{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,16]],"date-time":"2026-05-16T02:45:14Z","timestamp":1778899514156,"version":"3.51.4"},"reference-count":48,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2017,7,5]],"date-time":"2017-07-05T00:00:00Z","timestamp":1499212800000},"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>Natural convection heat transfer combined with entropy generation in a square cavity filled with a nanofluid under the effect of variable temperature distribution along left vertical wall has been studied numerically. Governing equations formulated in dimensionless non-primitive variables with corresponding boundary conditions taking into account the Brownian diffusion and thermophoresis effects have been solved by finite difference method. Distribution of streamlines, isotherms, local entropy generation as well as Nusselt number has been obtained for different values of key parameters. It has been found that a growth of the amplitude of the temperature distribution along the left wall and an increase of the wave number lead to an increase in the average entropy generation. While an increase in abovementioned parameters for low Rayleigh number illustrates a decrease in average Bejan number.<\/jats:p>","DOI":"10.3390\/e19070337","type":"journal-article","created":{"date-parts":[[2017,7,5]],"date-time":"2017-07-05T11:26:04Z","timestamp":1499253964000},"page":"337","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":45,"title":["Natural Convection and Entropy Generation in a Square Cavity with Variable Temperature Side Walls Filled with a Nanofluid: Buongiorno\u2019s Mathematical Model"],"prefix":"10.3390","volume":"19","author":[{"given":"Mikhail","family":"Sheremet","sequence":"first","affiliation":[{"name":"Department of Theoretical Mechanics, Tomsk State University, Tomsk 634050, Russia"},{"name":"Department of Nuclear and Thermal Power Plants, Tomsk Polytechnic University, Tomsk 634050, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Teodor","family":"Grosan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Babe\u015f-Bolyai University, Cluj-Napoca 400084, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ioan","family":"Pop","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Babe\u015f-Bolyai University, Cluj-Napoca 400084, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2017,7,5]]},"reference":[{"key":"ref_1","unstructured":"Minkowycz, W.J., Sparrow, E.M., Schneider, G.E., and Pletcher, R.H. (1988). Handbook of Numerical Heat Transfer, Wiley."},{"key":"ref_2","unstructured":"Yang, K.T. (1987). Natural Convection in Enclosure, Wiley."},{"key":"ref_3","unstructured":"Pop, I., and Ingham, D.B. (2001). Convective Heat Transfer: Mathematical and Computational Modelling of Viscous Fluids and Porous Media, Pergamon Press."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Bejan, A. (2013). Convective Heat Transfer, Wiley. [4th ed.].","DOI":"10.1002\/9781118671627"},{"key":"ref_5","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.","DOI":"10.1201\/9781315367637"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"683","DOI":"10.1080\/10407780500359943","article-title":"Natural convection in a square cavity with spatial side-wall temperature variation","volume":"49","author":"Saeid","year":"2006","journal-title":"Numer. Heat Transf. Part A"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1002\/fld.1650030305","article-title":"Natural convection of air in a square cavity: A bench mark numerical solution","volume":"3","year":"1983","journal-title":"Int. J. Numer. Methods Fluids"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1175","DOI":"10.1115\/1.3250619","article-title":"Natural convection in enclosures","volume":"110","author":"Ostrach","year":"1988","journal-title":"ASME J. Heat Transf."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1002\/fld.1650110206","article-title":"Finite volume multigrid prediction of laminar natural convection: Bench-mark solution","volume":"11","author":"Hortmann","year":"1990","journal-title":"Int. J. Numer. Methods Fluids"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"695","DOI":"10.1002\/fld.1650180705","article-title":"Natural convection flow in a square cavity revisited: Laminar and turbulent models with wall functions","volume":"18","author":"Barakos","year":"1994","journal-title":"Int. J. Numer. Methods Fluids"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"819","DOI":"10.1016\/0735-1933(94)90035-3","article-title":"Natural convection in an enclosure with nonuniform wall temperature","volume":"21","author":"Fu","year":"1994","journal-title":"Int. Commun. Heat Mass Transf."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1115\/1.2822289","article-title":"Effect of imposed wall temperature oscillations on the stability of natural convection in a square enclosure","volume":"117","author":"Xia","year":"1995","journal-title":"J. Heat Transf."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1080\/10407789508913703","article-title":"Natural convection in a square enclosure heated periodically from part of the bottom wall","volume":"27","author":"Lakhal","year":"1995","journal-title":"Numer. Heat Transf. Part A"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1080\/10407789108944836","article-title":"The Ra-Pr domain of laminar natural convection in an enclosure heated from the side","volume":"19","author":"Lage","year":"1991","journal-title":"Numer. Heat Transf. Part A"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"3157","DOI":"10.1016\/S0017-9310(98)00019-2","article-title":"Prediction of the resonance frequency of natural convection in an enclosure with time-periodic heating imposed on the sidewall","volume":"41","author":"Kwak","year":"1998","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"513","DOI":"10.1080\/10407780290059675","article-title":"Natural convection in a 2D enclosure with sinusoidal upper wall temperature","volume":"42","author":"Sarris","year":"2002","journal-title":"Numer. Heat Transf. Part A"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"411","DOI":"10.1007\/s11242-014-0375-7","article-title":"Natural convection in a square porous cavity with sinusoidal temperature distributions on both side walls filled with a nanofluid: Buongiorno\u2019s mathematical model","volume":"105","author":"Sheremet","year":"2014","journal-title":"Transp. Porous Media"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"072601","DOI":"10.1115\/1.4029816","article-title":"Natural convection in a wavy porous cavity with sinusoidal temperature distributions on both side walls filled with a nanofluid: Buongiorno\u2019s mathematical model","volume":"137","author":"Sheremet","year":"2015","journal-title":"J. Heat Transf."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"240","DOI":"10.1115\/1.2150834","article-title":"Convective transport in nanofluids","volume":"128","author":"Buongiorno","year":"2006","journal-title":"ASME J. Heat Transf."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"721","DOI":"10.1016\/0360-5442(80)90091-2","article-title":"Second law analysis in heat transfer","volume":"5","author":"Bejan","year":"1980","journal-title":"Energy Int. J."},{"key":"ref_21","unstructured":"Bejan, A. (1982). Entropy Generation through Heat and Fluid Flow, John Wiley & Sons."},{"key":"ref_22","unstructured":"Bejan, A. (1995). Entropy Generation Minimization: The Method of Thermodynamic Optimization of Finite-Size Systems and Finite-Time Processes, CRC Press."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"515","DOI":"10.1016\/j.ijheatmasstransfer.2013.03.004","article-title":"Entropy generation in steady MHD flow due to a rotating porous disk in a nanofluid","volume":"62","author":"Rashidi","year":"2013","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_24","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":"Numer. Heat Transf. Part A"},{"key":"ref_25","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_26","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_27","doi-asserted-by":"crossref","unstructured":"Sheremet, M.A., Oztop, H.F., Pop, I., and Abu-Hamdeh, N. (2016). 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_28","doi-asserted-by":"crossref","first-page":"958","DOI":"10.1108\/HFF-02-2016-0063","article-title":"Natural convection of nanofluid inside a wavy cavity with a non-uniform heating: Entropy generation analysis","volume":"27","author":"Sheremet","year":"2017","journal-title":"Int. J. Numer. Methods Heat Fluid Flow"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"244","DOI":"10.1016\/j.apt.2016.09.030","article-title":"Entropy generation due to natural convection of a nanofluid in a partially open triangular cavity","volume":"28","author":"Bondareva","year":"2017","journal-title":"Adv. Powder Technol."},{"key":"ref_30","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_31","doi-asserted-by":"crossref","first-page":"911","DOI":"10.1016\/j.rser.2011.09.012","article-title":"A review on entropy generation in natural and mixed convection heat transfer for energy systems","volume":"16","author":"Oztop","year":"2012","journal-title":"Renew. Sustain. Energy Rev."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"620","DOI":"10.1016\/j.ijheatmasstransfer.2015.01.116","article-title":"Numerical study on mixed convection and entropy generation of Cu\u2013water nanofluid in a differentially heated skewed enclosure","volume":"85","author":"Nayak","year":"2015","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1016\/j.jtice.2015.04.018","article-title":"Natural convection and entropy generation of nanofluid filled cavity having different shaped obstacles under the influence of magnetic field and internal heat generation","volume":"56","author":"Selimefendigil","year":"2015","journal-title":"J. Taiwan Inst. Chem. Eng."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"679","DOI":"10.1016\/j.powtec.2016.10.077","article-title":"Simulation of natural convection and entropy generation of non-Newtonian nanofluid in an inclined cavity using Buongiorno\u2019s mathematical model (Part II, entropy generation)","volume":"305","author":"Kefayati","year":"2017","journal-title":"Powder Technol."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"514","DOI":"10.1016\/j.ijheatmasstransfer.2013.06.010","article-title":"A review on entropy generation in nanofluid flow","volume":"65","author":"Mahian","year":"2013","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"380826","DOI":"10.1155\/2010\/380826","article-title":"Heat transfer in nanofluids","volume":"2010","author":"Manca","year":"2010","journal-title":"Adv. Mech. Eng."},{"key":"ref_37","unstructured":"Choi, S.U.S. (1995, January 12\u201317). Enhancing Thermal Conductivity of Fluids with Nanoparticles. Proceedings of the 1995 ASME International Mechanical Engineering Congress and Exposition, FED 231\/MD 66, San Francisco, CA, USA."},{"key":"ref_38","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_39","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_40","doi-asserted-by":"crossref","first-page":"1","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_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":"247","DOI":"10.1080\/10407782.2013.825510","article-title":"Natural Convection of Nanofluids in a Cavity with Nonuniform Temperature Distributions on Side Walls","volume":"65","author":"Sivasankaran","year":"2013","journal-title":"Numer. Heat Transf. Part A"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"214","DOI":"10.1016\/j.powtec.2016.12.025","article-title":"Transient natural convective heat transfer in a trapezoidal cavity filled with non-Newtonian nanofluid with sinusoidal boundary conditions on both sidewalls","volume":"308","author":"Alsabery","year":"2017","journal-title":"Powder Technol."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"835","DOI":"10.1016\/j.ijheatmasstransfer.2016.05.031","article-title":"Heatline visualization of conjugate natural convection in a square cavity filled with nanofluid with sinusoidal temperature variations on both horizontal walls","volume":"100","author":"Alsabery","year":"2016","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1016\/j.spmi.2011.01.002","article-title":"Computational analysis of non-isothermal temperature distribution on natural convection in nanofluid filled enclosures","volume":"49","author":"Oztop","year":"2011","journal-title":"Superlattices Microstruct."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"507","DOI":"10.1080\/01457630802186080","article-title":"Natural convection in a rectangular enclosure with sinusoidal temperature distributions on both side walls","volume":"54","author":"Deng","year":"2008","journal-title":"Numer. Heat Transf. Part A"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1080\/10407782.2012.715985","article-title":"Natural convection in a porous cavity with sinusoidal heating on both sidewalls","volume":"63","author":"Sivasankaran","year":"2013","journal-title":"Numer. Heat Transf. Part A"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"1","DOI":"10.3390\/e8010001","article-title":"Second law analysis in convective heat and mass transfer","volume":"8","author":"Magherbi","year":"2006","journal-title":"Entropy"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/7\/337\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:41:31Z","timestamp":1760208091000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/7\/337"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,7,5]]},"references-count":48,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2017,7]]}},"alternative-id":["e19070337"],"URL":"https:\/\/doi.org\/10.3390\/e19070337","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,7,5]]}}}