{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,10]],"date-time":"2026-06-10T06:24:40Z","timestamp":1781072680524,"version":"3.54.1"},"reference-count":36,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2018,9,4]],"date-time":"2018-09-04T00:00:00Z","timestamp":1536019200000},"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>The present research work explores the effects of suction\/injection and viscous dissipation on entropy generation in the boundary layer flow of a hybrid nanofluid (Cu\u2013Al2O3\u2013H2O) over a nonlinear radially stretching porous disk. The energy dissipation function is added in the energy equation in order to incorporate the effects of viscous dissipation. The Tiwari and Das model is used in this work. The flow, heat transfer, and entropy generation analysis have been performed using a modified form of the Maxwell Garnett (MG) and Brinkman nanofluid model for effective thermal conductivity and dynamic viscosity, respectively. Suitable transformations are utilized to obtain a set of self-similar ordinary differential equations. Numerical solutions are obtained using shooting and bvp4c Matlab solver. The comparison of solutions shows excellent agreement. To examine the effects of principal flow parameters like suction\/injection, the Eckert number, and solid volume fraction, different graphs are plotted and discussed. It is concluded that entropy generation inside the boundary layer of a hybrid nanofluid is high compared to a convectional nanofluid.<\/jats:p>","DOI":"10.3390\/e20090668","type":"journal-article","created":{"date-parts":[[2018,9,5]],"date-time":"2018-09-05T03:08:55Z","timestamp":1536116935000},"page":"668","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":90,"title":["Transpiration and Viscous Dissipation Effects on Entropy Generation in Hybrid Nanofluid Flow over a Nonlinear Radially Stretching Disk"],"prefix":"10.3390","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0920-6174","authenticated-orcid":false,"given":"Umer","family":"Farooq","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Jiangsu University, Zhenjiang 212013, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Muhammad Idrees","family":"Afridi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, COMSATS University Islamabad, Park Road, Tarlai Kalan, Islamabad 455000, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Muhammad","family":"Qasim","sequence":"additional","affiliation":[{"name":"Department of Mathematics, COMSATS University Islamabad, Park Road, Tarlai Kalan, Islamabad 455000, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6896-172X","authenticated-orcid":false,"given":"D. C.","family":"Lu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Jiangsu University, Zhenjiang 212013, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2018,9,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"718","DOI":"10.1115\/1.3451063","article-title":"A study of entropy generation in fundamental convective heat transfer","volume":"101","author":"Bejan","year":"1979","journal-title":"ASME J. Heat Transf."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1016\/j.molliq.2017.12.106","article-title":"Colloidal study of unsteady magnetohydrodynamic couple stress fluid flow over an isothermal vertical flat plate with entropy heat generation","volume":"252","author":"Reddy","year":"2018","journal-title":"J. Mol. Liq."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1016\/j.ijthermalsci.2017.09.014","article-title":"Entropy generation and heat transfer in boundary layer flow over a thin needle moving in a parallel stream in the presence of nonlinear Rosseland radiation","volume":"123","author":"Afridi","year":"2018","journal-title":"Int. J. Therm. Sci."},{"key":"ref_4","first-page":"1","article-title":"Entropy generation in three-dimensional flow of dissipative fluid","volume":"16","author":"Afridi","year":"2018","journal-title":"Int. J. Appl. Comput. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1283","DOI":"10.1016\/j.rinp.2017.12.032","article-title":"Irreversibility analysis in unsteady flow over a vertical plate with arbitrary wall shear stress and ramped wall temperature","volume":"8","author":"Khan","year":"2018","journal-title":"Results Phys."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1016\/j.cjph.2017.11.026","article-title":"Entropy Generation and MHD natural convection of a nanofluid in an inclined square porous cavity: Effects of a heat sink and source size and location","volume":"56","author":"Rashad","year":"2018","journal-title":"Chin. J. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Qasim, M., and Afridi, M.I. (2017). Effects of energy dissipation and variable thermal conductivity on entropy generation rate in mixed convection flow. ASME J. Therm. Sci. Eng. Appl.","DOI":"10.1115\/1.4038703"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1615\/JPorMedia.v19.i2.50","article-title":"A note on entropy generation in MHD flow over a vertical plate embedded in a porous medium with arbitrary shear stress and ramped temperature","volume":"19","author":"Khan","year":"2016","journal-title":"J. Porous Media"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"783","DOI":"10.1166\/jon.2018.1488","article-title":"Comparative study and entropy generation analysis of Cu\u2013H2O and Ag\u2013H2O nanofluids flow over a slendering stretching surface","volume":"7","author":"Afridi","year":"2018","journal-title":"J. Nanofluids"},{"key":"ref_10","first-page":"1","article-title":"A computational study of entropy generation and 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_11","first-page":"51","article-title":"Entropy generation in hydromagnetic and thermal boundary layer flow due to radial stretching sheet with Newtonian heating","volume":"2","author":"Das","year":"2015","journal-title":"J. Heat Mass Transf. Res."},{"key":"ref_12","first-page":"20","article-title":"Entropy generation on MHD flow of Powell-Eyring fluid between radially stretching rotating disk with diffusion-thermo and thermo-diffusion effects","volume":"11","author":"Khan","year":"2017","journal-title":"Acta Mech. Autom."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1515\/eng-2017-0025","article-title":"Entropy generation analysis and effects of slip conditions on micropolar fluid flow due to a rotating disk","volume":"7","author":"Khan","year":"2017","journal-title":"Open Eng."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1017\/S0022112062001196","article-title":"Effects of viscous dissipation in natural convection","volume":"14","author":"Gebhart","year":"1962","journal-title":"J. Fluid Mech."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"360","DOI":"10.1007\/s00231-004-0552-2","article-title":"Effect of viscous dissipation on the mixed convection heat transfer from an exponentially stretching surface","volume":"41","author":"Partha","year":"2005","journal-title":"Heat Mass Transf."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"631","DOI":"10.1016\/j.physleta.2007.08.005","article-title":"Effects of viscous dissipation and radiation on thermal boundary layer over a nonlinearly stretching sheet","volume":"372","author":"Cortell","year":"2008","journal-title":"Phys. Lett. A"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"267","DOI":"10.18869\/acadpub.jafm.68.224.20368","article-title":"Thermal radiation effects on MHD boundary layer slip flow past a permeable exponential stretching sheet in the presence of Joule heating and viscous dissipation","volume":"9","author":"Sreenivasulu","year":"2016","journal-title":"J. Appl. Fluid Mech."},{"key":"ref_18","unstructured":"Choi, S.U.S. (1995, January 12\u201317). Enhancing thermal conductivity of fluids with nanoparticles. Proceedings of the ASME International Mechanical Engineering Congress and Exposition, San Francisco, CA, USA."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1291","DOI":"10.1108\/HFF-12-2011-0258","article-title":"Effects of viscous dissipation and Newtonian heating on boundary-layer flow of nanofluids over a flat plate","volume":"23","author":"Makinde","year":"2013","journal-title":"Int. J. Numer. Methods Heat Fluid Flow"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"482","DOI":"10.1016\/j.molliq.2016.12.040","article-title":"Convection heat transfer in micropolar nanofluids with oxide nanoparticles in water, kerosene and engine oil","volume":"229","author":"Hussanan","year":"2017","journal-title":"J. Mol. Liq."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Sulochanan, C., Ashwinkumar, G.P., and Sandeep, N. (2017). Boundary layer analysis of persistent moving horizontal needle in magnetohydrodynamic ferrofluid: A numerical study. Alex. Eng. J.","DOI":"10.1016\/j.aej.2017.08.020"},{"key":"ref_22","unstructured":"Reddy, P.S., and Chamkha, A.J. (2016). Heat and mass transfer characteristics of nanofluid over horizontal circular cylinder. Ain Shams Eng. J."},{"key":"ref_23","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":"225","author":"Hussanan","year":"2018","journal-title":"J. Mol. Liq."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s40540-017-0021-2","article-title":"Double stratification effects on heat and mass transfer in unsteady MHD nanofluid flow over a flat surface","volume":"4","author":"Mutuku","year":"2017","journal-title":"Asia Pac. J. Comput. Eng."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1840015","DOI":"10.1142\/S0218348X18400157","article-title":"A fractal model for water flow through unsaturated porous rocks","volume":"26","author":"Xiao","year":"2018","journal-title":"Fractals"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.compfluid.2014.08.001","article-title":"Nanofluid flow with multimedia physical features for conjugate mixed convection and radiation","volume":"104","author":"Hsiao","year":"2014","journal-title":"Comput. Fluids"},{"key":"ref_27","first-page":"645","article-title":"The effects of perforation erosion on practical hydraulic-fracturing applications","volume":"22","author":"Long","year":"2017","journal-title":"Soc. Pet. Eng."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1411","DOI":"10.1016\/j.icheatmasstransfer.2005.07.005","article-title":"Free convection flow with thermal radiation and mass transfer past a moving vertical porous plate","volume":"32","author":"Makinde","year":"2005","journal-title":"Int. Commun. Heat Mass Transf."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1016\/j.powtec.2016.06.029","article-title":"Particle shape and radiation effects on Marangoni boundary layer flow and heat transfer of copper-water nanofluid driven by an exponential temperature","volume":"301","author":"Lin","year":"2016","journal-title":"Powder Technol."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1090","DOI":"10.1016\/j.ijheatmasstransfer.2015.07.067","article-title":"A new diffusion for laminar boundary layer flow of power law fluids past a flat surface with magnetic effect and suction or injection","volume":"90","author":"Lin","year":"2015","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"580","DOI":"10.1016\/j.applthermaleng.2015.12.138","article-title":"Stagnation electrical MHD nanofluid mixed convection with slip boundary on a stretching sheet","volume":"98","author":"Hsiao","year":"2016","journal-title":"Appl. Therm. Eng."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"931","DOI":"10.1016\/j.icheatmasstransfer.2012.05.014","article-title":"Free convection boundary layer flow over a horizontal cylinder of elliptic cross section in porous media saturated by a nanofluid","volume":"39","author":"Cheng","year":"2012","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1515\/ijnsns-2016-0037","article-title":"Numerical investigation of hydromagnetic Cu\u2013Al2O3\/water hybrid nanofluid flow over a permeable stretching sheet with suction","volume":"17","author":"Devi","year":"2015","journal-title":"Int. J. Nonlinear Sci. Numer. Simul."},{"key":"ref_34","first-page":"1","article-title":"Numerical analysis of unsteady conjugate natural convection of hybrid water-based nanofluid in a semicircular cavity","volume":"140","author":"Chamkha","year":"2018","journal-title":"ASME J. Heat Transf."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"490","DOI":"10.1139\/cjp-2015-0799","article-title":"Numerical investigation of three dimensional Cu\u2013Al2O3\/water hybrid nanofluid flow over a stretching sheet with effecting Lorentz force subject to Newtonian heating","volume":"94","author":"Devi","year":"2016","journal-title":"Can. J. Phys."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"29","DOI":"10.4028\/www.scientific.net\/DDF.377.29","article-title":"Analysis of Blasius flow of hybrid nanofluid over a convectively heated surface","volume":"377","author":"Olatundun","year":"2017","journal-title":"Defect Diffus. 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