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The new proposed methodology employs a second-order finite difference approach along with the deformation fields method to solve the integral constitutive equation and the marker particle method (known as marker-and-cell) to accurately capture the evolution of the fluid\u2019s free surface. The newly developed numerical method has proven its effectiveness in handling complex fluid flow scenarios, including confined flows and extrudate swell simulations of Boger fluids. Furthermore, a new semi-analytical solution for velocity and stress fields is derived, considering fully developed flows of a K\u2013BKZ\u2013PSM fluid in a pipe.<\/jats:p>","DOI":"10.3390\/polym15183705","type":"journal-article","created":{"date-parts":[[2023,9,8]],"date-time":"2023-09-08T07:57:17Z","timestamp":1694159837000},"page":"3705","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Numerical Simulation of Three-Dimensional Free Surface Flows Using the K\u2013BKZ\u2013PSM Integral Constitutive Equation"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3242-6340","authenticated-orcid":false,"given":"Juliana","family":"Bertoco","sequence":"first","affiliation":[{"name":"Center for Mathematics, Computing and Cognition \u2014 CMCC, Federal University of ABC \u2014 UFABC, Santo Andr\u00e9 09210-580, Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8009-4577","authenticated-orcid":false,"given":"Antonio","family":"Castelo","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics and Statistics, University of S\u00e3o Paulo \u2014 USP, S\u00e3o Carlos 13566-590, Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5477-3226","authenticated-orcid":false,"given":"Lu\u00eds L.","family":"Ferr\u00e1s","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering (Section of Mathematics), Faculty of Engineering of the University of Porto \u2014 FEUP, 4200-465 Porto, Portugal"},{"name":"Center of Mathematics \u2014 CMAT, University of Minho, 4800-058 Guimar\u00e3es, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0340-3556","authenticated-orcid":false,"given":"C\u00e9lio","family":"Fernandes","sequence":"additional","affiliation":[{"name":"Center of Mathematics \u2014 CMAT, University of Minho, 4800-058 Guimar\u00e3es, Portugal"},{"name":"Transport Phenomena Research Centre \u2014 CEFT, Faculty of Engineering at University of Porto \u2014 FEUP, 4200-465 Porto, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2023,9,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1016\/0377-0257(93)80031-6","article-title":"Numerical simulation of extrudate swell for Oldroyd-B fluids using the stream-tube analysis and a streamline approximation","volume":"50","author":"Clermont","year":"1993","journal-title":"J. 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