𝜕𝑝₀𝜕𝑥₂+𝑣𝑥,𝑦. (4.17) For the friction we getℱ_𝜀⟶_Ω𝑌𝜇|𝑉|ℎ+ℎ2|𝑉|𝑉⋅∇𝑝₀+∇_𝑦𝑝₁𝑑𝑦𝑑𝑥. (4.18) 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/𝜀)=ℎ³_𝜀(𝑥)/𝑥₂, 𝑎₂(𝑥,𝑥/𝜀)=𝑥₂ℎ³_𝜀(𝑥) and 𝑏_𝜀(𝑥)=6𝜇𝜔𝑥₂ℎ_𝜀(𝑥)𝑒₁. The friction torque is then given by (1.6), that is,𝒯_𝜀=_Ω𝜇𝜔𝑥³₂ℎ_𝜀𝑑𝑥+_Ωℎ_𝜀𝑥₂2𝜕𝑝_𝜀𝜕𝑥₁𝑑𝑥, (5.1) see, for example, [1]. In the same way as for the friction force we get that𝐼_𝜀∇𝑝_𝜀=−_Ω12𝑏_𝜀⋅∇𝑝_𝜀𝑑𝑥=−6𝜇𝜔_Ωℎ_𝜀𝑥₂2𝜕𝑝_𝜀𝜕𝑥₁𝑑𝑥. (5.2) Combining (1.6) and (5.2) gives𝒯_𝜀=_Ω𝜇𝜔𝑥³₂ℎ_𝜀1𝑑𝑥−𝐼6𝜇𝜔^𝜀∇𝑝_𝜀. (5.3) In the limit𝒯_𝜀1⟶𝐶−𝐼6𝜇𝜔⁰∇𝑝₀∶=𝒯₀, (5.4) where the constant 𝐶 is𝐶=𝜇𝜔_Ω𝑌𝑥³₂ℎ𝑥,𝑦𝑑𝑦𝑑𝑥. (5.5) 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,𝒯⁻₀≤𝒯₀≤𝒯⁺₀, (5.6) where𝒯⁻₀1=𝐶−𝐼6𝜇𝜔⁺∇𝑝⁺,𝒯⁺₀1=𝐶−𝐼6𝜇𝜔⁻∇𝑝⁻. (5.7) As for the friction this motivates that the average of 𝒯⁻₀ and 𝒯⁺₀ 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 where ℎ₀ and ℎ_𝑟 are of the formℎ₀(𝑥)=ℎ₁,0<𝑥₁≤𝐿₁2,0<𝑥₂<𝐿₂,ℎ₂,𝐿₁2<𝑥₁<𝐿₁,0<𝑥₂<𝐿₂,ℎ_𝑟𝑦=𝐴cos2𝜋𝑦₁cos2𝜋𝑦₂. (6.1) 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.

It 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 be ℎ_min=ℎ₀−𝐴 and ℎ_max=ℎ₀+𝐴 (𝐴 is the amplitude of the surface roughness). Denote the friction forces corresponding to ℎ_min and ℎ_max by ℱ_min respective ℱ_max. Since ℎ_min≤ℎ_max it is obvious that the homogenized friction force, ℱ₀, satisfies the estimates ℱ_max≤ℱ₀≤ℱ_min. In our example we get that ℱ_min=559.85 N and ℱ_max=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 ℎ_𝑟). 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 the 𝑥₁-direction, that is, 𝑉₂=0. From this it follows that 𝑏₂(𝑥,𝑥/𝜀)=0, which in turn implies that 𝑏^±₂=0 and 𝑐^±₂=0. Due to symmetry it also holds that 𝑎^±₁=𝑎^±₂. In Table 2 the coefficients for the bounds (3.2) are presented.

Table 2 

Table of input data for the numerical example.

Moreover, the constant 𝐾 in the bounds in (4.10) for ℱ⁻₀ and ℱ⁺₀ is 309.83. The bounds becomeℱ⁻₀1=𝐾+6𝜇|𝑉|_Ω12𝑏⁺⋅∇𝑝⁺−𝑐⁺₁ℱ𝑑𝑥=356.65N,₊₀1=𝐾+6𝜇|𝑉|_Ω12𝑏⁻⋅∇𝑝⁻−𝑐⁻₁𝑑𝑥=395.41N. (6.2) The new method presented in this paper is that we use the average of ℱ⁻₀+ℱ⁺₀ as an approximation of the friction. We get thatℱ⁻₀+ℱ⁺₀2=376.03N. (6.3) It should be observed that if we approximate the distance between the surfaces with ℎ₀ plus the mean value of ℎ_𝑟 and thereafter compute the friction, then the friction (denoted by ℱ₀₀) is underestimated.

Let us also compare the average of ℱ⁻₀+ℱ⁺₀ 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 that𝑐₀(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)=𝑏¹₀(𝑥)0, (6.4) where𝑎₀(𝑥)=7.8836⋅10⁻¹⁵,0<𝑥₁≤𝐿₁2,0<𝑥₂≤𝐿₂,9.4686⋅10⁻¹⁶,𝐿₁2<𝑥₁≤𝐿₁,0<𝑥₂≤𝐿₂,𝑏¹₀(𝑥)=2.3643⋅10⁻⁵,0<𝑥₁≤𝐿₁2,0<𝑥₂≤𝐿₂,1.1301⋅10⁻⁵,𝐿₁2<𝑥₁≤𝐿₁,0<𝑥₂≤𝐿₂. (6.5) After computing 𝑝₀ given by (2.7) we find that𝐼₀∇𝑝₀=_Ω12𝐴₀(𝑥)∇𝑝₀⋅∇𝑝₀−𝑏₀(𝑥)⋅∇𝑝₀−𝑐₀(𝑥)𝑑𝑥=−_Ω12𝑏₀(𝑥)⋅∇𝑝₀+𝑐₀(𝑥)𝑑𝑥=−79.653, (6.6) which implies thatℱ₀1=𝐾−|𝑉|𝐼6𝜇⁰∇𝑝₀=376.21N. (6.7)

As mentioned before, homogenization can be used to find good approximate answers to various problems for small values of 𝜀. This is due to various convergence results as 𝜀→0. However, it is natural to ask how small 𝜀 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 𝜀, much larger than in most typical applications. In the present example we have that ℱ_0.01=376.08 N, ℱ_0.005=376.15 N, and ℱ_0.004=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.

Acknowledgment

The authors thank the reviewers for the helpful comments.