{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,9]],"date-time":"2026-02-09T23:28:57Z","timestamp":1770679737743,"version":"3.49.0"},"reference-count":56,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2019,8,28]],"date-time":"2019-08-28T00:00:00Z","timestamp":1566950400000},"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>The revised Buongiorno\u2019s nanofluid model with the effect of induced magnetic field on steady magnetohydrodynamics (MHD) stagnation-point flow of nanofluid over a stretching or shrinking sheet is investigated. The effects of zero mass flux and suction are taken into account. A similarity transformation with symmetry variables are introduced in order to alter from the governing nonlinear partial differential equations into a nonlinear ordinary differential equations. These governing equations are numerically solved using the bvp4c function in Matlab solver, a very adequate finite difference method. The influences of considered parameters (    P r    , M,    \u03c7   ,     L e    ,     N b    ,     N t    , S, and    \u03bb   ) on velocity, induced magnetic, temperature, and concentration profiles together with the reduced skin friction and heat transfer rate are discussed. Results from these criterion exposed the existence of dual solutions when magnetic field and suction are applied for a specific range of    \u03bb   . The stability of the solutions obtained is carried out by performing a stability analysis.<\/jats:p>","DOI":"10.3390\/sym11091078","type":"journal-article","created":{"date-parts":[[2019,8,28]],"date-time":"2019-08-28T11:23:18Z","timestamp":1566991398000},"page":"1078","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Magnetohydrodynamics Stagnation-Point Flow of a Nanofluid Past a Stretching\/Shrinking Sheet with Induced Magnetic Field: A Revised Model"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3805-4974","authenticated-orcid":false,"given":"Mohamad Mustaqim","family":"Junoh","sequence":"first","affiliation":[{"name":"Institute for Mathematical Research, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia"}]},{"given":"Fadzilah","family":"Md Ali","sequence":"additional","affiliation":[{"name":"Institute for Mathematical Research, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia"},{"name":"Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia"}]},{"given":"Ioan","family":"Pop","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Babes-Bolyai University, R-400084 Cluj-Napoca, Romania"}]}],"member":"1968","published-online":{"date-parts":[[2019,8,28]]},"reference":[{"key":"ref_1","unstructured":"Fisher, E.G. (1976). Extrusion of Plastics, Wiley. [3rd ed.]."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1016\/0017-9310(67)90100-7","article-title":"Flow and heat transfer in the boundary layer on a continuous moving surface","volume":"10","author":"Tsou","year":"1967","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"26","DOI":"10.1002\/aic.690070108","article-title":"Boundary-layer behavior on continuous solid surfaces: I. Boundary-layer equations for two-dimensional and axisymmetric flow","volume":"7","author":"Sakiadis","year":"1961","journal-title":"AIChE J."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"645","DOI":"10.1007\/BF01587695","article-title":"Flow past a stretching plate","volume":"21","author":"Crane","year":"1970","journal-title":"J. Appl. Math. Phys. (ZAMP)"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1090\/S0033-569X-06-01002-5","article-title":"Viscous flow due to a shrinking sheet","volume":"64","author":"Wang","year":"2006","journal-title":"Q. Appl. Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1016\/j.ijnonlinmec.2007.12.021","article-title":"Stagnation flow towards a shrinking sheet","volume":"43","author":"Wang","year":"2008","journal-title":"Int. J. Non-Linear Mech."},{"key":"ref_7","unstructured":"Shercliff, J.A. (1965). A Textbook of Magnetohydrodynamics, Pergamon Press."},{"key":"ref_8","unstructured":"Branover, G.G., and Tinober, A.B. (1970). Magnetohydrodynamics of Incompressible Media (in Russian), Nauka."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Cramer, K.R., and Pai, S.I. (1973). Magneto Fluid Dynamics For Engineers and Applied Physicists, McGraw-Hill Book Company.","DOI":"10.1002\/eej.4390930120"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1143\/JPSJ.27.235","article-title":"Application of the Laplace transform to the solution of the boundary layer equations. III: Magnetohydrodynamic Falkner-Skan problem","volume":"27","author":"Apelblat","year":"1969","journal-title":"J. Phys. Soc. Jpn."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"521","DOI":"10.1002\/nme.1620060408","article-title":"Impulsively started viscous flows past a finite flat plate with and without an applied magnetic field","volume":"6","author":"Ingham","year":"1973","journal-title":"Int. J. Numer. Methods Eng."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1002\/zamm.19740540105","article-title":"Integration of the Magnetohydrodynamic boundary-layer equations by Meksyn\u2019s method","volume":"54","author":"Liron","year":"1974","journal-title":"J. Appl. Math. Mech. (ZAMM)"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"871","DOI":"10.1016\/0735-1933(93)90040-3","article-title":"Magnetohydrodynamic free convection flow over a wedge in the presence of a transverse magnetic field","volume":"20","author":"Watanabe","year":"1993","journal-title":"Int. Commun. Heat Mass Trans."},{"key":"ref_14","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_15","doi-asserted-by":"crossref","first-page":"101","DOI":"10.3329\/jname.v7i2.4370","article-title":"Effects of heat transfer and viscous dissipation on MHD free convection flow past an exponentially accelerated vertical plate with variable temperature","volume":"7","author":"Kishore","year":"2010","journal-title":"J. Nav. Archit. Mar. Eng."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"108","DOI":"10.1016\/j.jmmm.2015.07.057","article-title":"Thermally radiative three-dimensional flow of Jeffrey nanofluid with internal heat generation and magnetic field","volume":"397","author":"Shehzad","year":"2016","journal-title":"J. Magn. Magn. Mater."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"302","DOI":"10.1016\/j.molliq.2016.11.023","article-title":"Impacts of constructive and destructive chemical reactions in magnetohydrodynamic (MHD) flow of Jeffrey liquid due to nonlinear radially stretched surface","volume":"225","author":"Hayat","year":"2017","journal-title":"J. Mol. Liq."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1007\/s00231-007-0251-x","article-title":"Magnetic field effect on heat transfer and fluid flow characteristics of blood flow in multi-stenosis arteries","volume":"44","author":"Tashtoush","year":"2008","journal-title":"Heat Mass Trans."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1016\/j.aej.2013.02.003","article-title":"MHD boundary layer flow and heat transfer over an exponentially stretching sheet embedded in a thermally stratified medium","volume":"52","author":"Mukhopadhyay","year":"2013","journal-title":"Alexendria Eng. J."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1016\/j.ijheatmasstransfer.2016.09.060","article-title":"The effects of radiation optical properties on the unsteady 2D boundary layer MHD flow and heat transfer over a stretching plate","volume":"105","author":"Tian","year":"2017","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1186\/1687-2770-2013-32","article-title":"Dual solutions in MHD flow on a nonlinear porous shrinking sheet in a viscous fluid","volume":"1","author":"Ali","year":"2013","journal-title":"Bound. Value Probl."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1016\/j.jmmm.2018.06.020","article-title":"Magnetohydrodynamic rotating flow and heat transfer of ferrofluid due to an exponentially permeable stretching\/shrinking sheet","volume":"465","author":"Jusoh","year":"2018","journal-title":"J. Magn. Magn. Mater."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"022502","DOI":"10.1115\/1.4002602","article-title":"MHD mixed convection boundary layer flow toward a stagnation point on a vertical surface with induced magnetic field","volume":"133","author":"Ali","year":"2011","journal-title":"ASME J. Heat Transf."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1007\/s00231-010-0693-4","article-title":"MHD boundary layer flow and heat transfer over a stretching sheet with induced magnetic field","volume":"47","author":"Ali","year":"2011","journal-title":"Heat Mass Trans."},{"key":"ref_25","unstructured":"Choi, S.U.S., and Eastman, J.A. (1995, January 12\u201317). Enhancing thermal conductivity of fluids with nanoparticles. Proceedings of the 1995 International Mechanical Engineering Congress and Exposition, San Francisco, CA, USA."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"4568","DOI":"10.1063\/1.1454184","article-title":"Thermal conductivity enhancement of suspensions containing nanosized alumina particles","volume":"91","author":"Xie","year":"2002","journal-title":"J. Appl. Phys."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"567","DOI":"10.1115\/1.1571080","article-title":"Temperature dependence of thermal conductive enhancement for nanofluids","volume":"125","author":"Das","year":"2003","journal-title":"J. Heat Transf."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"4323","DOI":"10.1021\/jp057225m","article-title":"A new parameter to control heat transport in nanofluide: Surface charge state of the particle in suspension","volume":"110","author":"Lee","year":"2006","journal-title":"J. Phys. Chem. B"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"2651","DOI":"10.1016\/j.ijheatmasstransfer.2007.10.026","article-title":"Effective viscosities and thermal conductivities of aqueous nanofluids containing low volume concentrations of Al2O3 nanoparticles","volume":"51","author":"Lee","year":"2008","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_30","first-page":"363","article-title":"New temperature dependent thermal conductivity data for water based nanofluids","volume":"48","author":"Minsta","year":"2009","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1016\/j.ijheatmasstransfer.2017.01.124","article-title":"Heatline visualization of natural convection in a thick walled open cavity filled with a nanofluid","volume":"109","author":"Bondareva","year":"2017","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1646","DOI":"10.1016\/j.rser.2010.11.035","article-title":"A review on applications and challenges of nanofluids","volume":"15","author":"Saidur","year":"2011","journal-title":"Renew. Sustain. Energy Rev."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Das, S.K., Choi, S.U.S., Yu, W., and Pradeep, Y. (2008). Nanofluids: Science and Technology, Wiley.","DOI":"10.1002\/9780470180693"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Nield, D.A., and Bejan, A. (2013). Convection in Porous Media, Springer. [4th ed.].","DOI":"10.1007\/978-1-4614-5541-7"},{"key":"ref_35","unstructured":"Minkowycz, W.J., Sparrow, E.M., and Abraham, J.P. (2013). Nanoparticle Heat Transfer and Fluid Flow, CRC Press, Taylor and Francis Group."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Shenoy, A., Sheremet, M., and Pop, I. (2016). Convective Flow and Heat Transfer from Wavy Surfaces: Viscous Fluids, Porous Media and Nanofluids, CRC Press, Taylor and Francis Group.","DOI":"10.1201\/9781315367637"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"094312","DOI":"10.1063\/1.3245330","article-title":"A benchmark study on the thermal conductivity of nanofluids","volume":"106","author":"Buongiorno","year":"2009","journal-title":"J. Appl. Phys."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"3187","DOI":"10.1016\/j.ijheatmasstransfer.2009.02.006","article-title":"Review of convective heat transfer enhancement with nanofluids","volume":"52","author":"Pramuanjaroenkij","year":"2009","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"380826","DOI":"10.1155\/2010\/380826","article-title":"Heat transfer in nanofluids","volume":"2","author":"Manca","year":"2010","journal-title":"Adv. Mech. Eng."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"040801","DOI":"10.1115\/1.4002633","article-title":"Review of heat conduction in nanofluids","volume":"133","author":"Fan","year":"2011","journal-title":"ASME J. Heat Transf."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"582","DOI":"10.1016\/j.ijheatmasstransfer.2012.10.037","article-title":"A review of the applications of nanofluids in solar energy","volume":"57","author":"Mahian","year":"2013","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1016\/j.jtice.2016.05.014","article-title":"Nanofluid convective heat transfer using semi analytical and numerical approaches: A review","volume":"65","author":"Sheikholeslami","year":"2016","journal-title":"J. Taiwan Inst. Chem. Eng."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1016\/j.ijheatmasstransfer.2017.03.118","article-title":"Does mathematics contribute to the nanofluid debate?","volume":"111","author":"Myers","year":"2017","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"388","DOI":"10.1016\/j.molliq.2018.12.104","article-title":"Variable magnetic forces impact on magnetizable hybrid nanofluid heat transfer through a circular cavity","volume":"277","author":"Sheikholeslami","year":"2019","journal-title":"J. Mol. Liq."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1007\/s11012-012-9579-5","article-title":"Stability of dual solutions in stagnation-point flow and heat transfer over a porous shrinking sheet with thermal radiation","volume":"48","author":"Mahapatra","year":"2013","journal-title":"Meccanica"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1093\/qjmam\/39.1.15","article-title":"Existence and non-uniqueness of similarity solutions of a boundary-layer problem","volume":"39","author":"Hussaini","year":"1986","journal-title":"Quart. J. Mech. Appl. Math."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1007\/s002310000126","article-title":"Heat and mass transfer characteristics of the self-similar boundary-layer flows induced by continous surfaces stretched with rapidly decreasing velocities","volume":"38","author":"Magyari","year":"2001","journal-title":"Heat Mass Transf."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"4681","DOI":"10.1016\/j.ijheatmasstransfer.2006.02.056","article-title":"Final steady flow near a stagnation-point on a vertical surface in a porous medium","volume":"49","author":"Merrill","year":"2006","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1007\/BF00042775","article-title":"On dual solutions occuring in mixed convection in a porous medium","volume":"20","author":"Merkin","year":"1985","journal-title":"J. Eng. Math."},{"key":"ref_50","first-page":"10474","article-title":"MHD stagnation-point flow of a nanofluid past a stretching\/shrinking sheet with induced magnetic field","volume":"13","author":"Junoh","year":"2018","journal-title":"J. Eng. Appl. Sci."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"682","DOI":"10.1016\/j.ijheatmasstransfer.2013.06.054","article-title":"The Cheng\u2013Minkowycz problem for natural convective boundary layer flow in a porous medium saturated by a nanofluid: A revised model","volume":"65","author":"Kuznetsov","year":"2013","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_52","first-page":"496","article-title":"The magneto-hydrodynamic boundary layer in two-dimensional steady flow past a semi-infinite flat plate I. Uniform conditions at infinity","volume":"273","author":"Davies","year":"1962","journal-title":"Proc. R. Soc. A"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1016\/j.ijthermalsci.2009.07.015","article-title":"Natural convective boundary-layer flow of a nanofluid past a vertical plate","volume":"49","author":"Kuznetsov","year":"2010","journal-title":"Int. J. Therm. Sci."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1007\/s11242-008-9309-6","article-title":"Mixed convection boundary-layer flow near the stagnation point on a vertical surface in a porous medium: Brinkman model with slip","volume":"77","author":"Harris","year":"2009","journal-title":"Transp. Porous Media"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"68","DOI":"10.1016\/j.icheatmasstransfer.2013.06.005","article-title":"Magnetohydrodynamic stagnation-point flow towards a stretching\/shrinking sheet with slip effects","volume":"47","author":"Aman","year":"2013","journal-title":"Int. Commun. Heat Mass Transf."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"308","DOI":"10.1016\/j.ijheatmasstransfer.2010.09.041","article-title":"Slip effects on boundary layer stagnation-point flow and heat transfer towards a shrinking sheet","volume":"54","author":"Bhattacharyya","year":"2011","journal-title":"Int. J. Heat Mass Transf."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/9\/1078\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:14:39Z","timestamp":1760188479000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/9\/1078"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,8,28]]},"references-count":56,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2019,9]]}},"alternative-id":["sym11091078"],"URL":"https:\/\/doi.org\/10.3390\/sym11091078","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,8,28]]}}}