{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,30]],"date-time":"2026-01-30T09:38:23Z","timestamp":1769765903795,"version":"3.49.0"},"reference-count":41,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2020,7,13]],"date-time":"2020-07-13T00:00:00Z","timestamp":1594598400000},"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>In this research, we intend to develop a dynamical system for the magnetohydrodynamic (MHD) flow of an electrically conducting Casson nanofluid on exponentially shrinking and stretching surfaces, in the presence of a velocity and concertation slip effect, with convective boundary conditions. There are three main objectives of this article, specifically, to discuss the heat characteristics of flow, to find multiple solutions on both surfaces, and to do stability analyses. The main equations of flow are governed by the Brownian motion, the Prandtl number, and the thermophoresis parameters, the Schmid and Biot numbers. The shooting method and three-stage Lobatto IIIa formula have been employed to solve the equations. The ranges of the dual solutions are      f  w c 1   \u2264  f w      and      \u03bb c  \u2264 \u03bb    , while the no solution ranges are      f  w c 1   &gt;  f w      and      \u03bb c  &gt; \u03bb    . The results reveal that the temperature of the fluid increases with the extended values of the thermophoresis parameter, the Brownian motion parameter, and the Hartmann and Biot numbers, for both solutions. The presence of dual solutions depends on the suction parameter. In order to indicate that the first solution is physically relevant and stable, a stability analysis has been performed.<\/jats:p>","DOI":"10.3390\/sym12071162","type":"journal-article","created":{"date-parts":[[2020,7,22]],"date-time":"2020-07-22T05:10:30Z","timestamp":1595394630000},"page":"1162","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Stability Analysis of the Magnetized Casson Nanofluid Propagating through an Exponentially Shrinking\/Stretching Plate: Dual Solutions"],"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, Universiti Utara Malaysia, Sintok 06010, Kedah, 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, Universiti Utara Malaysia, Sintok 06010, Kedah, Malaysia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"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"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2080-8552","authenticated-orcid":false,"given":"El-Sayed M.","family":"Sherif","sequence":"additional","affiliation":[{"name":"Center of Excellence for Research in Engineering Materials (CEREM), King Saud University, P.O. Box 800, Al-Riyadh 11421, Saudi Arabia"},{"name":"Electrochemistry and Corrosion Laboratory, Department of Physical Chemistry, National Research Centre, El-Buhouth St., Dokki, Cairo 12622, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7260-639X","authenticated-orcid":false,"given":"Hany S.","family":"Abdo","sequence":"additional","affiliation":[{"name":"Center of Excellence for Research in Engineering Materials (CEREM), King Saud University, P.O. Box 800, Al-Riyadh 11421, Saudi Arabia"},{"name":"Mechanical Design and Materials Department, Faculty of Energy Engineering, Aswan University, Aswan 81521, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,7,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2666","DOI":"10.1016\/j.apt.2019.08.014","article-title":"Mixed convection of non-Newtonian nanofluid in an H-shaped cavity with cooler and heater cylinders filled by a porous material: Two phase approach","volume":"30","author":"Li","year":"2019","journal-title":"Adv. Powder Technol."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"2975","DOI":"10.1007\/s10973-017-6813-3","article-title":"Investigation into the effects of slip boundary condition on nanofluid flow in a double-layer microchannel","volume":"131","author":"Arabpour","year":"2018","journal-title":"J. Therm. Anal. 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