{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,13]],"date-time":"2026-01-13T22:10:24Z","timestamp":1768342224470,"version":"3.49.0"},"reference-count":48,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2019,3,1]],"date-time":"2019-03-01T00:00:00Z","timestamp":1551398400000},"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 internal average energy loss caused by entropy generation for steady mixed convective Poiseuille flow of a nanofluid, suspended with titanium dioxide (TiO2) particles in water, and passed through a wavy channel, was investigated. The models of thermal conductivity and viscosity of titanium dioxide of 21 nm size particles with a volume concentration of temperature ranging from 15 \u00b0C to 35 \u00b0C were utilized. The characteristics of the working fluid were dependent on electro-magnetohydrodynamics (EMHD) and thermal radiation. The governing equations were first modified by taking long wavelength approximations, which were then solved by a homotopy technique, whereas for numerical computation, the software package BVPh 2.0 was utilized. The results for the leading parameters, such as the electric field, the volume fraction of nanoparticles and radiation parameters for three different temperatures scenarios were examined graphically. The minimum energy loss at the center of the wavy channel due to the increase in the electric field parameter was noted. However, a rise in entropy was observed due to the change in the pressure gradient from low to high.<\/jats:p>","DOI":"10.3390\/e21030236","type":"journal-article","created":{"date-parts":[[2019,3,4]],"date-time":"2019-03-04T05:22:26Z","timestamp":1551676946000},"page":"236","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":92,"title":["Effects of Radiative Electro-Magnetohydrodynamics Diminishing Internal Energy of Pressure-Driven Flow of Titanium Dioxide-Water Nanofluid due to Entropy Generation"],"prefix":"10.3390","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2641-1575","authenticated-orcid":false,"given":"Ahmed","family":"Zeeshan","sequence":"first","affiliation":[{"name":"Department of Mathematics &amp; Statistics, FBAS, International Islamic University Islamabad, Islamabad 44000, Pakistan"}]},{"given":"Nasir","family":"Shehzad","sequence":"additional","affiliation":[{"name":"Department of Mathematics &amp; Statistics, FBAS, International Islamic University Islamabad, Islamabad 44000, Pakistan"}]},{"given":"Tehseen","family":"Abbas","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Education Lahore, Faisalabad Campus, Faisalabad 38000, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7805-8259","authenticated-orcid":false,"given":"Rahmat","family":"Ellahi","sequence":"additional","affiliation":[{"name":"Department of Mathematics &amp; Statistics, FBAS, International Islamic University Islamabad, Islamabad 44000, Pakistan"},{"name":"Center for Modeling &amp; Computer Simulation, Research Institute, King Fahd University of Petroleum &amp; Minerals, Dhahran 31261, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2019,3,1]]},"reference":[{"key":"ref_1","first-page":"167","article-title":"Hall current and heat transfer effects on MHD flow in a channel partially filled with a porous medium in a rotating system","volume":"33","author":"Chauhan","year":"2010","journal-title":"Turk. J. Eng. Environ. Sci."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1197","DOI":"10.1016\/j.cnsns.2009.05.051","article-title":"Hydromagnetic non-Darcy flow and heat transfer over a stretching sheet in the presence of thermal radiation and Ohmic dissipation","volume":"15","author":"Pal","year":"2010","journal-title":"Commun. Nonlinear Sci. Numer Simul."},{"key":"ref_3","first-page":"99","article-title":"Enhancing thermal conductivity of fluids with nanoparticles","volume":"231","author":"Choi","year":"1995","journal-title":"ASME Publ. Fed."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1474","DOI":"10.1016\/j.icheatmasstransfer.2011.07.007","article-title":"Investigation of turbulent convective heat transfer and pressure drop of TiO2\/water nanofluid in circular tube","volume":"38","author":"Sajadi","year":"2011","journal-title":"Int. Commun. Heat Mass Transf."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Ropers, M.H., Terrisse, H., Mercier-Bonin, M., and Humbert, B. (2017). Titanium Dioxide as Food Additive. Application of Titanium Dioxide, InTech.","DOI":"10.5772\/intechopen.68883"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"222","DOI":"10.1016\/j.cjph.2019.02.004","article-title":"Effects of coagulation on the two phase peristaltic pumping of magnetized Prandtl biofluid through an endoscopic annular geometry containing a porous medium","volume":"58","author":"Bhatti","year":"2019","journal-title":"Chinees J. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"724","DOI":"10.1016\/j.ijheatmasstransfer.2018.11.090","article-title":"Thermal management of water-based carbon nanotubes enclosed in a partially heated triangular cavity with heated cylindrical obstacle","volume":"131","author":"Haq","year":"2019","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1322","DOI":"10.1016\/j.ijheatmasstransfer.2018.11.020","article-title":"Heat transfer behavior of nanoparticle enhanced PCM solidification through an enclosure with V shaped fins","volume":"130","author":"Sheikholeslami","year":"2019","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_9","first-page":"401","article-title":"Water functionalized CuO nanoparticles filled in a partially heated trapezoidal cavity with inner heated obstacle: FEM approach","volume":"129","author":"Aman","year":"2019","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1016\/j.ijheatmasstransfer.2017.06.061","article-title":"New temperature, interfacial shell dependent dimensionless model for thermal conductivity of nanofluids","volume":"114","author":"Hosseini","year":"2017","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Nasiri, H., Jamalabadi, M.Y.A., Sadeghi, R., Safaei, M.R., Nguyen, T.K., and Shadloo, M.S. (2018). A smoothed particle hydrodynamics approach for numerical simulation of nano-fluid flows. J. Therm. Anal. Calorim., 1\u20139.","DOI":"10.1007\/s10973-018-7022-4"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Safaei, M.R., Ahmadi, G., Goodarzi, M.S., Shadloo, M.S., Goshayeshi, H.R., and Dahari, M. (2016). Heat transfer and pressure drop in fully developed turbulent flows of graphene nanoplatelets\u2013silver\/water nanofluids. Fluids, 1.","DOI":"10.3390\/fluids1030020"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1837","DOI":"10.1615\/HeatTransRes.2018025569","article-title":"Exploration of convective heat transfer and flow characteristics synthesis by Cu-Ag\/water hybrid-nanofluids","volume":"49","author":"Hassan","year":"2018","journal-title":"Heat Transf. Res."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Shehzad, N., Zeeshan, A., Ellahi, R., and Rashidi, S. (2018). Modelling study on internal energy loss due to entropy generation for non-Darcy Poiseuille flow of silver water nanofluid: An application of purification. Entropy, 20.","DOI":"10.3390\/e20110851"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Ellahi, R., Zeeshan, A., Hussain, F., and Abbas, T. (2018). Study of shiny film coating on multi-fluid flows of a rotating disk suspended with nano-sized silver and gold particles: A comparative analysis. Coatings, 8.","DOI":"10.3390\/coatings8120422"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"292","DOI":"10.1016\/j.molliq.2018.10.038","article-title":"Convective radiative plane Poiseuille flow of nanofluid through porous medium with slip: An application of Stefan blowing","volume":"273","author":"Alamri","year":"2019","journal-title":"J. Mol. Liq."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Ellahi, R., Zeeshan, A., Hussain, F., and Asadollahi, A. (2019). Peristaltic blood flow of couple stress fluid suspended with nanoparticles under the influence of chemical reaction and activation energy. Symmetry, 11.","DOI":"10.3390\/sym11020276"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"330","DOI":"10.1016\/S1164-0235(02)00075-4","article-title":"Analysis of mixed convection\u2014Radiation interaction in a vertical channel: Entropy generation","volume":"2","author":"Mahmud","year":"2002","journal-title":"Exergy Int. J."},{"key":"ref_19","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_20","doi-asserted-by":"crossref","first-page":"749","DOI":"10.1016\/j.ijheatmasstransfer.2013.02.044","article-title":"Natural convection heat transfer and entropy generation in wavy-wall enclosure containing water-based nanofluid","volume":"61","author":"Cho","year":"2013","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"649","DOI":"10.1016\/j.energy.2017.04.035","article-title":"Entropy generation on electro-osmotic flow pumping by a uniform peristaltic wave under magnetic environment","volume":"128","author":"Ranjit","year":"2017","journal-title":"Energy"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Sheremet, M.A., Grosan, T., and Pop, I. (2017). Natural convection and entropy generation in a square cavity with variable temperature side walls filled with a nanofluid: Buongiorno\u2019s mathematical model. Entropy, 19.","DOI":"10.3390\/e19070337"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Jamalabadi, M.Y.A., Safaei, M.R., Alrashed, A.A.A.A., Nguyen, T.K., and Filho, E.P.B. (2017). Entropy generation in thermal radiative loading of structures with distinct heaters. Entropy, 19.","DOI":"10.3390\/e19100506"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Darbari, B., Rashidi, S., and Abolfazli Esfahani, J. (2016). Sensitivity analysis of entropy generation in nanofluid flow inside a channel by response surface methodology. Entropy, 18.","DOI":"10.3390\/e18020052"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Bhatti, M.M., Abbas, T., Rashidi, M.M., and Ali, M.E.-S. (2016). Numerical simulation of entropy generation with thermal radiation on MHD carreau nanofluid towards a shrinking sheet. Entropy, 18.","DOI":"10.3390\/e18060200"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Bhatti, M.M., Abbas, T., Rashidi, M.M., Ali, M.E.-S., and Yang, Z. (2016). Entropy generation on MHD Eyring\u2013Powell nanofluid through a permeable stretching surface. Entropy, 18.","DOI":"10.3390\/e18060224"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Abbas, T., Ayub, M., Bhatti, M.M., Rashidi, M.M., and Ali, M.E.-S. (2016). Entropy generation on nanofluid flow through a horizontal Riga plate. Entropy, 18.","DOI":"10.3390\/e18060223"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"476","DOI":"10.1080\/16583655.2018.1483795","article-title":"Effects of MHD and slip on heat transfer boundary layer flow over a moving plate based on specific entropy generation","volume":"12","author":"Ellahi","year":"2018","journal-title":"J. Taibah Univ. Sci."},{"key":"ref_29","unstructured":"Liao, S.J. (2003). Beyond Perturbation: Introduction to Homotopy Analysis Method, Chapman and Hall\/CRC Press."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1451","DOI":"10.1016\/j.apm.2012.04.004","article-title":"The effects of MHD and temperature dependent viscosity on the flow of non-Newtonian nanofluid in a pipe: Analytical solutions","volume":"37","author":"Ellahi","year":"2013","journal-title":"Appl. Math. Model."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Hussain, F., Ellahi, R., and Zeeshan, A. (2018). Mathematical models of electro magnetohydrodynamic Multiphase Flows Synthesis with Nanosized Hafnium Particles. Appl. Sci., 8.","DOI":"10.3390\/app8020275"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Zeeshan, A., Ijaz, N., Abbas, T., and Ellahi, R. (2018). The sustainable characteristic of Bio-bi-phase flow of peristaltic transport of MHD Jeffery fluid in human body. Sustainability, 10.","DOI":"10.3390\/su10082671"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Tiwari, R.K., and Das, M.K. (2007). Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids. Int. J. Heat Mass Transf., 50.","DOI":"10.1016\/j.ijheatmasstransfer.2006.09.034"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"3371","DOI":"10.1007\/s00521-017-2924-9","article-title":"Convective Poiseuille flow of Al2O3-EG nanofluid in a porous wavy channel with thermal radiation","volume":"30","author":"Zeeshan","year":"2018","journal-title":"Neural Comput. Appl."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"134","DOI":"10.1016\/j.icheatmasstransfer.2013.05.001","article-title":"Forced convection heat and mass transfer flow of a nanofluid through a porous channel with a first order chemical reaction on the wall","volume":"46","author":"Matin","year":"2013","journal-title":"Int. Commun. Heat Mass Transf."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"706","DOI":"10.1016\/j.expthermflusci.2009.01.005","article-title":"Measurement of temperature-dependent thermal conductivity and viscosity of TiO2-water nanofluids","volume":"33","author":"Duangthongsuk","year":"2009","journal-title":"Exp. Therm. Fluid Sci."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"4410","DOI":"10.1016\/j.ijheatmasstransfer.2011.04.048","article-title":"A critical synthesis of thermophysical characteristics of nanofluids","volume":"54","author":"Khanafer","year":"2011","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1368","DOI":"10.1016\/j.icheatmasstransfer.2011.08.013","article-title":"Numerical investigations of flow and heat transfer enhancement in a corrugated channel using nanofluid","volume":"38","author":"Ahmed","year":"2011","journal-title":"Int. Commun. Heat Mass Transf."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"514","DOI":"10.1016\/j.ijheatmasstransfer.2013.06.010","article-title":"A review of entropy generation in nanofluid flow","volume":"65","author":"Mahian","year":"2013","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1166\/jon.2015.1166","article-title":"MHD convection of nanofluids: A review","volume":"4","author":"Chamkha","year":"2015","journal-title":"J. Nanofluids"},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Rosseland, S. (1931). Astrophysik und Atom-Theoretische Grundlagen, Springer Verlag.","DOI":"10.1007\/978-3-662-26679-3"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"607","DOI":"10.1016\/j.molliq.2018.05.103","article-title":"Structural impact of kerosene-Al2O3 nanoliquid on MHDpoiseuille flow with variable thermal conductivity: Application of cooling process","volume":"264","author":"Ellahi","year":"2018","journal-title":"J. Mol. Liq."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"020014","DOI":"10.1063\/1.4968063","article-title":"Unsteady natural convection flow of nanofluids past a semi-infinite isothermal vertical plate","volume":"1787","author":"Tippa","year":"2016","journal-title":"AIP Conf. Proc."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Raza, J., Rohni, A.M., and Omar, Z. (2016). Numerical investigation of copper-water (Cu-water) nanofluid with different shapes of nanoparticles in a channel with stretching wall: Slip effects. Math. Comput. Appl., 21.","DOI":"10.3390\/mca21040043"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"43","DOI":"10.18869\/acadpub.jafm.68.224.22695","article-title":"Mixed convection flow of couple stress fluid in a vertical channel with radiation and Soret effects","volume":"9","author":"Kaladhar","year":"2016","journal-title":"J. Appl. Fluid Mech."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1016\/j.ijheatmasstransfer.2014.07.092","article-title":"Conjugate natural convection in a square porous cavity filled by a nanofluid using Buongiorno\u2019s mathematical model","volume":"79","author":"Sheremet","year":"2014","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"4078","DOI":"10.1016\/j.cnsns.2009.03.008","article-title":"On the selection of auxiliary functions, operators, and con-vergence control parameters in the application of the Homotopy Analysis Method to nonlinear differential equations: A general approach","volume":"14","author":"Vajravelu","year":"2009","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"1173","DOI":"10.1016\/S0020-7462(02)00062-8","article-title":"An analytic approximate technique for free oscillations of positively damped systems with algebraically decaying amplitude","volume":"38","author":"Liao","year":"2003","journal-title":"Int. J. Non-Linear Mech."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/3\/236\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:35:41Z","timestamp":1760186141000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/3\/236"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,3,1]]},"references-count":48,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2019,3]]}},"alternative-id":["e21030236"],"URL":"https:\/\/doi.org\/10.3390\/e21030236","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,3,1]]}}}