Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 710214, 10 pages

http://dx.doi.org/10.1155/2013/710214

Research Article

Analysis of the Reynolds Equation for Lubrication in Case of Pressure-Dependent Viscosity

Eduard Marušić-Paloka and Sanja Marušić

Department of Mathematics, University of Zagreb, Bijenička Cesta 30, 10000 Zagreb, Croatia

Received 30 November 2012; Revised 5 March 2013; Accepted 10 March 2013

Academic Editor: Jyh-Hong Chou

Copyright © 2013 Eduard Marušić-Paloka and Sanja Marušić. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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].

Figure 1

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.

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