{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,12]],"date-time":"2025-11-12T03:27:37Z","timestamp":1762918057378,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2020,3,6]],"date-time":"2020-03-06T00:00:00Z","timestamp":1583452800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11271037"],"award-info":[{"award-number":["11271037"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The current study aims to probe the impacts of entropy in a hydromagnetic unsteady slip flow of viscous fluid past an exponentially stretching sheet. Appurtenant similarity variables are employed to transmute the governing partial differential equations into a system of non-linear differential equations, which are analytically solved by utilizing the homotopy analysis method (HAM). Moreover, a shooting technique with fourth\u2013fifth order Runge\u2013Kutta method is deployed to numerically solve the problem. The impact of the physical parameters that influence the flow and heat transmission phenomena are sketched, tabulated and discussed briefly. Additionally, the impact of these parameters on entropy generation is thoroughly discussed by plotting graphs of the local entropy generation number and the Bejan number.<\/jats:p>","DOI":"10.3390\/sym12030426","type":"journal-article","created":{"date-parts":[[2020,3,6]],"date-time":"2020-03-06T09:26:41Z","timestamp":1583486801000},"page":"426","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":16,"title":["Study of Entropy Generation with Multi-Slip Effects in MHD Unsteady Flow of Viscous Fluid Past an Exponentially Stretching Surface"],"prefix":"10.3390","volume":"12","author":[{"given":"Sajjad","family":"Haider","sequence":"first","affiliation":[{"name":"College of Applied Science, Beijing University of Technology, Beijing 100124, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Adnan","family":"Saeed Butt","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Govt postgraduate Gordon College, Rawalpindi 46000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yun-Zhang","family":"Li","sequence":"additional","affiliation":[{"name":"College of Applied Science, Beijing University of Technology, Beijing 100124, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Syed Muhammad","family":"Imran","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Govt postgraduate Gordon College, Rawalpindi 46000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Babar","family":"Ahmad","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Comsats University, Islamabad 44000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Asia","family":"Tayyaba","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Islamabad Campus, Preston University, Kohat 26000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,3,6]]},"reference":[{"key":"ref_1","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."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"744","DOI":"10.1002\/cjce.5450550619","article-title":"Heat and mass transfer on a stretching sheet with suction or blowing","volume":"55","author":"Gupta","year":"1977","journal-title":"Can. J. Chem. Eng."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"248","DOI":"10.1115\/1.3247387","article-title":"Heat transfer characteristics of a continuous stretching surface with variable temperature","volume":"107","author":"Grubka","year":"1985","journal-title":"Trans. ASME J. Heat Trans."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"568","DOI":"10.1016\/0022-247X(88)90172-2","article-title":"Heat transfer of a continuous, stretching surface with suction or blowing","volume":"135","author":"Chen","year":"1988","journal-title":"J. Math. Anal. Appl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1951","DOI":"10.1088\/0022-3727\/31\/16\/002","article-title":"Heat transfer over a stretching surface with variable surface heat flux","volume":"31","author":"Elbashbeshy","year":"1998","journal-title":"J. Phys. D Appl. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"577","DOI":"10.1088\/0022-3727\/32\/5\/012","article-title":"Heat and mass transfer in the boundary layers on an exponentially stretching continuous surface","volume":"32","author":"Magyari","year":"1999","journal-title":"J. Phys. D Appl. Phys."},{"key":"ref_7","first-page":"643","article-title":"Heat transfer over an exponentially stretching continuous surface with suction","volume":"53","author":"Elbashbeshy","year":"2001","journal-title":"Arch. Mech."},{"key":"ref_8","first-page":"289","article-title":"Thermal boundary layer on an exponentially stretching continuous surface in the presence of magnetic field effect","volume":"11","author":"Damseh","year":"2006","journal-title":"Int. J. Appl. Mech. Eng."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1139\/P09-047","article-title":"Viscous dissipation effect on mixed convection flow of a micro polar fluid over an exponentially stretching sheet","volume":"87","year":"2009","journal-title":"Can. J. Phys."},{"key":"ref_10","first-page":"710","article-title":"Numerical solution of the boundary layer flow over an exponentially stretching sheet with thermal radiation","volume":"4","author":"Bidin","year":"2009","journal-title":"Eur. J. Sci. Res."},{"key":"ref_11","first-page":"391","article-title":"MHD Boundary layer flow due to an exponentially stretching sheet with radiation effect","volume":"40","author":"Ishak","year":"2011","journal-title":"Sains Malays."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1773","DOI":"10.1007\/s00231-012-1024-8","article-title":"Effects of partial slip on boundary layer flow past a permeable exponential stretching sheet in presence of thermal radiation","volume":"48","author":"Mukhopadhyay","year":"2012","journal-title":"Heat Mass Transf."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"5010","DOI":"10.1016\/j.ijheatmasstransfer.2011.07.015","article-title":"Flow and heat transfer of a third grade fluid past an exponentially stretching sheet with partial slip boundary condition","volume":"54","author":"Sahoo","year":"2011","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2728","DOI":"10.1016\/j.cnsns.2011.11.011","article-title":"Stagnation-point flow and heat transfer over an exponentially shrinking sheet","volume":"17","author":"Bhattacharyya","year":"2012","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"054502","DOI":"10.1115\/1.4023618","article-title":"Casson Fluid Flow and Heat Transfer at an Exponentially Stretching Permeable Surface","volume":"80","author":"Mukhopadhyay","year":"2013","journal-title":"J. Appl. Mech."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"340","DOI":"10.1016\/j.proeng.2015.11.379","article-title":"Sarojamma, Study of thermophoresis on the MHD flow due to an exponentially stretching sheet in the presence of viscous dissipation","volume":"127","author":"Jyothi","year":"2015","journal-title":"Proc. Eng."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"593","DOI":"10.1016\/j.asej.2015.12.010","article-title":"Magnetohydrodynamic flow of a Casson fluid over an exponentially inclined permeable stretching surface with thermal radiation and chemical reaction","volume":"7","author":"Reddy","year":"2016","journal-title":"Ain Shams Eng. J."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"697","DOI":"10.1016\/j.asej.2015.10.012","article-title":"Non-similar solutions of mixed convection flow from an exponentially stretching surface","volume":"8","author":"Patil","year":"2017","journal-title":"Ain Shams Eng. J."},{"key":"ref_19","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_20","unstructured":"Bejan, A. (1996). Entropy Generation Minimization, CRC."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"661","DOI":"10.1007\/s11242-010-9584-x","article-title":"Thermal analysis of flow in a porous medium over a permeable stretching wall","volume":"85","author":"Tamayol","year":"2010","journal-title":"Transp. Porous Med."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"482","DOI":"10.1016\/j.ijnonlinmec.2010.01.007","article-title":"Entropy analysis for viscoelastic magnetohydrodynamic flow over a stretching surface","volume":"45","author":"Saouli","year":"2010","journal-title":"Int. J. Non-Linear Mech."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"451","DOI":"10.5560\/zna.2012-0055","article-title":"Entropy analysis of mixed convective magnetohydrodynamic flow of a viscoelastic fluid over a stretching sheet","volume":"67","author":"Butt","year":"2012","journal-title":"Z. Naturforsch"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1504\/IJEX.2013.055775","article-title":"Slip effects on entropy generation in MHD flow over a stretching surface in the presence of thermal radiation","volume":"13","author":"Butt","year":"2013","journal-title":"Int. J. Exergy"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"927","DOI":"10.1007\/s12206-013-0104-0","article-title":"Entropy analysis for nanofluid flow over a stretching sheet in the presence of heat generation\/absorption and partial slip","volume":"27","author":"Noghrehabadi","year":"2013","journal-title":"J. Mech. Sci. Technol."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1140\/epjp\/i2014-14013-7","article-title":"A computational study of entropy generation in magnetohydrodynamic flow and heat transfer over an unsteady stretching permeable sheet","volume":"129","author":"Butt","year":"2014","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"753","DOI":"10.18869\/acadpub.jafm.67.223.22916","article-title":"Tongji University Entropy Generation Analysis for Stagnation Point Flow in a Porous Medium over a Permeable Stretching Surface","volume":"8","author":"Rashidi","year":"2015","journal-title":"J. Appl. Fluid Mech."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Afridi, M.I., Qasim, M., Khan, I., and Shafie, S. (2017). Entropy generation in magnetohydrodynamic mixed convection flow over an inclined stretching sheet. Entropy, 19.","DOI":"10.3390\/e19010010"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1166\/jon.2019.1570","article-title":"Theoretical Investigation of Entropy Generation Effects in Magnetohydrodynamic Flow of Casson Nanofluid over an Unsteady Permeable Stretching Surface","volume":"8","author":"Butt","year":"2019","journal-title":"J. Nanofluids"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Riaz, A., Bhatti, M.M., Ellahi, R., Zeeshan, A., and Sait, S.M. (2020). Mathematical Analysis on an Asymmetrical Wavy Motion of Blood under the Influence Entropy Generation with Convective Boundary Conditions. Symmetry, 12.","DOI":"10.3390\/sym12010102"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Zaib, A., Khan, U., Khan, I., Seikh, A.H., and Sherif, E.-S.M. (2019). Khan Numerical Investigation of Aligned Magnetic Flow Comprising Nanoliquid over a Radial Stretchable Surface with Cattaneo\u2014Christov Heat Flux with Entropy Generation. Symmetry, 11.","DOI":"10.3390\/sym11121520"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/3\/426\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:04:53Z","timestamp":1760173493000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/3\/426"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,3,6]]},"references-count":31,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2020,3]]}},"alternative-id":["sym12030426"],"URL":"https:\/\/doi.org\/10.3390\/sym12030426","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2020,3,6]]}}}