{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T19:40:10Z","timestamp":1776800410619,"version":"3.51.2"},"reference-count":8,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2019,7,12]],"date-time":"2019-07-12T00:00:00Z","timestamp":1562889600000},"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":["POCI-01-0145-FEDER-016665,"],"award-info":[{"award-number":["POCI-01-0145-FEDER-016665,"]}],"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":["UID-MAT-00013\/2013"],"award-info":[{"award-number":["UID-MAT-00013\/2013"]}],"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":["UID\/MAT\/00297\/2013"],"award-info":[{"award-number":["UID\/MAT\/00297\/2013"]}],"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":["UID\/Multi\/04621\/2019"],"award-info":[{"award-number":["UID\/Multi\/04621\/2019"]}],"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"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Fluids"],"abstract":"<jats:p>This work presents new analytical and semi-analytical solutions for the pure Couette and Poiseuille\u2013Couette flows, described by the recently proposed (Ferr\u00e1s et al., A Generalised Phan-Thien\u2013Tanner Model, JNNFM 2019) viscoelastic model, known as the generalised Phan-Thien\u2013Tanner constitutive equation. This generalised version considers the Mittag\u2013Leffler function instead of the classical linear or exponential functions of the trace of the stress tensor, and provides one or two new fitting constants in order to achieve additional fitting flexibility. The analytical solutions derived in this work allow a better understanding of the model, and therefore contribute to improve the modelling of complex materials, and will provide an interesting challenge to computational rheologists, to benchmarking and to code verification.<\/jats:p>","DOI":"10.3390\/fluids4030129","type":"journal-article","created":{"date-parts":[[2019,7,12]],"date-time":"2019-07-12T11:49:38Z","timestamp":1562932178000},"page":"129","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["Semi-Analytical Solutions for the Poiseuille\u2013Couette Flow of a Generalised Phan-Thien\u2013Tanner Fluid"],"prefix":"10.3390","volume":"4","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4863-2616","authenticated-orcid":false,"given":"\u00c2ngela M.","family":"Ribau","sequence":"first","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 &amp; Departamento de matem\u00e1tica da Universidade do Minho, Campus de Azur\u00e9m, 4800-058 Guimar\u00e3es, Portugal"}]},{"given":"Maria L.","family":"Morgado","sequence":"additional","affiliation":[{"name":"Center for Computational and Stochastic Mathematics, Instituto Superior T\u00e9cnico, Universidade de Lisboa &amp; Department of Mathematics, University of Tr\u00e1s-os-Montes e Alto Douro, UTAD, 5001-801 Vila Real, Portugal"}]},{"given":"Magda","family":"Rebelo","sequence":"additional","affiliation":[{"name":"Centro de Matem\u00e1tica e Aplica\u00e7\u00f5es (CMA) and Departamento de Matem\u00e1tica, Faculdade de Ci\u00eancias e Tecnologia, Universidade NOVA de Lisboa, Quinta da Torre, 2829-516 Caparica, Portugal"}]},{"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":[[2019,7,12]]},"reference":[{"key":"ref_1","unstructured":"Bird, R.B., Armstrong, R.C., and Hassager, O. (1987). Dynamics of Polymeric Liquids. Volume 1: Fluid Mechanics, John Wiley & Sons."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"88","DOI":"10.1016\/j.jnnfm.2019.06.001","article-title":"A Generalised Phan-Thien-Tanner Model","volume":"269","author":"Morgado","year":"2019","journal-title":"J. Non-Newton. Fluid Mech."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"353","DOI":"10.1016\/0377-0257(77)80021-9","article-title":"A new constitutive equation derived from network theory","volume":"2","author":"Tanner","year":"1977","journal-title":"J. Non-Newton. Fluid Mech."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1122\/1.549481","article-title":"A Nonlinear Network Viscoelastic Model","volume":"22","year":"1978","journal-title":"J. Rheol."},{"key":"ref_5","unstructured":"Podlubny, I. (1998). Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Elsevier."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1016\/j.jnnfm.2014.07.004","article-title":"Annular flow of viscoelastic fluids: Analytical and numerical solutions","volume":"212","author":"Afonso","year":"2014","journal-title":"J. Non-Newton. Fluid Mech."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF01567859","article-title":"Description of the non-linear shear behaviour of a low density polyethylene melt","volume":"17","author":"Laun","year":"1978","journal-title":"Rheol. Acta"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1017\/S002211209900453X","article-title":"Analytical solution for fully developed channel and pipe flow of Phan-Thien-Tanner fluids","volume":"387","author":"Oliveira","year":"1999","journal-title":"J. Fluid Mech."}],"container-title":["Fluids"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2311-5521\/4\/3\/129\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:05:03Z","timestamp":1760187903000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2311-5521\/4\/3\/129"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,7,12]]},"references-count":8,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2019,9]]}},"alternative-id":["fluids4030129"],"URL":"https:\/\/doi.org\/10.3390\/fluids4030129","relation":{},"ISSN":["2311-5521"],"issn-type":[{"value":"2311-5521","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,7,12]]}}}