{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,20]],"date-time":"2026-01-20T09:15:50Z","timestamp":1768900550330,"version":"3.49.0"},"reference-count":45,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,10,2]],"date-time":"2022-10-02T00:00:00Z","timestamp":1664668800000},"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>Studies associated with ethylene glycol (EG) have great significance in various engineering sectors because EG is more useful as a cooling agent in various engines. Furthermore, fluid inspection using two distinct nanoparticles has applications in mechanical systems, electronic devices, medical apparatus, and the diagnosis and treatment of disease. Therefore, present comminution explored the entropy production in magnetized hybrid nanomaterials flowing via Darcy\u2013Forchheimer space with varying permeability. Hybrid nano liquid is synthesized by adding cobalt ferrite and gold nanoparticles to ethylene glycol and water. Effects of thermal radiation, Joule heating, heat sources, and an exponential heat source are considered in the energy expression. The assumed problem is modeled in the form of nonlinear PDEs. Such types of problems have mostly occurred in symmetrical phenomena and are applicable in engineering, physics, and applied mathematics. The obtained system is converted to ODEs using suitable substitution transformations. Resultant ODEs are numerically computed with the help of the NDSolve technique using Mathematica software. Their outcomes are displayed through figures and tables. Obtained results reveal that variable permeability and curvature parameters improve the velocity profile, while an exponential heat source (EHS) enhances the thermal effect. It is also observed that entropy optimization improves with the increment in magnetic parameter.<\/jats:p>","DOI":"10.3390\/sym14102057","type":"journal-article","created":{"date-parts":[[2022,10,11]],"date-time":"2022-10-11T03:32:56Z","timestamp":1665459176000},"page":"2057","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":32,"title":["Numerical Simulation of Entropy Optimization in Radiative Hybrid Nanofluid Flow in a Variable Features Darcy\u2013Forchheimer Curved Surface"],"prefix":"10.3390","volume":"14","author":[{"given":"Asif","family":"Hayat","sequence":"first","affiliation":[{"name":"Department of Mathematics, Abdul Wali Khan University, Mardan 23200, Pakistan"}]},{"given":"Ikram","family":"Ullah","sequence":"additional","affiliation":[{"name":"Department of Sciences and Humanities, National University of Computer and Emerging Sciences, Peshawar 25000, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6417-1181","authenticated-orcid":false,"given":"Hassan","family":"Khan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Abdul Wali Khan University, Mardan 23200, Pakistan"},{"name":"Department of Mathematics, Near East University TRNC, Mersin 99138, Turkey"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9595-6096","authenticated-orcid":false,"given":"Wajaree","family":"Weera","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3541-9704","authenticated-orcid":false,"given":"Ahmed","family":"Galal","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, College of Engineering in Wadi Alddawasir, Prince Sattam bin Abdulaziz University, Wadi Alddawasir 11991, Saudi Arabia"},{"name":"Production Engineering and Mechanical Design Department, Faculty of Engineering, Mansoura University, Mansoura P.O. Box 35516, Egypt"}]}],"member":"1968","published-online":{"date-parts":[[2022,10,2]]},"reference":[{"key":"ref_1","first-page":"1782","article-title":"Wasserbewegung durch boden","volume":"45","author":"Forchheimer","year":"1901","journal-title":"Z. Ver. Deutsch Ing."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"105416","DOI":"10.1016\/j.icheatmasstransfer.2021.105416","article-title":"Combined heat source and zero mass flux features on magnetized nanofluid flow by radial disk with the applications of Coriolis force and activation energy","volume":"126","author":"Ullah","year":"2021","journal-title":"Int. Commun. Heat Mass Transf."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Alotaibi, H., and Eid, M.R. (2021). Thermal analysis of 3D electromagnetic radiative nanofluid flow with suction\/blowing: Darcy\u2013Forchheimer scheme. 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