Change search
ReferencesLink to record
Permanent link

Direct link
Homogenization of some new mathematical models in lubrication theory
Luleå University of Technology, Department of Engineering Sciences and Mathematics.ORCID iD: 0000-0003-0799-5285
2016 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

We consider mathematical modeling of thin film flow between two rough surfaces which are in relative motion. For example such flows take place in different kinds of bearings and gears when a lubricant is used to reduce friction and wear between the surfaces. The mathematical foundations of lubrication theory is given by the Navier--Stokes equation, which describes the motion of viscous fluids. In thin domains several approximations are possible which lead to the so called Reynolds equation. This equation is crucial to describe the pressure in the lubricant film. When the pressure is found it is possible to predict vorous important physical quantities such as friction (stresses on the bounding surfaces), load carrying capacity and velocity field.

In hydrodynamic lubrication the effect of surface roughness is not negligible, because in practical situations the amplitude of the surface roughness are of the same order as the film thickness. Moreover, a perfectly smooth surface does not exist in reality due to imperfections in the manufacturing process. Therefore, any realistic lubrication model should account for the effects of surface roughness. This implies that the mathematical modeling leads to partial differential equations with coefficients that will oscillate rapidly in space and time. A direct numerical computation is therefore very difficult, since an extremely dense mesh is needed to resolve the oscillations due to the surface roughness. A natural approach is to do some type of averaging.

In this PhD thesis we use and develop modern homogenization theory to be able to handle the questions above. Especially, we use, develop and apply the method based on the multiple scale expansions and two-scale convergence. The thesis is based on five papers (A-E), with an appendix to paper A, and an extensive introduction, which puts these publications in a larger context.

In Paper A the connection between the Stokes equation and the Reynolds equation is investigated. More precisely, the asymptotic behavior as both the film thickness  and wavelength  of the roughness tend to zero is analyzed and described. Three different limit equations are derived. Time-dependent equations of Reynolds type are obtained in all three cases (Stokes roughness, Reynolds roughness and high frequency roughness regime). In paper C we extend the work done in Paper A where we compare the roughness regimes by numeric computations for the stationary case.

In paper B we present a mathematical model that takes into account cavitation, surfaces roughness and compressibility of the fluid. We compute the homogenized coefficients in the case of unidirectional roughness.In the paper D we derive a mathematical model of thin film flow between two close rough surfaces, which takes into account cavitation, surface roughness and pressure dependent density. Moreover, we use two-scale convergence to homogenize the model. Finally, in paper E we prove the existence of solutions to a frequently used mathematical model of thin film flow, which takes cavitation into account.

Place, publisher, year, edition, pages
Luleå: Luleå tekniska universitet, 2016.
Series
Doctoral thesis / Luleå University of Technology 1 jan 1997 → …, ISSN 1402-1544
Keyword [en]
lubrication theory, homogenization theory, Reynolds equation, cavitation, surfaces roughness, Stokes equation
National Category
Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:ltu:diva-59629ISBN: 978-91-7583-715-4ISBN: 978-91-7583-716-1 (pdf)OAI: oai:DiVA.org:ltu-59629DiVA: diva2:1034201
Public defence
2016-12-15, E246, LTU, Luleå, 10:00 (English)
Opponent
Supervisors
Available from: 2016-10-12 Created: 2016-10-10 Last updated: 2016-11-16Bibliographically approved

Open Access in DiVA

fulltext(5689 kB)14 downloads
File information
File name FULLTEXT01.pdfFile size 5689 kBChecksum SHA-512
1106ef73e09091b525f56bb968a95ac3e75a70f7f8a98af6cfc4da5f4113626a7fe085082c84dfc589ae76bc6f224cccdd7ca48d5427d933352e60068c8f7b97
Type fulltextMimetype application/pdf

Search in DiVA

By author/editor
Tsandzana, Afonso Fernando
By organisation
Department of Engineering Sciences and Mathematics
Mathematics

Search outside of DiVA

GoogleGoogle Scholar
Total: 14 downloads
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

Total: 43 hits
ReferencesLink to record
Permanent link

Direct link