{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,4]],"date-time":"2026-06-04T21:14:23Z","timestamp":1780607663819,"version":"3.54.1"},"reference-count":55,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2020,4,2]],"date-time":"2020-04-02T00:00:00Z","timestamp":1585785600000},"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 present work examines the effect of different magnetic nanoparticles and the heat transfer phenomena over the stretching sheet with thermal stratification and slips effect. The mixture of water (H     2    O) and ethylene glycol (C     2    H     6    O     2    ) is used as base fluid whereas the paramagnetic, diamagnetic, and ferromagnetic ferrites are taken as nanoparticles. In the presence of ferrite nanoparticles, the magnetic dipole has a significant effect in controlling the rate of heat transfer and the thermal boundary layers. By using suitable similarity transformations, the system of partial differential equations is transformed into nonlinear ordinary differential equations. The numerical solution of resulting equations is found out by using the variational finite element method. The effect of numerous emerging parameters on velocity, temperature, and micro-rotation velocity are represented graphically and analyzed numerically. It has been noticed that comparatively the diamagnetic ferrites have gained maximum thermal conductivity relative to the other nanoparticles. It was also observed that the thermal conduction of nanoparticles increases with the variation of volume fraction. Moreover, with increasing values of thermal stratification the thermal boundary layer thickness decreases and the heat transfer rate increases at the surface. Furthermore, the validation of code and the accuracy of the numerical technique has been confirmed by the assessment of current results with earlier studies.<\/jats:p>","DOI":"10.3390\/sym12040520","type":"journal-article","created":{"date-parts":[[2020,4,2]],"date-time":"2020-04-02T13:39:34Z","timestamp":1585834774000},"page":"520","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":51,"title":["The Impact of Nanoparticles Due to Applied Magnetic Dipole in Micropolar Fluid Flow Using the Finite Element Method"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1500-0463","authenticated-orcid":false,"given":"Liaqat","family":"Ali","sequence":"first","affiliation":[{"name":"School of Energy and Power, Xi\u2019an Jiaotong University, No. 28, Xianning West Road, Xi\u2019an 710049, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiaomin","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Energy and Power, Xi\u2019an Jiaotong University, No. 28, Xianning West Road, Xi\u2019an 710049, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5501-4181","authenticated-orcid":false,"given":"Bagh","family":"Ali","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Northwestern Polytechnical University, Dongxiang Road, Beilin District, Xi\u2019an 710129, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Saima","family":"Mujeed","sequence":"additional","affiliation":[{"name":"School of Management, Xi\u2019an Jiaotong University, No. 28, Xianning West Road, Xi\u2019an 710049, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7724-4340","authenticated-orcid":false,"given":"Sohaib","family":"Abdal","sequence":"additional","affiliation":[{"name":"School of Mathematics, Northwest University, No. 229 North Taibai Avenue, Xi\u2019an 7100069, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ali","family":"Mutahir","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering, School of Automation, Northwestern Polytechnical University, Xi\u2019an 710072, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2020,4,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1016\/j.icheatmasstransfer.2015.04.006","article-title":"Experimental investigation and development of new correlations for thermal conductivity of CuO\/EG\u2013water nanofluid","volume":"65","author":"Esfe","year":"2015","journal-title":"Int. Commun. Heat Mass Transf."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1089","DOI":"10.1007\/s11051-007-9352-1","article-title":"Thermal conductance of nanofluids: Is the controversy over?","volume":"10","author":"Keblinski","year":"2008","journal-title":"J. Nanoparticle Res."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"629","DOI":"10.1016\/j.rser.2009.10.004","article-title":"Enhancement of heat transfer using nanofluids\u2014An overview","volume":"14","author":"Godson","year":"2010","journal-title":"Renew. Sustain. Energy Rev."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1016\/j.physe.2015.04.017","article-title":"Ferromagnetic CNT suspended H2O+Cu nanofluid analysis through composite stenosed arteries with permeable wall","volume":"72","author":"Akbar","year":"2015","journal-title":"Phys. E Low-Dimens. Syst. Nanostruct."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2532","DOI":"10.1039\/b815548h","article-title":"Magnetic nanoparticles: Synthesis, functionalization, and applications in bioimaging and magnetic energy storage","volume":"38","author":"Frey","year":"2009","journal-title":"Chem. Soc. Rev."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"3705","DOI":"10.3390\/ijms12063705","article-title":"Application of magnetic nanoparticles to gene delivery","volume":"12","author":"Kami","year":"2011","journal-title":"Int. J. Mol. Sci."},{"key":"ref_7","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":"Z. Angew. Math. Phys. ZAMP"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2413","DOI":"10.1007\/s10973-019-08435-x","article-title":"Thermal analysis and thermo-hydraulic characteristics of zirconia\u2013water nanofluid under a convective boiling regime","volume":"139","author":"Sarafraz","year":"2020","journal-title":"J. Therm. Anal. Calorim."},{"key":"ref_9","unstructured":"Odenbach, S. (2008). Ferrofluids: Magnetically Controllable Fluids and Their Applications, Springer."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1016\/0020-7462(66)90025-4","article-title":"Some viscous flows of a saturated ferro-fluid under the combined influence of thermal and magnetic field gradients","volume":"1","author":"Neuringer","year":"1966","journal-title":"Int. J. Non-Linear Mech."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"494","DOI":"10.1515\/zna-2005-0705","article-title":"Effect of dust particles on thermal convection in a ferromagnetic fluid","volume":"60","author":"Sharma","year":"2005","journal-title":"Z. Naturforschung A"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"922","DOI":"10.1088\/0370-1298\/63\/8\/122","article-title":"The mechanism of colloid agglomeration in the formation of Bitter patterns","volume":"63","author":"Mee","year":"1950","journal-title":"Proc. Phys. Soc. Sect. A"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2361","DOI":"10.1016\/j.rinp.2017.06.007","article-title":"Mathematical analysis of ferromagnetic fluid embedded in a porous medium","volume":"7","author":"Nadeem","year":"2017","journal-title":"Results Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"494","DOI":"10.1016\/j.physleta.2016.11.042","article-title":"Numerical simulation of magnetic nanofluid natural convection in porous media","volume":"381","author":"Sheikholeslami","year":"2017","journal-title":"Phys. Lett. A"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"458","DOI":"10.1016\/j.joems.2015.06.001","article-title":"Finite element analysis of heat and mass transfer by MHD mixed convection stagnation-point flow of a non-Newtonian power-law nanofluid towards a stretching surface with radiation","volume":"24","author":"Madhu","year":"2016","journal-title":"J. Egypt. Math. Soc."},{"key":"ref_16","first-page":"3","article-title":"Ferrofluid Flow in the Presence of Magnetic Dipole","volume":"39","year":"2019","journal-title":"Tech. Mech."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Ali, B., Yu, X., Sadiq, M.T., Rehman, A.U., and Ali, L. (2020). A Finite Element Simulation of the Active and Passive Controls of the MHD Effect on an Axisymmetric Nanofluid Flow with Thermo-Diffusion over a Radially Stretched Sheet. Processes, 8.","DOI":"10.3390\/pr8020207"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1016\/j.expthermflusci.2005.03.021","article-title":"Experimental investigations on transport properties of magnetic fluids","volume":"30","author":"Li","year":"2005","journal-title":"Exp. Therm. Fluid Sci."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Ali, L., Liu, X., Ali, B., Mujeed, S., Abdal, S., and Khan, S.A. (2020). Analysis of Magnetic Properties of Nano-Particles Due to a Magnetic Dipole in Micropolar Fluid Flow over a Stretching Sheet. Coatings, 10.","DOI":"10.3390\/coatings10020170"},{"key":"ref_20","first-page":"674","article-title":"Heat and mass transfer in MHD flow of nanofluids through a porous media due to a permeable stretching sheet with viscous dissipation and chemical reaction effects","volume":"9","author":"Yirga","year":"2015","journal-title":"Int. J. Mech. Aerospace Ind. Mech. Manuf. Eng."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"423","DOI":"10.1016\/j.molliq.2016.10.006","article-title":"Impact of stratification and Cattaneo-Christov heat flux in the flow saturated with porous medium","volume":"224","author":"Nadeem","year":"2016","journal-title":"J. Mol. Liquids"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"854","DOI":"10.1016\/j.rinp.2016.12.027","article-title":"Heat transport phenomenon in the ferromagnetic fluid over a stretching sheet with thermal stratification","volume":"7","author":"Muhammad","year":"2017","journal-title":"Results Phys."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Ali, L., Liu, X., Ali, B., Mujeed, S., and Abdal, S. (2019). Finite Element Simulation of Multi-Slip Effects on Unsteady MHD Bioconvective Micropolar nanofluid Flow Over a Sheet with Solutal and Thermal Convective Boundary Conditions. Coatings, 9.","DOI":"10.3390\/coatings9120842"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"66","DOI":"10.1007\/s42452-019-1831-3","article-title":"Multiple slip effects on MHD unsteady viscoelastic nano-fluid flow over a permeable stretching sheet with radiation using the finite element method","volume":"2","author":"Khan","year":"2020","journal-title":"SN Appl. Sci."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1016\/j.ijnonlinmec.2016.04.004","article-title":"Flow of a micropolar fluid due to a porous stretching sheet and heat transfer","volume":"83","author":"Turkyilmazoglu","year":"2016","journal-title":"Int. J. Non-Linear Mech."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1108\/09615531311293470","article-title":"Series solution for heat transfer of continuous stretching sheet immersed in a micropolar fluid in the existence of radiation","volume":"23","author":"Shadloo","year":"2013","journal-title":"Int. J. Numer. Methods Heat Fluid Flow"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2613","DOI":"10.1002\/htj.21517","article-title":"Free convective MHD micropolar fluid flow with thermal radiation and radiation absorption: A numerical study","volume":"48","author":"Pradhan","year":"2019","journal-title":"Heat Transf. Asian Res."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"278","DOI":"10.1016\/j.powtec.2019.07.053","article-title":"Thermo-hydraulic performance of a biological nanofluid containing graphene nanoplatelets within a tube enhanced with rotating twisted tape","volume":"355","author":"Bahiraei","year":"2019","journal-title":"Powder Technol."},{"key":"ref_29","first-page":"1","article-title":"Theory of micropolar fluids","volume":"16","author":"Eringen","year":"1966","journal-title":"J. Math. Mech."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"639","DOI":"10.1016\/0020-7225(76)90006-9","article-title":"Self-similar solution of imcompressible micropolar boundary layer flow over a semi-infinite plate","volume":"14","author":"Ahmadi","year":"1976","journal-title":"Int. J. Eng. Sci."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1016\/0020-7225(83)90036-8","article-title":"Micropolar boundary layer flow at a stagnation point on a moving wall","volume":"21","author":"Gorla","year":"1983","journal-title":"Int. J. Eng. Sci."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.compfluid.2013.01.014","article-title":"MHD boundary layer flow and heat transfer of a nanofluid past a permeable stretching sheet with velocity, thermal and solutal slip boundary conditions","volume":"75","author":"Ibrahim","year":"2013","journal-title":"Comput. Fluids"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1016\/j.compfluid.2012.04.026","article-title":"Slip flow and convective heat transfer of nanofluids over a permeable stretching surface","volume":"64","author":"Das","year":"2012","journal-title":"Comput. Fluids"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"443","DOI":"10.1002\/fld.1825","article-title":"Slip effects and heat transfer analysis in a viscous fluid over an oscillatory stretching surface","volume":"59","author":"Abbas","year":"2009","journal-title":"Int. J. Numer. Methods Fluids"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Abdal, S., Ali, B., Younas, S., Ali, L., and Mariam, A. (2020). Thermo-Diffusion and Multislip Effects on MHD Mixed Convection Unsteady Flow of Micropolar Nanofluid over a Shrinking\/Stretching Sheet with Radiation in the Presence of Heat Source. Symmetry, 12.","DOI":"10.3390\/sym12010049"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"591","DOI":"10.18869\/acadpub.jafm.67.222.21973","article-title":"Ferromagnetic Liquid Flow due to Gravity-Aligned Stretching of an Elastic Sheet","volume":"8","author":"Titus","year":"2015","journal-title":"J. Appl. Fluid Mech."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"2002","DOI":"10.1016\/j.ijheatmasstransfer.2006.09.034","article-title":"Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids","volume":"50","author":"Tiwari","year":"2007","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1007\/BF01463158","article-title":"Flow of a heated ferrofluid over a stretching sheet in the presence of a magnetic dipole","volume":"128","author":"Andersson","year":"1998","journal-title":"Acta Mech."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1007\/s10973-018-7844-0","article-title":"Thermodynamic analysis of fly ash nanofluid for automobile (heavy vehicle) radiators","volume":"136","author":"Palaniappan","year":"2019","journal-title":"J. Therm. Anal. Calorim."},{"key":"ref_40","unstructured":"Domkundwar, A., and Domkundwar, V. (2007). Heat and Mass Transfer Data Book, Dhanpat Rai."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Ali, B., Naqvi, R.A., Nie, Y., Khan, S.A., Sadiq, M.T., Rehman, A.U., and Abdal, S. (2020). Variable Viscosity Effects on Unsteady MHD an Axisymmetric Nanofluid Flow over a Stretching Surface with Thermo-Diffusion: FEM Approach. Symmetry, 12.","DOI":"10.3390\/sym12020234"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"202","DOI":"10.1134\/S1810232818020078","article-title":"Finite element analysis of MHD flow of micropolar fluid over a shrinking sheet with a convective surface boundary condition","volume":"27","author":"Gupta","year":"2018","journal-title":"J. Eng. Thermophys."},{"key":"ref_43","first-page":"29","article-title":"Finite element solution to mixed convection in MHD flow of micropolar fluid along a moving vertical cylinder with variable conductivity","volume":"7","author":"Bhargava","year":"2011","journal-title":"Int. J. Appl. Math. Mech."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1640","DOI":"10.1166\/jon.2019.1721","article-title":"Stratification and Buoyancy Effect of Heat Transportation in Magnetohydrodynamics Micropolar Fluid Flow Passing over a Porous Shrinking Sheet Using the Finite Element Method","volume":"8","author":"Khan","year":"2019","journal-title":"J. Nanofluids"},{"key":"ref_45","unstructured":"Reddy, J.N. (1993). Solutions Manual for an Introduction to the Finite Element Method, McGraw-Hill."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1016\/j.jtice.2014.10.005","article-title":"Finite element modeling of a double-diffusive mixed convection flow of a chemically-reacting magneto-micropolar fluid with convective boundary condition","volume":"47","author":"Swapna","year":"2015","journal-title":"J. Taiwan Inst. Chem. Eng."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1615\/ComputThermalScien.2014008401","article-title":"Finite-element analysis of transient heat and mass transfer in microstructural boundary layer flow from a porous stretching sheet","volume":"6","author":"Gupta","year":"2014","journal-title":"Comput. Therm. Sci. Int. J."},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Ali, L., Liu, X., Ali, B., Mujeed, S., and Abdal, S. (2019). Finite Element Analysis of Thermo-Diffusion and Multi-Slip Effects on MHD Unsteady Flow of Casson Nano-Fluid over a Shrinking\/Stretching Sheet with Radiation and Heat Source. Appl. Sci., 9.","DOI":"10.3390\/app9235217"},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Ali, B., Nie, Y., Khan, S.A., Sadiq, M.T., and Tariq, M. (2019). Finite Element Simulation of Multiple Slip Effects on MHD Unsteady Maxwell Nanofluid Flow over a Permeable Stretching Sheet with Radiation and Thermo-Diffusion in the Presence of Chemical Reaction. Processes, 7.","DOI":"10.3390\/pr7090628"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"528","DOI":"10.1016\/j.molliq.2016.07.145","article-title":"Unsteady ferromagnetic liquid flow and heat transfer analysis over a stretching sheet with the effect of dipole and prescribed heat flux","volume":"223","author":"Majeed","year":"2016","journal-title":"J. Mol. Liq."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"935","DOI":"10.1007\/s11012-010-9353-5","article-title":"Flow and heat transfer over an unsteady stretching sheet in a micropolar fluid","volume":"46","author":"Bachok","year":"2011","journal-title":"Meccanica"},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Qasim, M., Khan, I., and Shafie, S. (2013). Heat transfer in a micropolar fluid over a stretching sheet with Newtonian heating. PLoS ONE, 8.","DOI":"10.1371\/journal.pone.0059393"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1016\/j.molliq.2018.01.138","article-title":"Microstructure and inertial characteristics of a magnetite ferrofluid over a stretching\/shrinking sheet using effective thermal conductivity model","volume":"255","author":"Hussanan","year":"2018","journal-title":"J. Mol. Liq."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"841","DOI":"10.1016\/j.commatsci.2009.04.021","article-title":"Finite element analysis of combined heat and mass transfer in hydromagnetic micropolar flow along a stretching sheet","volume":"46","author":"Kumar","year":"2009","journal-title":"Comput. Mater. Sci."},{"key":"ref_55","doi-asserted-by":"crossref","unstructured":"Muhammad, N., Nadeem, S., and Mustafa, M. (2018). Analysis of ferrite nanoparticles in the flow of ferromagnetic nanofluid. PLoS ONE, 13.","DOI":"10.1371\/journal.pone.0188460"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/4\/520\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:14:45Z","timestamp":1760174085000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/4\/520"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,4,2]]},"references-count":55,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2020,4]]}},"alternative-id":["sym12040520"],"URL":"https:\/\/doi.org\/10.3390\/sym12040520","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,4,2]]}}}