We study the lubrication process with incompressible fluid taking into account the dependence of the viscosity on the pressure. Assuming that the viscosity-pressure relation is given by the well-known Barus law, we derive an effective model using asymptotic analysis with respect to the film thickness. The key idea is to conveniently transform the governing system and then apply two-scale expansion technique.
Fluid film bearings are machine elements usually studied in the broader context of tribology. Simply speaking, they consist of two surfaces in relative motion, separated by a thin fluid film, that lubricates the device and minimizes the friction and, consequently, the wear of the device. In our case, the fluid is an incompressible liquid and the two surfaces are rigid. Such elements are very important in mechanical engineering since they provide the reliability of the system and are crucial factor in limiting the dissipation of energy, that is, increasing the efficiency. If a fluid film bearing is well designed, the wear is not an issue, since two surfaces are completely separated by lubricant. It is therefore important to understand the behavior of the fluid film in the bearing. For a systematic treatment of the fundamentals of fluid film lubrication and fluid film bearings we refer the reader to [
The foundations of the theoretical treatment of lubrication have been laid already by Rayleigh and Stokes and in particular by the famous work of Reynolds [
Several models have been used to describe the viscosity-pressure relation. Barus [
The exponential law (
The result from [
We consider an incompressible fluid flow in a three-dimensional thin domain
We denote by
The domain.
As mentioned in the Introduction section, the well-posedness of the above problem has been recently established by the first author. For reader’s convenience and in order to understand the concept of the solution and its properties, we provide the essential parts of the existence and uniqueness proof in the appendix.
Our main goal is to find an effective law describing the asymptotic behavior of the flow governed by (
It is important to emphasize that, by choosing
There are several methods that enable us to study the processes in thin domains and to find the asymptotic behavior of the fluid flow. The most sophisticated and precise approach is based on the fine asymptotic analysis with respect to the small parameter The problem originally posed in thin domain is rewritten on domain with unit thickness by introducing new, dilated (fast) variable. To accomplish that, the differential operator has to be replaced with the new one, containing the derivatives with respect to the fast variable. Consequently, the negative powers of the domain thickness appear, singularly perturbing the operator. On such, rescaled domain independent of the perturbation parameter Substituting the expansions in the governing equations and collecting equal powers of
We apply the above method to construct the asymptotic approximation for our transformed problem. We introduce the fast variable
We seek for an asymptotic expansion of the unknowns
So far, we have just constructed an asymptotic approximation
From the mathematical point of view, we have to ensure that the effective pressure
From (
In [
In this section we give additional important remarks regarding the considered problem and its possible generalizations.
Let
Let
Before proving the above assertion let us conveniently transform the governing system. For some
Let
We now define the new unknown
The idea is to construct the sequence
Assume that the viscosity function satisfies the Barus law (
Suppose that the problem has two solutions
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research was supported by the Ministry of Science, Education and Sports, Croatia, Grant 037-0372787-2797. The authors would like to thank the referees for valuable comments and suggestions that helped to correct and improve their paper.