{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,24]],"date-time":"2026-07-24T17:32:53Z","timestamp":1784914373774,"version":"3.55.0"},"reference-count":53,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2019,10,15]],"date-time":"2019-10-15T00:00:00Z","timestamp":1571097600000},"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 analysis in an unsteady magnetohydrodynamic (MHD) flow of generalized Casson fluid over a vertical plate is analyzed. The medium is porous, accepting Darcy\u2019s resistance. The plate is oscillating in its plane with a cosine type of oscillation. Sodium alginate (SA\u2013NaAlg) is taken as a specific example of Casson fluid. The fractional model of SA\u2013NaAlg fluid using the Atangana\u2013Baleanu fractional derivative (ABFD) of the non-local and non-singular kernel has been examined. The ABFD definition was based on the Mittag\u2013Leffler function, and promises an improved description of the dynamics of the system with the memory effects. Exact solutions in the case of ABFD are obtained via the Laplace transform and compared graphically. The influence of embedded parameters on the velocity field is sketched and discussed. A comparison of the Atangana\u2013Baleanu fractional model with an ordinary model is made. It is observed that the velocity and temperature profile for the Atangana\u2013Baleanu fractional model are less than that of the ordinary model. The Atangana\u2013Baleanu fractional model reduced the velocity profile up to 45.76% and temperature profile up to 13.74% compared to an ordinary model.<\/jats:p>","DOI":"10.3390\/sym11101295","type":"journal-article","created":{"date-parts":[[2019,10,16]],"date-time":"2019-10-16T03:32:54Z","timestamp":1571196774000},"page":"1295","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":29,"title":["MHD Flow and Heat Transfer in Sodium Alginate Fluid with Thermal Radiation and Porosity Effects: Fractional Model of Atangana\u2013Baleanu Derivative of Non-Local and Non-Singular Kernel"],"prefix":"10.3390","volume":"11","author":[{"given":"Arshad","family":"Khan","sequence":"first","affiliation":[{"name":"Institute of Computer Science and Information Technology, The University of Agriculture, Peshawar 25130, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3723-0796","authenticated-orcid":false,"given":"Dolat","family":"Khan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, City University of Science and Information Technology, Peshawar 25130, Khyber Pakhtunkhwa, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"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":[{"vocabulary":"crossref","role":"author"}]},{"given":"Muhammad","family":"Taj","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Azad Jammu and Kashmir, Muzaffarabad 13100, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Imran","family":"Ullah","sequence":"additional","affiliation":[{"name":"College of Civil Engineering, National University of Sciences and Technology Islamabad, Islamabad 44000, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Abdullah Mohammed","family":"Aldawsari","sequence":"additional","affiliation":[{"name":"Department of Chemistry, College of Arts and Science-Wadi Al-Dawaser, Prince Sattam bin Abdulaziz University, 11991 Al-kharj, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Phatiphat","family":"Thounthong","sequence":"additional","affiliation":[{"name":"Renewable Energy Research Centre, Department of Teacher Training in Electrical Engineering, Faculty of Technical Education, King Mongkut\u2019s University of Technology North Bangkok, Bangkok 10800, Thailand"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5769-4320","authenticated-orcid":false,"given":"Kottakkaran","family":"Sooppy Nisar","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Arts and Sciences, Wadi Aldawaser, Prince Sattam bin Abdulaziz University, 11991 Al-Kharj, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2019,10,15]]},"reference":[{"key":"ref_1","first-page":"e102","article-title":"A flow equation for the pigment oil suspensions of the printing ink type","volume":"Volume 84","author":"Casson","year":"1959","journal-title":"Rheology of Disperse Systems"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"689","DOI":"10.1615\/HeatTransRes.2015007148","article-title":"Numerical solution for the Blasius flow in Casson fluid with viscous dissipation and convective boundary conditions","volume":"46","author":"Qasim","year":"2015","journal-title":"Heat Transf. 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