{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,25]],"date-time":"2026-04-25T19:02:57Z","timestamp":1777143777200,"version":"3.51.4"},"reference-count":50,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2022,11,3]],"date-time":"2022-11-03T00:00:00Z","timestamp":1667433600000},"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>Heat transfer in a symmetrical cavity with two semi-cylinders was explored in this study. Several parameters, such as (103\u2264Ra\u2264106), (10\u22125\u2264Da\u226410\u22122), (0.02\u2264\u03d5\u22640.08), (0.2\u2264\u03b5\u22640.8), and (0\u2264Ha\u2264100) were selected and evaluated in this research. The outcome of the magnetic field and the temperature gradient on the nanofluid flow is considered. The geometric model is therefore described using a symmetry technique. The flow issue for the governing equations has been solved using the Galerkin finite element method (G-FEM), and these solutions are presented in dimensionless form. The equations for energy, motion, and continuity were solved using the application of the COMSOL Multiphysics\u00ae software computer package. According to the results, there is a difference in the occurrence of the magnetic parameter and an increase in heat transmission when the right wall is recessed inward. The heat transmission is also significantly reduced when the right wall is exposed to the outside. The number of Nusselt grows directly proportional to the number of nanofluids in the environment. In contrast, all porous media with low Darcy and Hartmann numbers, high porosity, and low volume fraction have high Nusselt numbers. It is found that double streamlines for the hot side and single cooling for Darcy, Rayleigh, and Hartmann numbers. A cold isotherm at various physical parameters is needed in the top cavity. Rayleigh\u2019s number and a solid volume fraction raise Darcy\u2019s number, increasing heat transmission inside the cavity and thermal entropy determines entropy components.<\/jats:p>","DOI":"10.3390\/sym14112312","type":"journal-article","created":{"date-parts":[[2022,11,4]],"date-time":"2022-11-04T04:00:51Z","timestamp":1667534451000},"page":"2312","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":23,"title":["Mathematical Entropy Analysis of Natural Convection of MWCNT\u2014Fe3O4\/Water Hybrid Nanofluid with Parallel Magnetic Field via Galerkin Finite Element Process"],"prefix":"10.3390","volume":"14","author":[{"given":"Djellouli","family":"Ghali","sequence":"first","affiliation":[{"name":"Laboratoire de Physique Plasmas des Mat\u00e9riaux Conducteur et Leur Applications\u2014LPPMCA, University of Sciences and Technology of Oran (USTO-MB), Bir El Djir 31000, Algeria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2702-374X","authenticated-orcid":false,"given":"Fares","family":"Redouane","sequence":"additional","affiliation":[{"name":"Laboratory of Industrial Engineering and Sustainable Development\u2014LGIDD, Ahmed Zabana University, Relizane 48000, Algeria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Roubi","family":"Abdelhak","sequence":"additional","affiliation":[{"name":"Laboratory of Industrial Technology Study and Research, University Saad Dahlab, Blida 09000, Algeria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Amine","family":"Belhadj Mahammed","sequence":"additional","affiliation":[{"name":"Laboratory of Industrial Engineering and Sustainable Development\u2014LGIDD, Ahmed Zabana University, Relizane 48000, Algeria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Chikr Djaoutsi","family":"Zineb","sequence":"additional","affiliation":[{"name":"Laboratory of Industrial Engineering and Sustainable Development\u2014LGIDD, Ahmed Zabana University, Relizane 48000, Algeria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9438-6132","authenticated-orcid":false,"given":"Wasim","family":"Jamshed","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Capital University of Science and Technology (CUST), Islamabad 44000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mohamed R.","family":"Eid","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, New Valley University, Al-Kharga 72511, Al-Wadi Al-Gadid, Egypt"},{"name":"Department of Mathematics, Faculty of Science, Northern Border University, Arar 1321, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0229-0705","authenticated-orcid":false,"given":"Sayed M.","family":"Eldin","sequence":"additional","affiliation":[{"name":"Center of Research, Faculty of Engineering, Future University in Egypt, New Cairo 11835, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0978-2187","authenticated-orcid":false,"given":"Awad","family":"Musa","sequence":"additional","affiliation":[{"name":"Department of Physics, College of Science and Humanities in Al-Aflaj, Prince Sattam bin Abdulaziz University, Al-Aflaj 11912, Saudi Arabia"},{"name":"Department of Physics, College of Science, Sudan University of Science and Technology, Khartoum 11115, Sudan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0857-0935","authenticated-orcid":false,"given":"Nor Ain Azeany","family":"Mohd Nasir","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Centre for Defense Foundation Studies, Universiti Pertahanan Nasional Malaysia, Kuala Lumpur 57000, Malaysia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,11,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1959","DOI":"10.1007\/s10973-020-09998-w","article-title":"A review on the properties, preparation, models and stability of hybrid nanofluids to optimize energy consumption","volume":"144","author":"Eshgarf","year":"2020","journal-title":"J. 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