{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,30]],"date-time":"2026-05-30T03:23:07Z","timestamp":1780111387533,"version":"3.54.0"},"reference-count":33,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2024,8,19]],"date-time":"2024-08-19T00:00:00Z","timestamp":1724025600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"King Khalid University","award":["RGP2\/115\/45"],"award-info":[{"award-number":["RGP2\/115\/45"]}]},{"name":"King Khalid University","award":["NBU-FFR-2024-2920-03"],"award-info":[{"award-number":["NBU-FFR-2024-2920-03"]}]},{"name":"King Khalid University","award":["PSAU\/2024\/R\/1445"],"award-info":[{"award-number":["PSAU\/2024\/R\/1445"]}]},{"name":"Northern Border University, Arar, KSA","award":["RGP2\/115\/45"],"award-info":[{"award-number":["RGP2\/115\/45"]}]},{"name":"Northern Border University, Arar, KSA","award":["NBU-FFR-2024-2920-03"],"award-info":[{"award-number":["NBU-FFR-2024-2920-03"]}]},{"name":"Northern Border University, Arar, KSA","award":["PSAU\/2024\/R\/1445"],"award-info":[{"award-number":["PSAU\/2024\/R\/1445"]}]},{"name":"Prince Sattam bin Abdulaziz University","award":["RGP2\/115\/45"],"award-info":[{"award-number":["RGP2\/115\/45"]}]},{"name":"Prince Sattam bin Abdulaziz University","award":["NBU-FFR-2024-2920-03"],"award-info":[{"award-number":["NBU-FFR-2024-2920-03"]}]},{"name":"Prince Sattam bin Abdulaziz University","award":["PSAU\/2024\/R\/1445"],"award-info":[{"award-number":["PSAU\/2024\/R\/1445"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>This analysis emphasizes the significance of radiation and chemical reaction effects on the boundary layer flow (BLF) of Casson liquid over a linearly elongating surface, as well as the properties of momentum, entropy production, species, and thermal dispersion. The mass diffusion coefficient and temperature-dependent models of thermal conductivity and species are used to provide thermal transportation. Nonlinear partial differential equations (NPDEs) that go against the conservation laws of mass, momentum, heat, and species transportation are the form arising problems take on. A set of coupled dimensionless partial differential equations (PDEs) are obtained from a set of convective differential equations by applying the proper non-similar transformations. Local non-similarity approaches provide an analytical approximation of the dimensionless non-similar system up to two degrees of truncations. The built-in Matlab (Version: 7.10.0.499 (R2010a)) solver bvp4c is used to perform numerical simulations of the local non-similar (LNS) truncations.<\/jats:p>","DOI":"10.3390\/e26080702","type":"journal-article","created":{"date-parts":[[2024,8,19]],"date-time":"2024-08-19T07:37:21Z","timestamp":1724053041000},"page":"702","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Analysis of Entropy Generation via Non-Similar Numerical Approach for Magnetohydrodynamics Casson Fluid Flow with Joule Heating"],"prefix":"10.3390","volume":"26","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5032-3877","authenticated-orcid":false,"given":"Hanen","family":"Louati","sequence":"first","affiliation":[{"name":"Mathematics Department, Faculty of Science, Northern Border University, Arar 73213, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0007-7296-7631","authenticated-orcid":false,"given":"Sajid","family":"Khan","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, University of Wah, Wah Cantt 47040, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0003-8803-6170","authenticated-orcid":false,"given":"Muavia","family":"Mansoor","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Wah, Wah Cantt 47040, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shreefa O.","family":"Hilali","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Sciences and Arts (Majardah), King Khalid University, Magardah 61937, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8759-0469","authenticated-orcid":false,"given":"Ameni","family":"Gargouri","sequence":"additional","affiliation":[{"name":"Mathematics Department, College of Humanities and Science in Al Aflaj, Prince Sattam Bin Abdulaziz University, Alkharj 11912, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2024,8,19]]},"reference":[{"key":"ref_1","first-page":"13","article-title":"Magnetohydrodynamic radiative casson fluid flow over a stretching sheet with heat source\/sink","volume":"55","author":"Vijayaragavan","year":"2016","journal-title":"Adv. Phys. Theo. Appl."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"895","DOI":"10.1016\/S0895-7177(02)00058-4","article-title":"A mathematical study of peristaltic transport of a casson fluid","volume":"35","author":"Mernone","year":"2002","journal-title":"Math. Comput. Model."},{"key":"ref_3","first-page":"45","article-title":"Heat and mass transfer in magnetohydrodynamic Casson fluid over an exponentially permeable stretching surface","volume":"19","author":"Raju","year":"2016","journal-title":"Eng. Sci. Technol. Int. J."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"100873","DOI":"10.1016\/j.csite.2021.100873","article-title":"On thermally corrugated porous enclosure (TCPE) equipped with casson liquid suspension: Finite element thermal analysis","volume":"25","author":"Rehman","year":"2021","journal-title":"Case Stud. Therm. Eng."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"101433","DOI":"10.1016\/j.csite.2021.101433","article-title":"Thermal analysis of Casson rheological fluid with gold nanoparticles under the impact of gravitational and magnetic forces","volume":"28","author":"Hussain","year":"2021","journal-title":"Case Stud. Therm. Eng."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"111198","DOI":"10.1016\/j.chaos.2021.111198","article-title":"Development of mathematical modeling of multi-phase flow of Casson rheological fluid: Theoretical approach","volume":"150","author":"Nazeer","year":"2021","journal-title":"Chaos Solitons Fractals"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"3575","DOI":"10.1002\/htj.21789","article-title":"Effect of Joule heating on MHD non-Newtonian fluid flow past an exponentially stretching curved surface","volume":"49","author":"Kempannagari","year":"2020","journal-title":"Heat Transf."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"154","DOI":"10.1016\/j.imu.2017.08.003","article-title":"Viscous and Joule\u2019s dissipation on Casson fluid over a chemically reacting stretching sheet with inclined magnetic field and multiple slips","volume":"9","author":"Gopal","year":"2017","journal-title":"Inform. Med. Unlocked"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"633","DOI":"10.1007\/s40010-021-00764-7","article-title":"Stagnation-Point Flow of the Williamson Nanofluid Containing Gyrotactic Micro-organisms","volume":"91","author":"Muthtamilselvan","year":"2021","journal-title":"Proc. Natl. Acad. Sci. India Sect. A Phys. Sci."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Al-Zubaidi, A., Nazeer, M., Subia, G.S., Hussain, F., Saleem, S., and Ghafar, M. (2021). Mathematical modeling and simulation of MHD electro-osmotic flow of Jeffrey fluid in convergent geometry. Waves Random Complex Media, 1\u201317.","DOI":"10.1080\/17455030.2021.2000672"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2150204","DOI":"10.1142\/S0217979221502040","article-title":"A comparative study of MHD fluid-particle suspension induced by metachronal wave under the effects of lubricated walls","volume":"35","author":"Nazeer","year":"2021","journal-title":"Int. J. Mod. Phys. B"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Nazeer, M., Saleem, S., Hussain, F., Zia, Z., Khalid, K., and Feroz, N. (2022). Heat transmission in a magnetohydrodynamic multiphase flow induced by metachronal propulsion through porous media with thermal radiation. Proc. Inst. Mech. Eng. Part E J. Process Mech. Eng., 09544089221075299.","DOI":"10.1177\/09544089221075299"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1515\/jnet-2018-0022","article-title":"Impact of non-linear radiation on MHD non-aligned stagnation point flow of micropolar fluid over a convective surface","volume":"43","author":"Kumar","year":"2018","journal-title":"J. Non-Equilib. Thermodyn."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Khan, N.M., Chu, Y.M., Khan, M.I., Kadry, S., and Qayyum, S. (2020). Modeling and dual solutions for magnetized mixed convective stagnation point flow of upper convected Maxwell fluid model with second-order velocity slip. Math. Methods Appl. Sci.","DOI":"10.1002\/mma.6824"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"917","DOI":"10.1016\/j.icheatmasstransfer.2011.04.020","article-title":"Dual solutions in boundary layer stagnation-point flow and mass transfer with chemical reaction past a stretching\/shrinking sheet","volume":"38","author":"Bhattacharyya","year":"2011","journal-title":"Int. Commun. Heat Mass Transf."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"084702","DOI":"10.1088\/0256-307X\/28\/8\/084702","article-title":"Dual solutions in unsteady stagnation-point flow over a shrinking sheet","volume":"28","author":"Bhattacharyya","year":"2011","journal-title":"Chin. Phys. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"169674","DOI":"10.1155\/2013\/169674","article-title":"MHD stagnation-point flow of Casson fluid and heat transfer over a stretching sheet with thermal radiation","volume":"2013","author":"Bhattacharyya","year":"2013","journal-title":"J. Thermodyn."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1016\/j.asej.2012.07.002","article-title":"Heat transfer analysis in unsteady boundary layer stagnation-point flow towards a shrinking\/stretching sheet","volume":"4","author":"Bhattacharyya","year":"2013","journal-title":"Ain Shams Eng. J."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"094702","DOI":"10.1088\/0256-307X\/28\/9\/094702","article-title":"Slip effects on an unsteady boundary layer stagnation-point flow and heat transfer towards a stretching sheet","volume":"28","author":"Bhattacharyya","year":"2011","journal-title":"Chin. Phys. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1007\/BF01176953","article-title":"Magnetohydrodynamic stagnation-point flow towards a stretching sheet","volume":"152","author":"Mahapatra","year":"2001","journal-title":"Acta Mech."},{"key":"ref_21","unstructured":"Bejan, A. (1994). Entropy Generation Through Heat and Fluid Flow, Wiley and Sons."},{"key":"ref_22","unstructured":"Usha, R., and Uma, R. (2013). Entropy generation in a Casson fluid flow in a stenosed artery. J. Mech. Med. Biol., 13."},{"key":"ref_23","first-page":"1202","article-title":"Heat transfer and entropy generation in Casson fluid flow over a stretching sheet with thermal radiation and viscous dissipation","volume":"223","author":"Rizvi","year":"2016","journal-title":"J. Mol. Liq."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Shevchuk, I.V. (2009). Convective Heat and Mass Transfer in Rotating Disk Systems, Springer.","DOI":"10.1007\/978-3-642-00718-7"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"104","DOI":"10.1023\/B:HITE.0000020097.59838.02","article-title":"A Self-Similar Solution of Navier\u2013Stokes and Energy Equations for Rotating Flows Between a Cone and a Disk","volume":"42","author":"Shevchuk","year":"2004","journal-title":"High Temp."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Gul, A., Khan, I., Shafie, S., Khalid, A., and Khan, A. (2015). Heat transfer in MHD mixed convection flow of a ferrofluid along a vertical channel. PLoS ONE, 10.","DOI":"10.1371\/journal.pone.0141213"},{"key":"ref_27","first-page":"309","article-title":"Unsteady MHD free convection flow of Casson fluid past over an oscillating vertical plate embedded in a porous medium","volume":"18","author":"Khalid","year":"2015","journal-title":"Eng. Sci. Technol. Int. J."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"490","DOI":"10.1186\/s11671-015-1144-4","article-title":"Energy transfer in mixed convection MHD flow of nanofluid containing different shapes of nanoparticles in a channel filled with saturated porous medium","volume":"10","author":"Gul","year":"2015","journal-title":"Nanoscale Res. Lett."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1936","DOI":"10.2514\/3.6029","article-title":"Local nonsimilarity boundary-layer solutions","volume":"8","author":"Sparrow","year":"1970","journal-title":"AIAA J."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"291","DOI":"10.2514\/2.6767","article-title":"Impingement Heat Transfer over a Rotating Disk: Integral Method","volume":"17","author":"Shevchuk","year":"2003","journal-title":"J. Thermophys. Heat Transf."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"318","DOI":"10.1504\/IJEX.2016.077429","article-title":"Entropy analysis in MHD nanofluid flow near a convectively heated stretching surface","volume":"20","author":"Butt","year":"2016","journal-title":"Int. J. Exergy"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"709","DOI":"10.1016\/j.ijheatmasstransfer.2017.04.121","article-title":"Simulation of natural convection and entropy generation of non-Newtonian nanofluid in a porous cavity using Buongiorno\u2019s mathematical model","volume":"112","author":"Kefayati","year":"2017","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"150","DOI":"10.1504\/IJEX.2020.105481","article-title":"Entropy generation effects in MHD Casson nanofluid past a permeable stretching surface","volume":"31","author":"Butt","year":"2020","journal-title":"Int. J. Exergy"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/8\/702\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:38:40Z","timestamp":1760110720000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/8\/702"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,8,19]]},"references-count":33,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2024,8]]}},"alternative-id":["e26080702"],"URL":"https:\/\/doi.org\/10.3390\/e26080702","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,8,19]]}}}