{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,7]],"date-time":"2026-03-07T10:24:55Z","timestamp":1772879095394,"version":"3.50.1"},"reference-count":29,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2021,7,1]],"date-time":"2021-07-01T00:00:00Z","timestamp":1625097600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["PTDC\/EMS-ENE\/3362\/2014"],"award-info":[{"award-number":["PTDC\/EMS-ENE\/3362\/2014"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["SFRH\/BPD\/100353\/2014"],"award-info":[{"award-number":["SFRH\/BPD\/100353\/2014"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["UIDB\/00013\/2020 and UIDP\/00013\/2020"],"award-info":[{"award-number":["UIDB\/00013\/2020 and UIDP\/00013\/2020"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["UIDB\/00532\/2020 and UIDP\/00532\/2020"],"award-info":[{"award-number":["UIDB\/00532\/2020 and UIDP\/00532\/2020"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Fluids"],"abstract":"<jats:p>This work presents a detailed numerical investigation on the required development length (L=L\/B) in laminar Newtonian fluid flow in microchannels with rectangular cross section and different aspect ratios (AR). The advent of new microfluidic technologies shifted the practical Reynolds numbers (Re) to the range of unitary (and even lower) orders of magnitude, i.e., creeping flow conditions. Therefore, accurate estimations of L at Re\u2264O(1) are important for microsystem design. At such low Reynolds numbers, in which inertial forces are less dominant than viscous forces, flow characteristics become necessarily different from those at the macroscale where Re is typically much larger. A judicious choice of mesh refinement and adequate numerical methods allowed obtaining accurate results and a general correlation for estimating L, valid in the ranges 0\u2264Re\u22642000 and 0.1\u2264AR\u22641, thus covering applications in both macro and microfluidics.<\/jats:p>","DOI":"10.3390\/fluids6070240","type":"journal-article","created":{"date-parts":[[2021,7,1]],"date-time":"2021-07-01T12:03:27Z","timestamp":1625141007000},"page":"240","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":27,"title":["Hydrodynamic Entrance Length for Laminar Flow in Microchannels with Rectangular Cross Section"],"prefix":"10.3390","volume":"6","author":[{"given":"Germ\u00e1n","family":"Ferreira","sequence":"first","affiliation":[{"name":"AIMEN\u2014Northwest Association of Metallurgical Research, C\/. Pol\u00edgono Industrial de Cataboi SUR-PPI-2 (Sector 2) Parcela 3, O Porri\u00f1o, 36418 Pontevedra, Spain"}]},{"given":"Artur","family":"Sucena","sequence":"additional","affiliation":[{"name":"Centro de Estudos de Fen\u00f3menos de Transporte, Departamento de Engenharia Mec\u00e2nica, Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias, s\/n, 4200-465 Porto, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5477-3226","authenticated-orcid":false,"given":"Lu\u00eds L.","family":"Ferr\u00e1s","sequence":"additional","affiliation":[{"name":"Centro de Matem\u00e1tica, Departamento de Matem\u00e1tica da Universidade do Minho, Campus de Azur\u00e9m, 4800-058 Guimar\u00e3es, Portugal"}]},{"given":"Fernando T.","family":"Pinho","sequence":"additional","affiliation":[{"name":"Centro de Estudos de Fen\u00f3menos de Transporte, Departamento de Engenharia Mec\u00e2nica, Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias, s\/n, 4200-465 Porto, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2825-0709","authenticated-orcid":false,"given":"Alexandre M.","family":"Afonso","sequence":"additional","affiliation":[{"name":"Centro de Estudos de Fen\u00f3menos de Transporte, Departamento de Engenharia Mec\u00e2nica, Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias, s\/n, 4200-465 Porto, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2021,7,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"815","DOI":"10.1016\/j.ijheatmasstransfer.2005.11.007","article-title":"Characterization of liquid flows in microfluidic systems","volume":"49","author":"Bayraktar","year":"2006","journal-title":"Int. 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