{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,13]],"date-time":"2026-01-13T02:11:19Z","timestamp":1768270279820,"version":"3.49.0"},"reference-count":43,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2020,10,5]],"date-time":"2020-10-05T00:00:00Z","timestamp":1601856000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This article aims to study numerically the rotating, steady, and three-dimensional (3D) flow of a hybrid nanofluid over an exponentially shrinking sheet with the suction effect. We considered water as base fluid and alumina (Al2O3), and copper (Cu) as solid nanoparticles. The system of governing partial differential equations (PDEs) was transformed by an exponential similarity variable into the equivalent system of ordinary differential equations (ODEs). By applying a three-stage Labatto III-A method that is available in bvp4c solver in the Matlab software, the resultant system of ODEs was solved numerically. In the case of the hybrid nanofluid, the heat transfer rate improves relative to the viscous fluid and regular nanofluid. Two branches were obtained in certain ranges of the involved parameters. The results of the stability analysis revealed that the upper branch is stable. Moreover, the results also indicated that the equations of the hybrid nanofluid have a symmetrical solution for different values of the rotation parameter (\u03a9).<\/jats:p>","DOI":"10.3390\/sym12101637","type":"journal-article","created":{"date-parts":[[2020,10,5]],"date-time":"2020-10-05T08:35:57Z","timestamp":1601886957000},"page":"1637","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":18,"title":["Rotating 3D Flow of Hybrid Nanofluid on Exponentially Shrinking Sheet: Symmetrical Solution and Duality"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5203-5588","authenticated-orcid":false,"given":"Liaquat Ali","family":"Lund","sequence":"first","affiliation":[{"name":"School of Quantitative Sciences, University Utara Malaysia, Sintok, Kedah 06010, Malaysia"},{"name":"KCAET Khairpur Mir\u2019s, Sindh Agriculture University, Tandojam Sindh 70060, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zurni","family":"Omar","sequence":"additional","affiliation":[{"name":"School of Quantitative Sciences, University Utara Malaysia, Sintok, Kedah 06010, Malaysia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2776-9376","authenticated-orcid":false,"given":"Sumera","family":"Dero","sequence":"additional","affiliation":[{"name":"Faculty of Engineering and Technology, University of Sindh, Jamshoro 76080, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0286-7244","authenticated-orcid":false,"given":"Dumitru","family":"Baleanu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Cankaya University, 06790 Ankara, Turkey"},{"name":"Institute of Space Sciences, 077125 Magurele, Romania"},{"name":"Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40447, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2056-9371","authenticated-orcid":false,"given":"Ilyas","family":"Khan","sequence":"additional","affiliation":[{"name":"Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City 72915, Vietnam"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,10,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"26","DOI":"10.1002\/aic.690070108","article-title":"Boundary-layer behavior on continuous solid surfaces: I. 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