{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,9]],"date-time":"2026-05-09T20:45:50Z","timestamp":1778359550322,"version":"3.51.4"},"reference-count":38,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2020,3,23]],"date-time":"2020-03-23T00:00:00Z","timestamp":1584921600000},"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 article, the magnetohydrodynamic (MHD) flow of Casson nanofluid with thermal radiation over an unsteady shrinking surface is investigated. The equation of momentum is derived from the Navier\u2013Stokes model for non-Newtonian fluid where components of the viscous terms are symmetric. The effect of Stefan blowing with partial slip conditions of velocity, concentration, and temperature on the velocity, concentration, and temperature distributions is also taken into account. The modeled equations of partial differential equations (PDEs) are transformed into the equivalent boundary value problems (BVPs) of ordinary differential equations (ODEs) by employing similarity transformations. These similarity transformations can be obtained by using symmetry analysis. The resultant BVPs are reduced into initial value problems (IVPs) by using the shooting method and then solved by using the fourth-order Runge\u2013Kutta (RK) technique. The numerical results reveal that dual solutions exist in some ranges of different physical parameters such as unsteadiness and suction\/injection parameters. The thickness of the velocity boundary layer is enhanced in the second solution by increasing the magnetic and velocity slip factor effect in the boundary layer. Increment in the Prandtl number and Brownian motion parameter is caused by a reduction of the thickness of the thermal boundary layer and temperature. Moreover, stability analysis performed by employing the three-stage Lobatto IIIA formula in the BVP4C solver with the help of MATLAB software reveals that only the first solution is stable and physically realizable.<\/jats:p>","DOI":"10.3390\/sym12030487","type":"journal-article","created":{"date-parts":[[2020,3,24]],"date-time":"2020-03-24T13:04:04Z","timestamp":1585055044000},"page":"487","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":60,"title":["Effects of Stefan Blowing and Slip Conditions on Unsteady MHD Casson Nanofluid Flow Over an Unsteady Shrinking Sheet: 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":"Jawad","family":"Raza","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Institute of Southern Punjab (ISP), Multan 66000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"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":[{"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-Behoth St. 33, Dokki, Cairo 12622, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,3,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"105005","DOI":"10.1016\/j.cmpb.2019.105005","article-title":"Numerical simulation of electroosmosis regulated peristaltic transport of Bingham nanofluid","volume":"2019","author":"Tanveer","year":"2019","journal-title":"Comput. Methods Programs Biomed."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1016\/j.cmpb.2019.05.012","article-title":"Theoretical investigation of Ree\u2013Eyring nanofluid flow with entropy optimization and Arrhenius activation energy between two rotating disks","volume":"177","author":"Hayat","year":"2019","journal-title":"Comput. 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