{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,8]],"date-time":"2026-04-08T02:51:24Z","timestamp":1775616684083,"version":"3.50.1"},"reference-count":173,"publisher":"Annual Reviews","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2021,1,5]]},"abstract":"<jats:p>Complex fluids exist in nature and are continually engineered for specific applications involving the addition of macromolecules to a solvent, among other means. This imparts viscoelasticity to the fluid, a property responsible for various flow instabilities and major modifications to the fluid dynamics. Recent developments in the numerical methods for the simulation of viscoelastic fluid flows, described by continuum-level differential constitutive equations, are surveyed, with a particular emphasis on the finite-volume method. This method is briefly described, and the main benchmark flows currently used in computational rheology to assess the performance of numerical methods are presented. Outstanding issues in numerical methods and novel and challenging applications of viscoelastic fluids, some of which require further developments in numerical methods, are discussed.<\/jats:p>","DOI":"10.1146\/annurev-fluid-010719-060107","type":"journal-article","created":{"date-parts":[[2020,11,3]],"date-time":"2020-11-03T14:07:20Z","timestamp":1604412440000},"page":"509-541","source":"Crossref","is-referenced-by-count":257,"title":["Numerical Methods for Viscoelastic Fluid Flows"],"prefix":"10.1146","volume":"53","author":[{"given":"M.A.","family":"Alves","sequence":"first","affiliation":[{"name":"1CEFT (Centro de Estudos de Fen\u00f3menos de Transporte), Departamento de Engenharia Qu\u00edmica, Faculdade de Engenharia, Universidade do Porto, 4200-465 Porto, Portugal; email: mmalves@fe.up.pt"}]},{"given":"P.J.","family":"Oliveira","sequence":"additional","affiliation":[{"name":"2C-MAST (Center for Mechanical and Aerospace Science and Technologies), Departamento de Engenharia Electromec\u00e2nica, Faculdade de Engenharia, Universidade da Beira Interior, 6201-001 Covilh\u00e3, Portugal; email: pjpo@ubi.pt"}]},{"given":"F.T.","family":"Pinho","sequence":"additional","affiliation":[{"name":"3CEFT, Departamento de Engenharia Mec\u00e2nica, Faculdade de Engenharia, Universidade do Porto, 4200-465 Porto, Portugal; email: fpinho@fe.up.pt"}]}],"member":"22","reference":[{"key":"B1","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1016\/S0377-0257(00)00196-8","article-title":"A cell-vertex finite volume\/element method on triangles for abrupt contraction viscoelastic flows","volume":"98","year":"2001","journal-title":"J. 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