#### Abstract

We study the Reynolds equation, describing the ow of a lubricant, in case of pressure-dependent viscosity. First we prove the existence and uniqueness of the solution. Then, we study the asymptotic behavior of the solution in case of periodic roughness via homogenization method. Some interesting nonlocal effects appear due to the nonlinearity.

#### 1. Introduction

The Reynolds equations [1] describe the flow of a thin film of lubricant separating two rigid surfaces in relative motion. Controlling the flow of lubricant is an important engineering issue since inappropriate lubrication would increase the friction and wear, finally resulting in degrading the performance of the device. In his classical paper from 1848 [2] Stokes predicted that the viscosity of the fluid can depend on the pressure. Those effects for various liquids have been measured in many engineering papers, starting from the beginning of the 20th century (see, e.g., [3]). That effect is usually neglected as it becomes important only in case of high pressure. Most fluid-lubricated bearings operate with high pressure and in such a flow regime the dependence of the viscosity on the pressure becomes important. According to Szeri [4] the idea of pressure-dependent viscosity was introduced in lubrication theory by Gatcombe in 1945 [5]. Several models have been used to describe that relation since. The most popular is probably the exponential law
usually called the Barus formula [6]. Here and are the constants depending on the lubricant. The formula seems to be reasonable for mineral oil, unless the pressure is very high (larger then 0.5 MPa). The coefficient typically ranges between 1 and 10^{−8}. The lower end of the range corresponds to * paraffinic* and the upper end corresponds to the * naphthenic* oils (see Jones [7]). That formula is still frequently used by engineers. The simplest viscosity-pressure relation is given by the the power law
In case of the two above-mentioned laws explicit solutions of the equations of motion, for some particular situations like unidirectional and plane-parallel flows, were found in [8]. Discussion on other possibilities for the viscosity-pressure formula and some historical remarks on the subject can be found in the same paper. Several engineering papers can be found discussing other possible laws and their consistency. We mention, for instance, [9, 10].

We do not make any assumption on the particular form of the function . Some technical assumptions, like smoothness, will be needed for the proofs.

We study the stationary version of the Reynolds equation. Unless the velocity of relative motion is time dependent, steady approximation is reasonable in most cases (see, e.g., [4, Chapter 2.2]).

Our first goal is to prove that the problem is well posed. Secondly, we investigate the asymptotic behavior of the solution in case of periodically distributed asperities. Using the homogenization approach, we find the macroscopic Reynolds pressure. Interesting nonlocal effects appear due to the nonlinearity caused by the pressure-dependent viscosity.

#### 2. Position of the Problem

The fluid domain is bounded by two rigid surfaces. The simplified mathematical model can be written in the following form. Let be a bounded domain and let be a bounded, strictly positive smooth function such that Function describes the shape of the slide. By we denote a very small parameter representing the domain thickness. Using the shape function we define the fluid domain (Figure 1) by We then consider the stationary flow through a domain . We want to describe the situation with a lower-dimensional model. The velocity of the relative motion of two surfaces is the constant vector denoted by . The unknowns in the model are (the velocity) and (the pressure). We recall that the stationary motion of the incompressible viscous laminar flow is governed by the stationary Navier-Stokes equations. Thus we write the following system: where is the symmetric part of the velocity gradient. It is important to notice that in such system the pressure is not defined only up to a constant, as in the classical Navier-Stokes system with constant viscosity. Under certain technical assumptions, if the given data are not too large, the existence of the solution for such system was discussed in [11, 12]. Neglecting the effects of inertia, we get the Stokes system with pressure-dependent viscosity studied in [13].

If the thickness of the domain is small, the solution can be fairly approximated by the solution of the Reynolds equations [4, 14] Indeed, if we derive a formal asymptotic expansion of the solution to the system (5) in powers of , then the solution of the Reynolds equation (7) makes the first term of the expansion (see, e.g., [4]). Here and in the sequel the differential operatorsdivand are taken only with respect to variable; that is, It leads to an elliptic equation of the form The goal of this paper is to study that equation.

We assume that the function is of class and for any value of . In real life the viscosity increases with pressure, but such an assumption is not necessary for our study.

#### 3. Existence of the Solution

##### 3.1. Transformed Equation

Equation (10) is a quasilinear elliptic PDE, but it can be linearized by simple trick. To do so we rewrite the equation using the function We choose . Function is strictly increasing, since and thus it is bijective. Furthermore has the same sign as ; that is, for we have , for obviously , and finally , if and only if .

We introduce the new unknown function At this point we assume that the integral is divergent, that is, As a consequence as well as Deriving (13) we obtain and the problem can be written as That is a linear elliptic equation for and it has a unique solution. To get the existence and uniqueness of the solution we quote Theorem 8.34 from classical book of Gilbarg and Trudinger [15]. For simplicity, here and in the sequel, we assume that and, consequently, are defined on whole . We combine that with the maximum principle from the appendix, and it gives the following.

Theorem 1. *Under the* assumption that the boundary is of class and that , the problem (18), (19) has a unique solution
*
Furthermore
**
where
**
and if . Otherwise
**with .*

*Proof. *The existence follows directly from Theorem 8.34. from Gilbarg and Trudinger [15]. If , then (21) follows directly from the weak maximum principle (see, e.g., [15]). In case the problem can be solved by quadratures and the solution given by (25) can be easily estimated to get (23). In the remaining case (21) follows from the special variant of the maximum principle proved in the appendix.

*Remark 2. *In case (18) is an ODE (we take and , without losing generality)
and it can be solved by quadratures

##### 3.2. Back to the Original Equation

Now, our goal is not to find the auxiliary function but to find the pressure . Since we have introduced as we should have . In order to do so we have to make sure that for any . Since is strictly increasing and we have assumed that (14) holds, if we define due to (16) we obviously have for any Thus So, to fulfill the condition we need to have

That condition is not necessarily fulfilled.

In view of (21) that condition reduces to where is defined by (23).

We have proved the following theorem.

Theorem 3. *Suppose that the conditions of Theorem 1 hold, and that in addition (31) is fulfilled. Then is the unique solution of (10) and (11). *

*Remark 4. *It is important to notice that even though does depend on , the effective pressure does not. For the purpose of this remark, we denote to stress the dependence on the parameter which is of interest here. We start by
It is obvious from the definition of that
As , deriving with respect to we arrive at

Deriving (13) we get . Using the rule for deriving the inverse function, we have
Thus

#### 4. Homogenization

In this section we want to study the effects of rugosities of surfaces on lubrication process. The idea of finding the macroscopic effects of roughness on lubrication process, via homogenization, is quite old and well studied. Case of constant viscosity for incompressible and compressible flows as well as non-Newtonian, deformation dependent, viscosities were investigated. The subject was treated by several authors and we here mention [16–18]. The case of pressure-dependent viscosity brings some new interesting nonlocal effects.

We assume that the function , describing the form of the fluid domain, is periodic with small period , with . To stress that dependence we denote it by . More precisely, we denote by , the period. We further assume that , is periodic with period and smooth. Then we take of the form Thus, the function describes the form of periodically distributed rugosities.

To emphasize that the relative velocity of bearing surfaces is large, we assume that it also depends on , the same parameter that is taken for description of rugosities. In that case our equation reads

##### 4.1. One-Dimensional Case

If , the above problem is posed on an interval . With an appropriate boundary condition It forms a boundary value problem for nonlinear ODE: To study the asymptotic behavior of the solution with respect to we linearize the problem using the transformation . To simplify, in this section we choose and, dropping the index in and , we denote

Theorem 5. *Let**and let *. *Suppose that there exists a limit**and that*, *for * *large enough and * *defined in* (27), *the following condition holds:**with**Then*

*Proof. *Equation (40), with boundary conditions (39), can be solved by reduction to quadratures, after the substitution
with strictly increasing function defined by (12). The problem for now reads
It is easy to see that (49) has a unique solution given by (25). Since , (25) now reduces to
where, for ,
Now , the solution to the problem (40), exists if (45) is fulfilled. The second term in (50) thus obviously tends to , as . The last term is more interesting. The denominator tends to
As for its numerator, we have

Suppose that
and denote
Obviously the function is periodic with period 1 so that, due to the standard periodicity lemma (see, e.g., [19]), as ,

By direct computation
Thus

Now, denoting
we have
and thus

However, we are not interested in convergence of the auxiliary function but in the convergence of the pressure . Since (45) is assumed to be true, we can define , where , and we have

Deriving the expression on the right-hand side, we obtain the effective pressure drop in the form As we can see, the pressure drop is not constant, as for the Newtonian flow. The interesting effect appears if because, in that case, the expressions for the pressure and for the pressure drop are nonlocal due to the integral with respect to . That phenomenon is entirely due to the fact that the viscosity is depending on the pressure.

##### 4.2. Two-Dimensional Case

We suppose here that the function is constructed from positive, smooth, -periodic function , , in the same way as before, that is, by taking We have seen in the previous section that interesting effects happen only if we assume that In that case our equation reads As we did in the existence analysis and in the previous section, we linearize the equation by substitution where the function is defined by (12). Now satisfies

We postulate the asymptotic expansion in the form All functions are assumed to be -periodic in variable.

Plugging that in (69) and collecting, formally, terms with equal powers of , we get : : 1: .

Denoting we have

*Remark 6. *The same computation can be done in one-dimensional case and it gives
Constants are chosen in a way that boundary conditions are met, and it follows that
Then
That is a very good approximation of our exact solution (50).

It is important to notice that the choice of constants was determined from the exterior boundary condition. So, we should expect the same in two-dimensional case. However the treatment of boundary conditions in two-dimensional case is much more complicated and the boundary layer is to be expected.

The derived asymptotic expansion should be justified by proving the convergence. And we need the strong convergence (with corrector, of course) for in order to get the convergence for . The form of the approximation suggests that the boundary layer phenomenon should appear on the exterior boundary since term cannot satisfy the Dirichlet condition on . To get the error estimate and the strong convergence we need to handle that boundary layer. Thus, at this point we simplify the domain and the boundary condition, in order to be able to avoid it. We assume that Now and is 1-periodic.

In that case we can compute and , explicitly and we can impose exterior condition on . Indeed is exactly the same as in the monodimensional case; that is, it is given by (73) and (75). Obviously so that As for the last term Finally, the function satisfies the boundary value problem It can be solved using the Fourier method, and we get Since the approximation now satisfies the boundary conditions on , it is easy to see that follows from the maximum principle. Assuming that, for large enough, we have Finally We have proved that.

Theorem 7. *Let * *be the solution to the problem* (67), (79), and (80) *and let *, *be defined by* (87) *and* (42), *respectively*. *If* (90) *holds*, *then*

*Remark 8. *It is important to notice that satisfies
and thus we would expect it to be the limit of in analogy with the linear case. However .

If is small, we can expand in powers of and we get
Thus
It can be, formally, written as

#### Appendix

#### The Maximum Principle

Our goal is to derive maximum principles for the linear Reynolds equation, with sharp explicit constants, in order to solve the nonlinear Reynolds equation with pressure-dependent viscosity. We assume, without losing generality, that . Indeed, we can always choose the coordinate system in a way that the first coordinate axis has a direction of the velocity of relative motion .

The lower bound for is of no interest, just the upper bound. Function is the solution to the boundary value problem We assume that if , then cannot have a maximum point in the domain and, thus, However it is not realistic to assume that does not change the sign. To find the upper bound in the general case we use the procedure from the DeGiorgi theorem. The main result of the section is as follows.

Theorem A.1. *Let be the solution to the problem (A.1). Then
*

*Proof. *The function satisfies
Next we introduce the embedding constant for denoted , such that
That constant can be estimated as
See, for example, [20, Lemma 1]. Next we define the sequence
Easy computation yields
Let
We test (A.5) with and get
For the left-hand side, we get the lower bound
We estimate the terms on the right-hand side using the same idea
Thus, it remains to estimate . We have

Combining with (A.14) and (A.13), we get
We recall that
and define
as well as
Then (A.16) implies

Taking the logarithm, we arrive at
We first notice that
and then
Since the function is decreasing for , we have
Then,
Finally
Now it remains to estimate . From the definition we see that
To estimate , we proceed as before and test (A.5) with . We get
Thus
Finally, testing (A.5) with , we get
so that
Combining (A.29) with (A.31) and (A.26) gives
Since
we have arrived to
Since , we get

Finally (A.8) implies (A.3).

#### Acknowledgment

This work was supported by MZOS grant 037-0372787-2797.