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New insights on lubrication theory for compressible fluids
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.ORCID iD: 0000-0001-7029-1112
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.ORCID iD: 0000-0003-1963-6829
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.ORCID iD: 0000-0002-3556-328x
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.ORCID iD: 0000-0001-8211-3671
2019 (English)In: International Journal of Engineering Science, ISSN 0020-7225, E-ISSN 1879-2197, Vol. 145, article id 103170Article in journal (Refereed) Published
Abstract [en]

The fact that the film is thin is in lubrication theory utilised to simplify the full Navier–Stokes system of equations. For incompressible and iso-viscous fluids, it turns out that the inertial terms are small enough to be neglected. However, for a compressible fluid, we show that the influence of inertia depends on the (constitutive) density-pressure relationship and may not always be neglected. We consider a class of iso-viscous fluids obeying a power-law type of compressibility, which in particular includes both incompressible fluids and ideal gases. We show by scaling and asymptotic analysis, that the degree of compressibility determines whether the terms governing inertia may or may not be neglected. For instance, for an ideal gas, the inertial terms remain regardless of the film height-to-length ratio. However, by means of a specific modified Reynolds number that we define we show that the magnitudes of the inertial terms rarely are large enough to be influential. In addition, we consider fluids obeying the well-known Dowson and Higginson density-pressure relationship and show that the inertial terms can be neglected, which allows for obtaining a Reynolds type of equation. Finally, some numerical examples are presented in order to illustrate our theoretical results.

Place, publisher, year, edition, pages
Elsevier, 2019. Vol. 145, article id 103170
Keywords [en]
Thin film approximation, Reynold’s equation, Compressible flow, Navier–Stokes equations, Dimension reduction, Asymptotic analysis
National Category
Mathematical Analysis Tribology (Interacting Surfaces including Friction, Lubrication and Wear)
Research subject
Machine Elements; Mathematics
Identifiers
URN: urn:nbn:se:ltu:diva-76138DOI: 10.1016/j.ijengsci.2019.103170ISI: 000496842000009Scopus ID: 2-s2.0-85072601607OAI: oai:DiVA.org:ltu-76138DiVA, id: diva2:1355166
Note

Validerad;2019;Nivå 2;2019-09-27 (johcin)

Available from: 2019-09-27 Created: 2019-09-27 Last updated: 2019-12-09Bibliographically approved

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Almqvist, AndreasBurtseva, EvgeniyaRàfols, Francesc PérezWall, Peter

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Mathematical AnalysisTribology (Interacting Surfaces including Friction, Lubrication and Wear)

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