{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,6]],"date-time":"2026-02-06T03:35:43Z","timestamp":1770348943149,"version":"3.49.0"},"reference-count":42,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2016,5,24]],"date-time":"2016-05-24T00:00:00Z","timestamp":1464048000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this article, entropy generation with radiation on non-Newtonian Carreau nanofluid towards a shrinking sheet is investigated numerically. The effects of magnetohydrodynamics (MHD) are also taken into account. Firstly, the governing flow problem is simplified into ordinary differential equations from partial differential equations with the help of similarity variables. The solution of the resulting nonlinear differential equations is solved numerically with the help of the successive linearization method and Chebyshev spectral collocation method. The influence of all the emerging parameters is discussed with the help of graphs and tables. It is observed that the influence of magnetic field and fluid parameters oppose the flow. It is also analyzed that thermal radiation effects and the Prandtl number show opposite behavior on temperature profile. Furthermore, it is also observed that entropy profile increases for all the physical parameters.<\/jats:p>","DOI":"10.3390\/e18060200","type":"journal-article","created":{"date-parts":[[2016,5,24]],"date-time":"2016-05-24T21:53:40Z","timestamp":1464126820000},"page":"200","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":105,"title":["Numerical Simulation of Entropy Generation with Thermal Radiation on MHD Carreau Nanofluid towards a Shrinking Sheet"],"prefix":"10.3390","volume":"18","author":[{"given":"Muhammad","family":"Bhatti","sequence":"first","affiliation":[{"name":"Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China"}]},{"given":"Tehseen","family":"Abbas","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Quaid-I-Azam University, Islamabad 44000, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6309-8688","authenticated-orcid":false,"given":"Mohammad","family":"Rashidi","sequence":"additional","affiliation":[{"name":"Shanghai Key Lab of Vehicle Aerodynamics and Vehicle Thermal Management Systems, Tongji University, Shanghai 201804, China"},{"name":"ENN-Tongji Clean Energy Institute of advanced studies, Shanghai 200072, Tongji University, China"}]},{"given":"Mohamed","family":"Ali","sequence":"additional","affiliation":[{"name":"Mechanical Engineering Department, College of Engineering, King Saud University, P. O. Box 800, Riyadh 11421, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2016,5,24]]},"reference":[{"key":"ref_1","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_2","doi-asserted-by":"crossref","first-page":"6786","DOI":"10.1016\/j.ijheatmasstransfer.2012.06.086","article-title":"A new frontier of nanofluid research\u2013application of nanofluids in heat pipes","volume":"55","author":"Liu","year":"2012","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"72","DOI":"10.1016\/j.jmmm.2016.02.075","article-title":"Magnetic field analysis in a suspension of gyrotactic microorganisms and nanoparticles over a stretching surface","volume":"410","author":"Akbar","year":"2016","journal-title":"J. Magn. Magn. Mater."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1016\/j.ijheatmasstransfer.2015.08.013","article-title":"Unsteady flow of nanofluid with double stratification and magnetohydrodynamics","volume":"92","author":"Hayat","year":"2016","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"726","DOI":"10.1109\/TNANO.2015.2435899","article-title":"Study of natural convection MHD nanofluid by means of single and multi-walled carbon nanotubes suspended in a salt-water solution","volume":"14","author":"Ellahi","year":"2015","journal-title":"IEEE Trans. Nanotechnol."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Ellahi, R., Hassan, M., Zeeshan, A., and Khan, A.A. (2015). The shape effects of nanoparticles suspended in HFE-7100 over wedge with entropy generation and mixed convection. Appl. Nanosci., 1\u201311.","DOI":"10.1007\/s13204-015-0481-z"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Garoosi, F., Hoseininejad, F., and Rashidi, M.M. (2016). Numerical study of heat transfer performance of nanofluids in a heat exchanger. Appl. Thermal Eng.","DOI":"10.1016\/j.applthermaleng.2016.03.015"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1016\/j.ijthermalsci.2012.07.008","article-title":"Natural convection heat transfer inside vertical circular enclosure filled with water-based Al2O3 nanofluids","volume":"63","author":"Ali","year":"2013","journal-title":"Int. J. Therm. Sci."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1289","DOI":"10.1080\/01457632.2013.793115","article-title":"The effect of Alumina-water nanofluid on natural convection heat transfer inside vertical circular enclosure heated from above","volume":"34","author":"Ali","year":"2013","journal-title":"Heat Transf. Eng."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1186\/1556-276X-7-139","article-title":"Nanofluid impingement jet heat transfer","volume":"7","author":"Zeitoun","year":"2012","journal-title":"Nanoscale Res. Lett."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"154","DOI":"10.1080\/15567265.2012.749963","article-title":"The effect of particle concentration on cooling of a circular horizontal surface using nanofluid jets","volume":"17","author":"Zeitoun","year":"2013","journal-title":"Nanoscale Microscale Thermophys. Eng."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"164","DOI":"10.1504\/IJNP.2009.028749","article-title":"Nanofluids forced convection heat transfer inside circular tubes","volume":"2","author":"Ali","year":"2009","journal-title":"Int. J. Nanopart."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2109","DOI":"10.1016\/j.applthermaleng.2008.01.005","article-title":"Thermophysical and electrokinetic properties of nanofluids\u2014A critical review","volume":"28","author":"Murshed","year":"2008","journal-title":"Appl. Thermal Eng."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"718","DOI":"10.1590\/S0103-97332005000400018","article-title":"Ferrofluids: Properties and applications","volume":"35","author":"Scherer","year":"2005","journal-title":"Braz. J. Phys."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"3280","DOI":"10.1063\/1.858685","article-title":"The use of magnetohydrodynamic effects to investigate fluid flow in electrically conducting melts","volume":"5","author":"Baumgartl","year":"1993","journal-title":"Phys. Fluids A Fluid Dyn. (1989\u20131993)"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"805","DOI":"10.1002\/cnm.1420","article-title":"Magnetohydrodynamic biorheological transport phenomena in a porous medium: A simulation of magnetic blood flow control and filtration","volume":"27","author":"Rashidi","year":"2011","journal-title":"Int. J. Numer. Methods Biomed. Eng."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"917","DOI":"10.1016\/j.snb.2007.10.064","article-title":"MEMS-based micropumps in drug delivery and biomedical applications","volume":"130","author":"Nisar","year":"2008","journal-title":"Sens. Actuators B Chem."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"674","DOI":"10.1016\/j.ijheatmasstransfer.2015.10.014","article-title":"Non-uniform heat source\/sink and Soret effects on MHD non-Darcian convective flow past a stretching sheet in a micropolar fluid with radiation","volume":"93","author":"Mabood","year":"2016","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"901","DOI":"10.1016\/j.asej.2014.02.007","article-title":"Free convective heat and mass transfer for MHD fluid flow over a permeable vertical stretching sheet in the presence of the radiation and buoyancy effects","volume":"5","author":"Rashidi","year":"2014","journal-title":"Ain Shams Eng. J."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"549","DOI":"10.1016\/j.molliq.2015.12.110","article-title":"Effect of magnetic dipole on viscous ferro-fluid past a stretching surface with thermal radiation","volume":"215","author":"Zeeshan","year":"2016","journal-title":"J. Mol. Liq."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1016\/j.jmmm.2016.01.087","article-title":"MHD heat and mass transfer flow over a permeable stretching\/shrinking sheet with radiation effect","volume":"407","author":"Yasin","year":"2016","journal-title":"J. Magn. Magn. Mater."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"216","DOI":"10.1016\/j.molliq.2016.02.071","article-title":"Analytical study for unsteady nanofluid MHD Flow impinging on heated stretching sheet","volume":"219","author":"Mabood","year":"2016","journal-title":"J. Mol. Liq."},{"key":"ref_23","first-page":"227","article-title":"Unsteady MHD radiative flow and heat transfer of a dusty nanofluid over an exponentially stretching surface","volume":"19","author":"Sandeep","year":"2016","journal-title":"Eng. Sci. Technol. Int. J."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1016\/j.amc.2015.07.038","article-title":"Approximate solutions for steady boundary layer MHD viscous flow and radiative heat transfer over an exponentially porous stretching sheet","volume":"269","author":"Ene","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"892","DOI":"10.1016\/j.ijheatmasstransfer.2013.11.058","article-title":"Investigation of entropy generation in MHD and slip flow over a rotating porous disk with variable properties","volume":"70","author":"Rashidi","year":"2014","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1","DOI":"10.3390\/e18040123","article-title":"Entropy Generation on MHD Casson Nanofluid Flow over a Porous Stretching\/Shrinking Surface","volume":"18","author":"Qing","year":"2016","journal-title":"Entropy"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"3245","DOI":"10.1016\/j.ijheatmasstransfer.2004.02.005","article-title":"Magnetohydrodynamic free convection and entropy generation in a square porous cavity","volume":"47","author":"Mahmud","year":"2004","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"300","DOI":"10.1016\/S1164-0235(02)00065-1","article-title":"Entropy generation in a porous channel with hydromagnetic effect","volume":"2","author":"Tasnim","year":"2002","journal-title":"Exergy"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1140\/epjp\/i2013-13051-y","article-title":"Entropy effects in hydromagnetic free convection flow past a vertical plate embedded in a porous medium in the presence of thermal radiation","volume":"128","author":"Butt","year":"2013","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"786","DOI":"10.1080\/10407782.2012.672890","article-title":"Analysis of the magnetic effect on entropy generation in an inclined channel partially filled with porous medium","volume":"61","author":"Komurgoz","year":"2012","journal-title":"Num. Heat Transf. Part A"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"90","DOI":"10.3390\/e18030090","article-title":"Analysis of Entropy Generation in the Flow of Peristaltic Nanofluids in Channels with Compliant Walls","volume":"18","author":"Abbas","year":"2016","journal-title":"Entropy"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1411","DOI":"10.3390\/e17031411","article-title":"Entropy generation analysis for a CNT suspension nanofluid in plumb ducts with peristalsis","volume":"17","author":"Akbar","year":"2015","journal-title":"Entropy"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1155\/2014\/413213","article-title":"Parametric analysis of entropy generation in magneto-hemodynamic flow in a semi-porous channel with OHAM and DTM","volume":"11","author":"Rashidi","year":"2014","journal-title":"Appl. Bion. Biomech."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"117","DOI":"10.3390\/e18040117","article-title":"Entropy Generation on MHD Blood Flow of Nanofluid Due to Peristaltic Waves","volume":"18","author":"Rashidi","year":"2016","journal-title":"Entropy"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"396","DOI":"10.1016\/j.ijheatmasstransfer.2014.11.066","article-title":"Three-dimensional natural convection in a porous enclosure filled with a nanofluid using Buongiorno\u2019s mathematical model","volume":"82","author":"Sheremet","year":"2015","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"66","DOI":"10.1016\/j.icheatmasstransfer.2015.07.007","article-title":"Unsteady free convection in a porous open wavy cavity filled with a nanofluid using Buongiorno's mathematical model","volume":"67","author":"Sheremet","year":"2015","journal-title":"Int. Commun. Heat Mass Transf."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1016\/j.jtice.2015.12.015","article-title":"Magnetic field effect on the unsteady natural convection in a wavy-walled cavity filled with a nanofluid: Buongiorno\u2019s mathematical model","volume":"61","author":"Sheremet","year":"2016","journal-title":"J. Taiwan Inst. Chem. Eng."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Sheremet, M.A., Oztop, H.F., Pop, I., and Abu-Hamdeh, N. (2015). Analysis of Entropy Generation in Natural Convection of Nanofluid inside a Square Cavity Having Hot Solid Block: Tiwari and Das\u2019 Model. Entropy, 18.","DOI":"10.3390\/e18010009"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/j.aej.2016.01.015","article-title":"Numerical simulation of Fluid flow over a shrinking porous sheet by Successive linearization method","volume":"55","author":"Bhatti","year":"2016","journal-title":"Alex. Eng. J."},{"key":"ref_40","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_41","doi-asserted-by":"crossref","first-page":"622","DOI":"10.1016\/j.ijnonlinmec.2006.03.002","article-title":"Non-orthogonal stagnation point flow towards a stretching sheet","volume":"41","author":"Lok","year":"2006","journal-title":"Int. J. Non-Linear Mech."},{"key":"ref_42","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."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/18\/6\/200\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T19:24:24Z","timestamp":1760210664000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/18\/6\/200"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,5,24]]},"references-count":42,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2016,6]]}},"alternative-id":["e18060200"],"URL":"https:\/\/doi.org\/10.3390\/e18060200","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,5,24]]}}}