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

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 𝜇=𝜇₀exp𝛼𝑝 (1) usually called the Barus formula [6]. Here 𝜇₀ and 𝛼 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 𝛼 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 𝜇𝑝=𝛼𝑝^𝛾. (2) 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 𝑝↦𝜇(𝑝). 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 𝒪⊂𝐑^𝑛 be a bounded domain and let ℎ∶𝒪→𝐑 be a bounded, strictly positive smooth function such that 0<ℎ₀≤ℎ(𝑥)≤ℎ₁. (3) Function ℎ describes the shape of the slide. By 𝜀≪1 we denote a very small parameter representing the domain thickness. Using the shape function ℎ we define the fluid domain (Figure 1) by Ω_𝜀=𝑋=𝑥,𝑥_𝑛+1∈𝐑^𝑛+1;𝑥=𝑥₁,…,𝑥_𝑛∈𝒪,0<𝑥_𝑛+1<𝜀ℎ(𝑥). (4) We then consider the stationary flow through a domain Ω_𝜀. 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 𝐕=𝑣₁𝐞₁+⋯+𝑣_𝑛𝐞_𝑛. The unknowns in the model are 𝑢 (the velocity) and 𝑝 (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: −div𝜇𝑝𝐃𝐮+(𝐮∇)𝐮+∇𝑝=0,div𝐮=0inΩ_𝜀,𝐮=𝐕for𝑥_𝑛+1=0,𝐮=0for𝑥_𝑛+1=𝜀ℎ(𝑥), (5) where 𝐃𝐮=(1/2)(∇𝐮+∇𝐮^𝑡) 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 div𝜇𝑝𝐃𝐮+∇𝑝=0,div𝐮=0inΩ_𝜀, (6) 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] 𝐯𝑥,𝑥_𝑛+1=1𝑝𝑥12𝜇^𝑛+1𝜀ℎ−𝑥_𝑛+1(𝑥)+𝑥∇𝑝1−^𝑛+1𝜀ℎ𝐕,div₀^𝜀ℎ𝐯𝑑𝑥_𝑛+1=0. (7)(8) Indeed, if we derive a formal asymptotic expansion of the solution to the system (5) in powers of 𝜀, 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 𝑥 variable; that is, div𝐛=𝜕𝑏₁𝜕𝑥₁+⋯+𝜕𝑏_𝑛𝜕𝑥_𝑛,∇𝜙=𝜕𝜙𝜕𝑥₁𝐞₁+⋯+𝜕𝜙𝜕𝑥_𝑛𝐞_𝑛. (9) It leads to an elliptic equation of the form ℎdiv³𝜇𝑝∇𝑝=6𝐕⋅∇ℎin𝒪𝑝=𝑞on𝜕𝒪. (10)(11) The goal of this paper is to study that equation.

We assume that the function 𝑝↦𝜇(𝑝) is of class 𝐶¹(𝐑) and 𝜇>0 for any value of 𝑝. 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 𝑀_𝜎𝑝=^𝑝_𝜎𝑑𝑡𝜇(𝑡). (12) We choose 𝜎≤𝑞. Function 𝑀_𝜎 is strictly increasing, since 𝑀^′_𝜎(𝑝)=(1/𝜇(𝑝))>0 and thus it is bijective. Furthermore 𝑀_𝜎(𝑝) has the same sign as 𝑝−𝜎; that is, for 𝑝>𝜎 we have 𝑀_𝜎(𝑝)>0, for 𝑝<𝜎 obviously 𝑀_𝜎(𝑝)<0, and finally 𝑀_𝜎(𝑝)=0, if and only if 𝑝=𝜎.

We introduce the new unknown function 𝑤(𝑥)=𝑀_𝜎𝑝(𝑥)=_𝜎^𝑝(𝑥)𝑑𝑡𝜇(𝑡). (13) At this point we assume that the integral ∫₀^−∞(𝑑𝑡/𝜇(𝑡)) is divergent, that is, ⁰_−∞𝑑𝑡𝜇(𝑡)=+∞. (14) As a consequence lim_{𝜎→−∞}𝑀_𝜎𝑝=+∞,∀𝑝∈𝐑, (15) as well as lim_{𝑝→−∞}𝑀_𝜎𝑝=−∞,∀𝜎∈𝐑. (16) Deriving (13) we obtain 1𝜇𝑝∇𝑝=∇𝑤 (17) and the problem can be written as divℎ³∇𝑤=6𝐕⋅∇ℎin𝒪,𝑤=𝑀_𝜎𝑞on𝜕𝒪. (18)(19) That is a linear elliptic equation for 𝑤 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 𝑞 and, consequently, 𝑀_𝜎(𝑞) are defined on whole 𝒪. We combine that with the maximum principle from the appendix, and it gives the following.

Theorem 1. Under the assumption that the boundary 𝜕𝒪 is of class 𝐶^1,𝛼 and that ℎ∈𝐶^𝛼(𝒪), 𝑞∈𝐶^1,𝛼(𝒪) the problem (18), (19) has a unique solution 𝑤∈𝐶^1,𝛼𝒪. (20) Furthermore 𝑤(𝑥)≤𝑀𝑞+𝒵, (21) where 𝑞=𝑞_{𝐿∞(𝜕𝜔)} (22) and 𝒵=0 if 𝐕⋅∇h<0. Otherwise 6|𝐕∫𝒵=|¹₀𝑑𝑡ℎ(𝑡)²−ℎ³₀ℎ⁵₁+ℎ₀ℎ₁³∫^𝑞1_𝑞0𝑑𝑠𝜇(𝑠)3if𝑛=1_(8/5)32^(28/5)(2𝜋)^(1/4)ℎ³₀×𝑑6|𝐕|ℎ₁|𝒪|^1/5+ℎ³₁∇𝑀_𝜎𝑞_𝐿5(𝒪)|𝒪|^(7/5)if𝑛=2, (23)
with 𝑑=diam𝒪.

Proof. The existence follows directly from Theorem 8.34. from Gilbarg and Trudinger [15]. If 𝐕⋅∇ℎ<0, then (21) follows directly from the weak maximum principle (see, e.g., [15]). In case 𝑛=1 the problem can be solved by quadratures and the solution given by (25) can be easily estimated to get (23). In the remaining case 𝑛=2 (21) follows from the special variant of the maximum principle proved in the appendix.

Remark 2. In case 𝑛=1 (18) is an ODE (we take 𝒪=]0,1[  and 𝑞₀>𝑞₁, without losing generality) ℎ³𝑤^′′=6𝑉ℎ^′][,𝑤(𝑖)in0,1=𝑀_𝜎𝑞_𝑖for𝑖=0,1, (24) and it can be solved by quadratures 𝑤(𝑥)=𝑀_𝜎𝑞₀+^𝑥₀1ℎ(𝑡)³ℎ(𝑡)−∫6𝑉¹⁰𝑑𝑟/ℎ(𝑟)²∫¹₀𝑑𝑠/ℎ(𝑠)³+𝑀_𝜎𝑞₁−𝑀_𝜎𝑞₀∫¹₀𝑑𝑠/ℎ(𝑠)³𝑑𝑡. (25)

3.2. Back to the Original Equation

Now, our goal is not to find the auxiliary function 𝑤 but to find the pressure 𝑝. Since we have introduced 𝑤 as 𝑤=𝑀_𝜎𝑝, (26) we should have 𝑝=𝑀_𝜎⁻¹(𝑤). In order to do so we have to make sure that 𝑤(𝑥)∈Im𝑀_𝜎 for any 𝑥∈𝒪. Since 𝑀_𝜎 is strictly increasing and we have assumed that (14) holds, if we define 𝑀⁺_𝜎=lim_𝑠→+∞𝑀_𝜎(𝑠)=_𝜎^+∞𝑑𝑠𝜇(𝑠), (27) due to (16) we obviously have for any 𝑝∈𝐑−∞=lim_{𝑠→−∞}𝑀_𝜎(𝑠)≤𝑀_𝜎𝑝≤𝑀⁺_𝜎. (28) Thus Im𝑀_𝜎=−∞,𝑀⁺_𝜎. (29) So, to fulfill the condition 𝑤(𝑥)∈Im𝑀_𝜎 we need to have 𝑤(𝑥)≤𝑀⁺_𝜎,∀𝑥∈𝒪. (30)

That condition is not necessarily fulfilled.

In view of (21) that condition reduces to 𝒵≤^+∞_𝑞𝑑𝑡𝜇(𝑡), (31) where 𝒵=𝒵(𝐕,ℎ,𝒪,𝜇,𝑞) 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 𝑝=𝑀_𝜎⁻¹(𝑤) is the unique solution of (10) and (11).

Remark 4. It is important to notice that even though 𝑤  does depend on 𝜎, the effective pressure 𝑝  does not. For the purpose of this remark, we denote 𝐻_𝜎(𝑤)=𝑀_𝜎⁻¹(𝑤)   to stress the dependence on the parameter 𝜎  which is of interest here. We start by 𝑝=𝐻_𝜎(𝑤). (32) It is obvious from the definition of 𝑀_𝜎  that 𝜕𝑀_𝜎𝑝=1𝜕𝑝𝜇𝑝,𝜕𝑀_𝜎𝑝1𝜕𝜎=−𝜇(𝜎). (33) As 𝑀_𝜎(𝐻_𝜎(𝑤))=𝑤, deriving with respect to 𝜎 we arrive at 𝜕𝐻_𝜎(𝑤)𝜕𝜎=−𝜕𝑀_𝜎/𝜕𝜎𝑝𝜕𝑀_𝜎𝑝=𝜇𝑝/𝜕𝑝𝜇(𝜎)=𝜇𝐻_𝜎(𝑤)𝜇(𝜎). (34)
Deriving (13) we get (𝜕𝑤/𝜕𝜎)=−1/𝜇(𝜎). Using the rule for deriving the inverse function, we have 𝜕𝐻_𝜎(𝑤)𝑝𝐻𝜕𝑤=𝜇=𝜇_𝜎(𝑤). (35) Thus 𝜕𝑝=𝜕𝜎𝜕𝐻_𝜎(𝑤)𝜕𝑤𝜕𝑤(𝜎)+𝜕𝜎𝜕𝐻_𝜎(𝑤)𝜇𝐻𝜕𝜎=−^𝜎(𝑤)𝜇(𝜎)+𝜇𝐻_𝜎(𝑤)𝜇(𝜎)=0. (36)

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 ℎ, describing the form of the fluid domain, is periodic with small period 1/𝑚, with 𝑚∈𝐍. To stress that dependence we denote it by ℎ_𝑚. More precisely, we denote by 𝑌=]0,1[^𝑛,𝑛=1,2, the period. We further assume that ℎ∶𝐑^𝑛→[𝑑₀,+∞[, 𝑑₀>0 is periodic with period 𝑌 and smooth. Then we take ℎ_𝑚 of the form ℎ_𝑚(𝑥)=ℎ(𝑚𝑥). (37) Thus, the function ℎ describes the form of periodically distributed rugosities.

To emphasize that the relative velocity of bearing surfaces 𝐕 is large, we assume that it also depends on 𝑚, the same parameter that is taken for description of rugosities. In that case our equation reads ℎdiv^3𝑚𝜇𝑝_𝑚∇𝑝_𝑚=6𝐕_𝑚⋅∇ℎ_𝑚in𝒪. (38)

4.1. One-Dimensional Case

If 𝑛=1, the above problem is posed on an interval 𝒪=]0,1[. With an appropriate boundary condition 𝑝_𝑚(0)=𝑞₀,𝑝_𝑚(1)=𝑞₁. (39) It forms a boundary value problem for nonlinear ODE: ℎ³_𝑚𝜇𝑝_𝑚𝑝^′_𝑚^′=6𝑉_𝑚ℎ^′_𝑚in𝒪. (40) To study the asymptotic behavior of the solution with respect to 𝑚 we linearize the problem using the transformation 𝑤_𝑚=𝑀_𝜎(𝑝_𝑚). To simplify, in this section we choose 𝜎=𝑞₀ and, dropping the index 𝜎 in 𝑀_𝜎 and 𝑀⁺_𝜎, we denote 𝑀𝑝=^𝑝_𝑞0𝑑𝑡𝜇(𝑡),𝑀⁺=_𝑞^+∞₀𝑑𝑡𝜇(𝑡). (41)

Theorem 5. Let𝜒₀𝑦=6¹₀𝑑𝑠ℎ(𝑠)³⁻¹×1ℎ^3𝑦0𝑑𝑣ℎ(𝑣)²−1ℎ^2𝑦0𝑑𝑣ℎ(𝑣)³,⟨⋅⟩=¹₀⋅𝑑𝑦, (42)(43)and let 𝐻=𝑀⁻¹. Suppose that there exists a limit𝑉=lim_𝑚→∞𝑉_𝑚𝑚 (44)and that, for 𝑚  large enough and 𝑀⁺  defined in (27), the following condition holds:𝑏^𝑚₃(𝑥)𝑏^𝑚₃(1)𝑀𝑞₁+6𝑉_𝑚𝑏^𝑚₃(1)𝑏^𝑚₂(𝑥)−𝑏^𝑚₃(𝑥)𝑏^𝑚₂(1)𝑏^𝑚₃(1)≤𝑀⁺, (45)with𝑏^𝑚_𝛼(𝑥)=^𝑥₀𝑑𝑡ℎ_𝑚(𝑡)^𝛼,𝛼=2,3. (46)Then𝑝_𝑚⇀𝑝₀=¹₀𝐻𝑥𝑀𝑞₁+𝑉𝜒₀𝑦𝑑𝑦weak^∗in𝐿^∞(0,1). (47)

Proof. Equation (40), with boundary conditions (39), can be solved by reduction to quadratures, after the substitution 𝑤_𝑚=𝑀𝑝_𝑚, (48) with 𝑀_𝜎 strictly increasing function defined by (12). The problem for 𝑤_𝑚 now reads ℎ³_𝑚𝑤^′_𝑚^′=6𝑉_𝑚ℎ^′_𝑚][,𝑤in0,1_𝑚(0)=𝑀𝑞₀=0,𝑤_𝑚(1)=𝑀𝑞₁. (49) It is easy to see that (49) has a unique solution given by (25). Since 𝑀(𝑞₀)=0, (25) now reduces to 𝑤_𝑚=𝑏^𝑚₃(𝑥)𝑏^𝑚₃(1)𝑀_𝜎𝑞₁+6𝑉_𝑚𝑏^𝑚₃(1)𝑏^𝑚₂(𝑥)−𝑏^𝑚₃(𝑥)𝑏^𝑚₂(1)𝑏^𝑚₃(1), (50) where, for 𝛼=2,3, 𝑏^𝑚_𝛼(𝑥)=^𝑥₀𝑑𝑡ℎ_𝑚(𝑡)^𝛼=1𝑚₀^𝑚𝑥𝑑𝑠ℎ(𝑠)^𝛼⟶𝑥¹₀𝑑𝑠ℎ(𝑠)^𝛼,as𝑚⟶+∞. (51) Now 𝑝_𝑚, the solution to the problem (40), exists if (45) is fulfilled. The second term in (50) thus obviously tends to 𝑥𝑀(𝑞₁), as 𝑚→+∞. The last term is more interesting. The denominator tends to 1ℎ³=¹₀𝑑𝑠ℎ(𝑠)³. (52) As for its numerator, we have 𝑉_𝑚𝑏^𝑚₃(1)𝑏^𝑚₂(𝑥)−𝑏^𝑚₃(𝑥)𝑏^𝑚₂(1)=𝑉_𝑚¹₀^𝑥₀1ℎ_𝑚(𝑡)³ℎ_𝑚(𝑠)²−1ℎ_𝑚(𝑡)²ℎ(𝑠)³_𝑚=𝑉𝑑𝑠𝑑𝑡_𝑚𝑚1ℎ^30𝑚𝑥𝑑𝑣ℎ(𝑣)²−1ℎ^20𝑚𝑥𝑑𝑣ℎ(𝑣)³. (53)
Suppose that 𝑉=lim_𝑚→+∞𝑉_𝑚𝑚 (54) and denote 𝐺𝑦=1ℎ^3𝑦0𝑑𝑣ℎ(𝑣)²−1ℎ^2𝑦0𝑑𝑣ℎ(𝑣)³. (55) Obviously the function 𝐺 is periodic with period 1 so that, due to the standard periodicity lemma (see, e.g., [19]), as 𝑚→+∞, 𝐺(𝑚𝑥)⇀⟨𝐺⟩=¹₀𝐺𝑦𝑑𝑦,weak^∗in𝐿^∞(0,1). (56)
By direct computation ⟨𝐺⟩=𝑦ℎ³1ℎ²−1ℎ³𝑦ℎ²=¹₀𝑠−𝑡ℎ(𝑠)³ℎ(𝑡)²𝑑𝑠𝑑𝑡. (57) Thus 𝑉_𝑚𝑏^𝑚₃(1)𝑏^𝑚₂(𝑥)−𝑏^𝑚₃(𝑥)𝑏^𝑚₂(1)⇀𝑉⟨𝐺⟩. (58)
Now, denoting 𝜒₀𝑦=𝐺𝑦𝑏₃(1) (59) we have 𝑤_𝑚≈𝑥𝑀𝑞₁+𝑉𝜒₀(𝑚𝑥), (60) and thus 𝑤_𝑚⇀𝑤₀=𝑥𝑀𝑞₁+𝑉𝜒₀weak^∗in𝐿^∞(0,1). (61)
However, we are not interested in convergence of the auxiliary function 𝑤_𝑚 but in the convergence of the pressure 𝑝_𝑚. Since (45) is assumed to be true, we can define 𝑝_𝑚=𝐻(𝑤_𝑚), where 𝐻=𝑀⁻¹, and we have 𝑝_𝑚⇀𝑝₀=¹₀𝐻𝑥𝑀𝑞₁+𝑉𝜒₀𝑦𝑑𝑦weak^∗in𝐿^∞(0,1). (62)

Deriving the expression on the right-hand side, we obtain the effective pressure drop in the form 𝑝^′₀(𝑥)=𝑀𝑞₁¹₀𝜇𝑥𝑀𝑞₁+𝑉𝜒₀𝑦=𝑑𝑦_𝑞1𝑞0𝑑𝜏𝜇(𝜏)¹₀𝜇𝑥𝑀𝑞₁+𝑉𝜒₀𝑦=𝑑𝑦_𝑞1𝑞0𝑑𝜏𝜇(𝜏)¹₀𝜇𝑥^𝑞1_𝑞0𝑑𝑠𝜇(𝑠)+𝑉𝜒₀𝑦𝑑𝑦. (63)(64) As we can see, the pressure drop is not constant, as for the Newtonian flow. The interesting effect appears if 𝑉≠0 because, in that case, the expressions for the pressure and for the pressure drop are nonlocal due to the integral with respect to 𝑦. 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 ℎ_𝑚 is constructed from positive, smooth, 𝑌-periodic function ℎ∶𝐑²→[𝑑₀,+∞[, 𝑑₀>0, 𝑌=]0,1[² in the same way as before, that is, by taking ℎ_𝑚(𝑥)=ℎ(𝑚𝑥). (65) We have seen in the previous section that interesting effects happen only if we assume that 𝐕_𝑚=𝑚𝐕. (66) In that case our equation reads ℎdiv^3𝑚𝜇𝑝_𝑚∇𝑝_𝑚=6𝐕_𝑚⋅∇ℎ_𝑚in𝒪. (67) As we did in the existence analysis and in the previous section, we linearize the equation by substitution 𝑤_𝑚=𝑀_𝜎𝑝_𝑚, (68) where the function 𝑀_𝜎 is defined by (12). Now 𝑤_𝑚 satisfies divℎ³𝑀∇𝑤_𝑚=6𝐕_𝑚⋅∇ℎ_𝑚in𝒪. (69)

We postulate the asymptotic expansion in the form 𝑤_𝑚(𝑥)≈𝑤₀+1𝑥,𝑦𝑚𝑤₁+1𝑥,𝑦𝑚²𝑤₂𝑥,𝑦+⋯,𝑦=𝑚𝑥. (70) All functions are assumed to be 𝑌-periodic in 𝑦 variable.

Plugging that in (69) and collecting, formally, terms with equal powers of 𝑚, we get 𝑚²: div_𝑦(ℎ³∇_𝑦𝑤₀)=6𝐕⋅∇_𝑦ℎ 𝑚: div_𝑦(ℎ³∇_𝑦𝑤₁)+div_𝑦(ℎ³∇_𝑥𝑤₀)+ℎ³div_𝑥∇_𝑦𝑤₀=0 1: div_𝑦(ℎ³∇_𝑦𝑤₂)+div_𝑦(ℎ³∇_𝑥𝑤₁)+ℎ³div_𝑥∇_𝑦𝑤₁ +ℎ³Δ_𝑥𝑤₀=0.

Denoting 𝑉=|𝐕|,𝐕₀=𝐕𝑉 (71) we have 𝑤₀𝑥,𝑦=𝑣₀(𝑥)+𝑉𝜒₀𝑦,div_𝑦ℎ³∇_𝑦𝜒₀=6𝐕₀⋅∇_𝑦𝑤ℎin𝑌,₁𝑥,𝑦=²_𝑘=1𝜒_𝑘𝑦𝜕𝑣₀𝜕𝑥_𝑘(𝑥)+𝑣₁(𝑥),div_𝑦ℎ³∇_𝑦𝜒_𝑘+𝑦_𝑘=0,_2𝑘,ℓ=1𝐚_𝑘ℓ𝜕²𝑣₀𝜕𝑥_𝑘𝜕𝑥_ℓ𝐚=0in𝒪,_𝑘ℓ=_𝑌ℎ³𝜕𝜕𝑦_ℓ𝜒_𝑘+𝑦_𝑘𝑑𝑦. (72)

Remark 6. The same computation can be done in one-dimensional case and it gives ℎ³𝜒^′₀^′=6ℎ^′⇒𝜒₀=6^𝑦₀𝑑𝑡ℎ(𝑡)²+𝐶₀^𝑦₀𝑑𝑡ℎ(𝑡)³+𝐶₁,𝑣₀=𝑥𝑀𝑞₁. (73)(74) Constants 𝐶₀,𝐶₁ are chosen in a way that boundary conditions 𝜒₀(0)=𝜒₀(1)=0  are met, and it follows that 𝐶₁=0,𝐶₀=−6¹₀𝑑𝑡ℎ(𝑡)³⁻¹¹⁰𝑑𝑡ℎ(𝑡)². (75) Then 𝑤_𝑚≈𝑤₀(𝑥,𝑚𝑥)=𝑣₀(𝑥)+𝜒₀(𝑚𝑥)=𝑥𝑀_𝜎𝑞₁+6𝑉₀^𝑚𝑥𝑑𝑡ℎ(𝑡)²−¹₀𝑑𝑡ℎ(𝑡)³⁻¹×¹₀𝑑𝑡ℎ(𝑡)^20𝑚𝑥𝑑𝑡ℎ(𝑡)³=𝑥𝑀_𝜎𝑞₁𝑏+6𝑉^𝑚2(𝑥)𝑏₃(1)−𝑏₂(1)𝑏^𝑚₃(𝑥)𝑏₃(1)=𝑥𝑀_𝜎𝑞₁+𝑉𝜒₀(𝑚𝑥). (76) That is a very good approximation of our exact solution (50).
It is important to notice that the choice of constants 𝐶₀,𝐶₁ 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 𝑤_𝑚 in order to get the convergence for 𝑝_𝑚. The form of the approximation 𝑤_𝑚≈𝑣₀(𝑥)+𝑉𝜒₀(𝑚𝑥)+⋯. (77) suggests that the boundary layer phenomenon should appear on the exterior boundary 𝜕𝒪 since 𝜒₀ term cannot satisfy the Dirichlet condition on 𝜕𝒪. 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 ][𝑝𝒪=0,1×𝐑,_𝜀0,𝑥₂=0,𝑝_𝜀1,𝑥₂=𝑞𝑥₂,𝑥₂↦𝑝_𝜀𝑥₁,𝑥₂ℎ𝑦is1-periodic,₁,𝑦₂=ℎ𝑦₁,𝑞is1-periodic. (78)(79)(80)(81)(82) Now 𝑤_𝜀=𝑀_𝜎𝑝_𝜀,with𝑀_𝜎𝑝=^𝑝_𝜎𝑑𝑠𝜇(𝑠)⇒𝑤_𝜀0,𝑥₂𝑤=0,_𝜀1,𝑥₂=𝑀_𝜎𝑞𝑥₂ (83) and 𝑥₂↦𝑤_𝜀(𝑥₁,𝑥₂) is 1-periodic.

In that case we can compute 𝜒₀ and 𝜒_𝑘,𝑘=1,2, explicitly and we can impose exterior condition on 𝜒₀. Indeed 𝜒₀ is exactly the same as in the monodimensional case; that is, it is given by (73) and (75). Obviously 𝜒₂=0 so that 𝐚₂₂=¹₀ℎ(𝑠)³𝑑𝑠=ℎ³,𝐚₁₂=𝐚₂₁=0. (84) As for the last term 𝜒₁=−𝑦₁+¹₀𝑑𝑠ℎ(𝑠)^3−1𝑦10𝑑𝑠ℎ(𝑠)³,𝐚₁₁=¹₀𝑑𝑠ℎ(𝑠)³⁻¹=11/ℎ³. (85) Finally, the function 𝑣₀ satisfies the boundary value problem 11/ℎ³𝜕²𝑣₀𝜕𝑥²₁+ℎ³𝜕²𝑣₀𝜕𝑥²₂𝑣=0in𝒪,₀0,𝑥₂𝑣=0,₀1,𝑥₂=𝑀_𝜎𝑞𝑥₂,𝑣₀is1-periodicin𝑥₂. (86) It can be solved using the Fourier method, and we get 𝑣₀𝑥₁,𝑥₂=^∞_𝑘=1shℎ³ℎ⁻³𝑘𝜋𝑥₁×𝛼_𝑘sin𝑘𝜋𝑥₂+𝛽_𝑘cos𝑘𝜋𝑥₂,𝛼_𝑘=2¹₀𝑀_𝜎𝑞(𝑡𝛽)sin𝑘𝜋𝑡𝑑𝑡,_𝑘=2¹₀𝑀_𝜎𝑞(𝑡)cos𝑘𝜋𝑡𝑑𝑡. (87) Since the approximation w_𝑚≈𝑣₀(𝑥)+𝑉𝜒₀𝑚𝑥₁+1𝑚𝜒₁𝑚𝑥₁𝜕𝑣₀𝜕𝑥₁(𝑥) (88) now satisfies the boundary conditions on 𝜕𝒪, it is easy to see that 𝑤_𝑚−𝑣₀(𝑥)+𝑉𝜒₀(𝑚𝑥)_{𝐿∞(𝒪)}1≤𝐶𝑚 (89) follows from the maximum principle. Assuming that, for 𝑚 large enough, 𝑣₀(𝑥)+𝑉𝜒₀(𝑚𝑥)≤𝑀⁺ (90) we have 𝑝_𝑚−𝐻𝑣₀(𝑥)+𝑉𝜒₀(𝑚𝑥)_{𝐿∞(𝒪)}⟶0. (91) Finally 𝑝_𝑚⇀𝑝₀=¹₀𝐻𝑣₀(𝑥)+𝑉𝜒₀𝑦₁𝑑𝑦₁. (92) We have proved that.

Theorem 7. Let 𝑝_𝑚  be the solution to the problem (67), (79), and (80) and let 𝑣₀, 𝜒₀  be defined by (87) and (42), respectively. If (90) holds, then𝑝_𝑚−𝐻𝑣₀(𝑥)+𝑉𝜒₀(𝑚𝑥)_{𝐿∞(𝒪)}𝑝⟶0,_𝑚⇀𝑝₀=¹₀𝐻𝑣₀(𝑥)+𝑉𝜒₀𝑦₁𝑑𝑦₁. (93)

Remark 8. It is important to notice that 𝜋₀=𝐻(𝑝₀)   satisfies ²_𝑖,𝑘=1𝜕𝜕𝑥_𝑖𝐚_𝑖𝑘𝜇𝜋₀𝜕𝜋₀𝜕𝑥_𝑘=0in𝒪,𝜋₀=𝑞on𝜕𝒪, (94) and thus we would expect it to be the limit of 𝑝_𝑚  in analogy with the linear case. However 𝑝₀≠𝜋₀.
If 𝑉=lim_𝑚→∞𝑚|𝐕_𝑚|  is small, we can expand 𝐻(𝑣₀(𝑥)+𝑉𝜒₀(𝑚𝑥))   in powers of 𝑉 and we get 𝑝_𝑚(𝑥)≈𝐻𝑣₀(𝑥)+𝑉𝜒₀()𝑣𝑚𝑥=𝐻₀(𝑥)+𝑉𝐻^′𝑣₀(𝑥)𝜒₀𝑦+𝑂𝑉²=𝜋₀(𝑥)+𝑉𝜇𝜋₀(𝑥)𝜒₀𝑦+𝑂𝑉². (95) Thus 𝑝₀=𝜋₀+𝜇𝜋₀𝜒₀𝑉+𝑂𝑉². (96) It can be, formally, written as 𝑝_𝑚(𝑥)≈𝜋₀(𝑥)+𝐕_𝑚𝑚𝜇𝜋₀(𝑥)𝜒₀()𝐕𝑚𝑥+𝑂^𝑚2𝑚². (97)

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 𝐕=(1/6)𝑣𝐢. Indeed, we can always choose the coordinate system in a way that the first coordinate axis 𝑥 has a direction of the velocity of relative motion 𝐕.

The lower bound for 𝑤(𝑥,𝑦) is of no interest, just the upper bound. Function 𝑤(𝑥,𝑦) is the solution to the boundary value problem divℎ³∇𝑤=𝑣𝜕ℎ𝜕𝑥in𝒪⊂𝐑²,𝑤=𝑀_𝜎𝑞on𝜕𝒪. (A.1) We assume that if 𝑣(𝜕ℎ/𝜕𝑥)>0, then 𝑤 cannot have a maximum point in the domain 𝒪 and, thus, 𝑤𝑥,𝑦≤max_{(𝑥,𝑦)∈𝜕𝒪}𝑀_𝜎𝑞𝑥,𝑦. (A.2) However it is not realistic to assume that (𝜕ℎ/𝜕𝑥) 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 𝑤 be the solution to the problem (A.1). Then 𝑤𝑥,𝑦≤𝑀_𝜎𝑞_{𝐿∞(𝜕𝒪)}+𝒵𝑣,ℎ,𝒪,𝜇,𝑞,𝒵=3^(8/5)×32^(28/5)(2𝜋)^(1/4)ℎ³₀diam𝒪|𝑣|ℎ₁|𝒪|^1/5+ℎ³₁∇𝑀_𝜎𝑞_𝐿5(𝒪)×|𝒪|^(7/5). (A.3)(A.4)

Proof. The function 𝑧=𝑤−𝐺 satisfies divℎ³∇𝑧=𝑣𝜕ℎℎ𝜕𝑥+div³∇𝑀_𝜎𝑞in𝒪⊂𝐑²,𝑧=0on𝜕𝒪. (A.5)(A.6) Next we introduce the embedding constant for 𝑊₀^1,2(𝒪)⊂𝐿^𝑟(𝒪) denoted 𝑀_𝑟, such that |𝑣|_𝐿^𝑟_(𝒪)≤𝑀_𝑟|∇𝑣|_𝐿²_(𝒪),∀𝑣∈𝐻¹₀(𝒪). (A.7) That constant can be estimated as 𝑀_𝑟≤12(diam𝒪)²|𝒪|^1/𝑟−1/2𝑟+22^{((𝑟+2)/2𝑟)}√2𝜋. (A.8) See, for example, [20,  Lemma  1].  Next we define the sequence 𝜆_𝑘+1𝜆=3^𝑘2+1,𝜆₁=2. (A.9) Easy computation yields 𝜆_𝑘3=82^𝑘−1−6. (A.10) Let 𝑧⁺𝑥,𝑦=max𝑧𝑥,𝑦,0. (A.11) We test (A.5) with (𝑧⁺)^1+𝜆𝑘+1 and get _𝒪ℎ³∇𝑧⁺∇𝑧^{+1+𝜆𝑘+1}=1+𝜆_𝑘+11+𝜆_𝑘+1/2^2𝒪ℎ³∇𝑧^{+1+(𝜆𝑘+1/2)2}=_𝒪𝜕𝑣ℎ𝑧𝜕𝑥^{+1+𝜆𝑘+1}+ℎ³∇𝑀_𝜎𝑞∇𝑧^{+1+𝜆𝑘+1}. (A.12) For the left-hand side, we get the lower bound ℎ³₀1+𝜆_𝑘+11+𝜆_𝑘+1/2²∇𝑧^{+1+(𝜆𝑘+1/2)2}_𝐿2(𝒪). (A.13) We estimate the terms on the right-hand side using the same idea _𝒪ℎ³∇𝑀_𝜎𝑞∇𝑧^{+1+𝜆𝑘+1}≤ℎ³₁∇𝑀_𝜎𝑞_𝐿5(𝒪)∇𝑧^{+1+𝜆𝑘+1}_{𝐿5/4(𝒪)},_𝒪𝜕ℎ𝑣𝑧𝜕𝑥^{+1+𝜆𝑘+1}≤ℎ₁|𝑣||𝒪|^1/5∇𝑧^{+1+𝜆𝑘+1}_{𝐿5/4(𝒪)}. (A.14) Thus, it remains to estimate |∇(𝑧⁺)^1+𝜆𝑘+1|_𝐿^5/4_(𝒪). We have ∇𝑧^{+1+𝜆𝑘+1}_𝐿^5/4_5/4(𝒪)=1+𝜆_𝑘+1^5/4_𝒪𝑧^{+(5𝜆𝑘+1/4)}∇𝑧^+5/4=1+𝜆_𝑘+11+𝜆_𝑘+1/2^5/4_𝒪∇𝑧^{+1+(𝜆𝑘+1/2)5/4}𝑧^{+(5𝜆𝑘+1/8)}≤Ḧ8oldersinequalitywith𝑝=5,𝑝^′=83≤1+𝜆_𝑘+11+𝜆_𝑘+1/2^5/4_𝒪∇𝑧^{+1+(𝜆𝑘+1/2)25/8}×_𝒪𝑧^{+(5/3)𝜆𝑘+13/8}=()5duetoA.93𝜆_𝑘+1𝜆=51+^𝑘2,38=14𝜆_𝑘+12+𝜆_𝑘=1+𝜆_𝑘+11+𝜆_𝑘+1/2^5/4∇𝑧^{+1+(𝜆𝑘+1/2)}_𝐿^5/4_2(𝒪)×𝑧^{+1+(𝜆𝑘/2)(𝜆𝑘+1/(2+𝜆𝑘}_𝐿^))(5/4)_5(𝒪). (A.15)
Combining with (A.14) and (A.13), we get ∇𝑧^{+1+(𝜆𝑘+1/2)}_𝐿2(𝒪)≤ℎ₀⁻³𝜆1+^𝑘+12𝑧^{+1+(𝜆𝑘/2)((𝜆𝑘+1)/(2+𝜆𝑘}_𝐿⁾⁾_5(𝒪)×ℎ³₁∇𝑀_𝜎𝑞_𝐿5(𝒪)+ℎ₁|𝑣||𝒪|^1/5≤𝑀^{((𝜆𝑘+1)/(2+𝜆𝑘}₅⁾⁾ℎ₀⁻³𝜆1+^𝑘+12×∇𝑧^{+1+(𝜆𝑘/2)((𝜆𝑘+1)/(2+𝜆𝑘}_𝐿⁾⁾_2(𝒪)×ℎ³₁∇𝑀_𝜎𝑞_𝐿5(𝒪)+ℎ₁|𝑣||𝒪|^1/5. (A.16) We recall that 𝜆_𝑘+12+𝜆_𝑘=32 (A.17) and define 𝛼=𝑀₅^(3/2)ℎ₀⁻³ℎ³₁∇𝑀_𝜎𝑞_𝐿5(𝒪)+ℎ₁|𝑣||𝒪|^1/5 (A.18) as well as 𝜎_𝑘=1𝛼∇𝑧^{+1+(𝜆𝑘/2)(2/(2+𝜆𝑘}_𝐿⁾⁾_2(𝒪). (A.19) Then (A.16) implies 𝜎_𝑘+1≤𝜆1+^𝑘+12^{(2/(2+𝜆𝑘))}𝜎^{((𝜆𝑘+1)/(2+𝜆𝑘+1}_𝑘⁾⁾. (A.20)
Taking the logarithm, we arrive at log𝜎_𝑘+1≤log1+𝜆_𝑘+1/21+𝜆_𝑘+1+𝜆/22+𝜆^𝑘+1_𝑘+1log𝜎_𝑘. (A.21) We first notice that 𝜆_𝑘+12+𝜆_𝑘+1<1 (A.22) and then 𝜆1+^𝑘23=42^𝑘−13−2>2^𝑘. (A.23) Since the function 𝑥↦(log𝑥/𝑥) is decreasing for 𝑥>𝑒, we have log1+𝜆_𝑘/21+𝜆_𝑘≤)/2log(3/2^𝑘(3/2)^𝑘3=𝑘log223^𝑘,∀𝑘≥3. (A.24) Then, log𝜎_𝑘+1≤()3𝑘+1log223^𝑘+log𝜎_𝑘3≤log2^𝑘+1_𝑗=2𝑗23^𝑗+log𝜎₁3≤8log2+log𝜎₁. (A.25) Finally 𝜎_𝑘+1≤32⁸𝜎₁. (A.26) Now it remains to estimate 𝜎₁. From the definition we see that 𝜎₁=1𝛼∇𝑧⁺²_𝐿^1/2_2(𝒪). (A.27) To estimate 𝜎₁, we proceed as before and test (A.5) with (𝑧⁺)³. We get 34_𝒪ℎ³∇𝑧⁺²²=_𝒪𝜕𝑧𝑣ℎ⁺³𝜕𝑥+ℎ³∇𝑀_𝜎𝑞∇𝑧⁺³≤|𝑣|ℎ₁|𝒪|^1/5+ℎ³₁∇𝑀_𝜎𝑞_𝐿5(𝒪)∇𝑧⁺³_{𝐿5/4(𝒪)}≤|𝑣|ℎ₁|𝒪|^1/5+ℎ³₁∇𝑀_𝜎𝑞_𝐿5(𝒪)×32∇𝑧⁺²_𝐿2(𝒪)𝑧⁺_{𝐿10/3(𝒪)}. (A.28) Thus ∇𝑧⁺²_𝐿2(𝒪)≤2ℎ³₀|𝑣|ℎ₁|𝒪|^1/5+ℎ³₁∇𝑀_𝜎𝑞_𝐿5(𝒪)𝑀_(10/3)∇𝑧⁺_𝐿2(𝒪). (A.29) Finally, testing (A.5) with 𝑧⁺, we get _𝒪ℎ³∇𝑧⁺²=_𝒪𝑣ℎ𝜕𝑧⁺𝜕𝑥+ℎ³∇𝑀_𝜎𝑞∇𝑧⁺ (A.30) so that ∇𝑧⁺_𝐿2(𝒪)≤1ℎ³₀|𝑣|ℎ₁|𝒪|^1/2+ℎ³₁∇𝑀_𝜎𝑞_𝐿2(𝒪)≤|𝒪|^3/10|𝑣|ℎ₁|𝒪|^1/5+ℎ³₁∇𝑀_𝜎𝑞_𝐿5(𝒪). (A.31) Combining (A.29) with (A.31) and (A.26) gives 𝜎_𝑘+1≤32⁸1𝛼√2ℎ³₀×|𝑣|ℎ₁|𝒪|^1/5+ℎ³₁∇𝑀_𝜎𝑞_𝐿5(𝒪)𝑀₍₎^1/2_10/3|𝒪|^3/2. (A.32) Since 𝜎_𝑘+1≥1𝛼𝑀^(2/(2+𝜆k₂⁾⁾𝑧⁺_{𝐿𝑘2+𝜆(𝒪)}, (A.33) we have arrived to 𝑧⁺_{𝐿𝑘2+𝜆(𝒪)}≤32⁸𝑀^{(2/(2+𝜆𝑘}₂⁾⁾√2ℎ³₀×|𝑣|ℎ₁|𝒪|^1/5+ℎ³₁∇𝑀_𝜎𝑞_𝐿5(𝒪)𝑀₍₎^1/2_10/3|𝒪|^3/2. (A.34) Since lim_𝑘→∞𝜆_𝑘=∞, we get 𝑧⁺_{𝐿∞(𝒪)}≤32⁸√2ℎ³₀×|𝑣|ℎ₁|𝒪|^1/5+ℎ³₁∇𝑀_𝜎𝑞_𝐿5(𝒪)𝑀₍₎^1/2_10/3|𝒪|^3/2. (A.35)
Finally (A.8) implies (A.3).

Acknowledgment

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