{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,28]],"date-time":"2026-03-28T06:44:10Z","timestamp":1774680250350,"version":"3.50.1"},"reference-count":25,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2019,5,28]],"date-time":"2019-05-28T00:00:00Z","timestamp":1559001600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia - Portugal","award":["UTAP-EXPL\/MAT\/0017\/2017"],"award-info":[{"award-number":["UTAP-EXPL\/MAT\/0017\/2017"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Fluids"],"abstract":"<jats:p>We present dimensionally reduced Reynolds type equations for steady lubricating flows of incompressible non-Newtonian fluids with shear-dependent viscosity by employing a rigorous perturbation analysis on the governing equations of motion. Our analysis shows that, depending on the strength of the power-law character of the fluid, the novel equation can either present itself as a higher-order correction to the classical Reynolds equation or as a completely new nonlinear Reynolds type equation. Both equations are applied to two classic problems: the flow between a rolling rigid cylinder and a rigid plane and the flow in an eccentric journal bearing.<\/jats:p>","DOI":"10.3390\/fluids4020098","type":"journal-article","created":{"date-parts":[[2019,5,28]],"date-time":"2019-05-28T11:18:09Z","timestamp":1559042289000},"page":"98","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Lubrication Approximation for Fluids with Shear-Dependent Viscosity"],"prefix":"10.3390","volume":"4","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9269-8335","authenticated-orcid":false,"given":"Bruno M.M.","family":"Pereira","sequence":"first","affiliation":[{"name":"Instituto Superior de Engenharia de Lisboa, Instituto Polit\u00e9cnico de Lisboa, Rua Conselheiro Em\u00eddio Navarro, 1959-007 Lisboa, Portugal"}]},{"given":"Gon\u00e7alo A.S.","family":"Dias","sequence":"additional","affiliation":[{"name":"Center for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior T\u00e9cnico, Universidade de Lisboa, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal"}]},{"given":"Filipe S.","family":"Cal","sequence":"additional","affiliation":[{"name":"Instituto Superior de Engenharia de Lisboa, Instituto Polit\u00e9cnico de Lisboa, Rua Conselheiro Em\u00eddio Navarro, 1959-007 Lisboa, Portugal"}]},{"given":"Kumbakonam R.","family":"Rajagopal","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, Texas A&amp;M University, College Station, TX 77843, USA"}]},{"given":"Juha H.","family":"Videman","sequence":"additional","affiliation":[{"name":"Center for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior T\u00e9cnico, Universidade de Lisboa, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2019,5,28]]},"reference":[{"key":"ref_1","first-page":"159","article-title":"On the theory of lubrication and its application to Mr Tower\u2019s experiments","volume":"177","author":"Reynolds","year":"1886","journal-title":"Philos. Trans. R. Soc. Lond."},{"key":"ref_2","unstructured":"Szeri, A.Z. (2010). Fluid Film Lubrication, Cambridge University Press. [2nd ed.]."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1243\/13506501JET79","article-title":"Reynolds equations for common generalized Newtonian models and an approximate Reynolds\u2013Carreau equation","volume":"220","author":"Bair","year":"2006","journal-title":"Proc. Inst. Mech. Eng. Part J J. 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Sci."}],"container-title":["Fluids"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2311-5521\/4\/2\/98\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:54:02Z","timestamp":1760187242000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2311-5521\/4\/2\/98"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,5,28]]},"references-count":25,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2019,6]]}},"alternative-id":["fluids4020098"],"URL":"https:\/\/doi.org\/10.3390\/fluids4020098","relation":{},"ISSN":["2311-5521"],"issn-type":[{"value":"2311-5521","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,5,28]]}}}