{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,6]],"date-time":"2025-11-06T06:22:43Z","timestamp":1762410163179,"version":"build-2065373602"},"reference-count":49,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2022,12,21]],"date-time":"2022-12-21T00:00:00Z","timestamp":1671580800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Princess Nourah bint Abdulrahman University Researchers Supporting Project","award":["PNURSP2022R59"],"award-info":[{"award-number":["PNURSP2022R59"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This communication predominately discusses the rheological attributes of the Darcy\u2013Forchheimer flow of a nanoliquid over a stretchy sheet with a magnetic impact. The present model considers the two diverse nanoparticles, such as Cu and Ag, and water as a base liquid. The heat equation accounts for the consequences of thermal radiation and a nonlinear heat sink\/source when evaluating heat transmission phenomena. The current mechanical system is represented by higher-order PDEs, which are then remodeled into nonlinear higher-order ODEs that employ appropriate symmetry variables. The current mathematical systems are numerically computed by implementing the bvp4c technique. The characteristic attitudes of the related pertinent factors on the non-dimensional profiles are sketched via the figures, tables, and charts. The analysis predicts that the speed of the nanoliquid particles becomes slower when there is more presence of a magnetic field and injection\/suction parameters. The growing amount of radiation is also pointed out, and the Eckert number corresponds to enriching the thermal profile.<\/jats:p>","DOI":"10.3390\/sym15010016","type":"journal-article","created":{"date-parts":[[2022,12,22]],"date-time":"2022-12-22T01:33:12Z","timestamp":1671672792000},"page":"16","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Thermally Radiative Darcy\u2013Forchheimer Flow of Cu\/Ag Nanoliquid in Water Past a Heated Stretchy Sheet with Magnetic and Viscous Dissipation Impacts"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9262-8123","authenticated-orcid":false,"given":"S.","family":"Divya","sequence":"first","affiliation":[{"name":"Department of Mathematics, Dr. N.G.P. Arts and Science College, Coimbatore 641048, Tamil Nadu, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3283-4870","authenticated-orcid":false,"given":"Nazek","family":"Alessa","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, College of Science, Princess Nourah Bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0528-3591","authenticated-orcid":false,"given":"S.","family":"Eswaramoorthi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Dr. N.G.P. Arts and Science College, Coimbatore 641048, Tamil Nadu, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6435-2916","authenticated-orcid":false,"given":"Karuppusamy","family":"Loganathan","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Manipal University Jaipur, Jaipur 302034, Rajasthan, India"}]}],"member":"1968","published-online":{"date-parts":[[2022,12,21]]},"reference":[{"key":"ref_1","unstructured":"Choi, S.U.S., and Eastman, J.A. (1995, January 12\u201317). Enhancing thermal conductivity of fluids with nanoparticle, Developments and Applications of Non-Newtonian Flows. Proceedings of the ASME International Mechanical Engineering Congress and Exhibition, San Francisco, CA, USA."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"429","DOI":"10.1016\/j.joems.2014.04.005","article-title":"Entropy generation analysis of magneto hydrodynamic flow of a nanofluid over a stretching sheet","volume":"23","author":"Govindaraju","year":"2015","journal-title":"J. Egypt. Math. Soc."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2199","DOI":"10.1016\/j.aej.2017.08.010","article-title":"Nanofluid flow on the stagnation point of a permeable non-linearly stretching\/shrinking sheet","volume":"57","author":"Malvandi","year":"2018","journal-title":"Alex. Eng. J."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"101462","DOI":"10.1016\/j.csite.2021.101462","article-title":"An effect of MHD and radiation on CNTs-Water based nanofluids due to a stretching sheet in a Newtonian fluid","volume":"28","author":"Mahabaleshwar","year":"2021","journal-title":"Case Stud. Therm. Eng."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"819","DOI":"10.1016\/j.aej.2016.02.001","article-title":"MHD flow over a permeable stretching\/shrinking sheet of a nanofluid with suction\/injection","volume":"55","author":"Sandeep","year":"2016","journal-title":"Alex. Eng. J."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"3189","DOI":"10.1016\/j.aej.2017.11.009","article-title":"Combined impact of viscosity variation and Lorentz force on slip flow of radiative nanofluid towards a vertical stretching surface with convective heat and mass transfer","volume":"57","author":"Iqbal","year":"2018","journal-title":"Alex. Eng. J."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1016\/j.aej.2019.12.010","article-title":"Transpiration effects on hybrid nanofluid flow and heat transfer over a stretching\/shrinking sheet with uniform shear flow","volume":"59","author":"Waini","year":"2020","journal-title":"Alex. Eng. J."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"101348","DOI":"10.1016\/j.csite.2021.101348","article-title":"Numerical computation of melting heat transfer in nonlinear radiative flow of hybrid nanofluids due to permeable stretching curved surface","volume":"27","author":"Waqas","year":"2021","journal-title":"Case Stud. Therm. Eng."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"100867","DOI":"10.1016\/j.csite.2021.100867","article-title":"Numerical computation of buoyancy and radiation effects on MHD micropolar nanofluid flow over a stretching\/shrinking sheet with heat source","volume":"25","author":"Rehman","year":"2021","journal-title":"Case Stud. Therm. Eng."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"101674","DOI":"10.1016\/j.csite.2021.101674","article-title":"Melting heat transfer of a magnetized water-based hybrid nanofluid flow past over a stretching\/shrinking wedge","volume":"30","author":"Kakar","year":"2022","journal-title":"Case Stud. Therm. Eng."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"7319988","DOI":"10.1155\/2022\/7319988","article-title":"Entropy and heat transfer analysis for MHD flow of-water-based nanofluid on a heated 3D plate with nonlinear radiation","volume":"2022","author":"Eswaramoorthi","year":"2022","journal-title":"Math. Probl. Eng."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"228","DOI":"10.1016\/j.csite.2018.04.008","article-title":"A numerical investigation on ethylene glycol-titanium dioxide nanofluid convective flow over a stretching sheet in presence of heat generation\/absorption","volume":"12","author":"Hosseinzadeh","year":"2018","journal-title":"Case Stud. Therm. Eng."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"915","DOI":"10.1016\/j.aej.2020.10.020","article-title":"Stability analysis of MHD hybrid nanofluid flow over a stretching\/shrinking sheet with quadratic velocity","volume":"60","author":"Zainal","year":"2021","journal-title":"Alex. Eng. J."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"709","DOI":"10.1016\/j.asej.2015.05.015","article-title":"MHD flow of dusty nanofluid over a stretching surface with volume fraction of dust particles","volume":"7","author":"Sandeep","year":"2016","journal-title":"Ain Shams Eng. J."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"206","DOI":"10.1016\/j.jppr.2017.07.001","article-title":"Cu-water nanofluid flow induced by a vertical stretching sheet in presence of a magnetic field with convective heat transfer","volume":"6","author":"Das","year":"2017","journal-title":"Propuls. Power Res."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"5252918","DOI":"10.1155\/2022\/5252918","article-title":"Heat Transfer Analysis on Carboxymethyl Cellulose Water-Based Cross Hybrid Nanofluid Flow with Entropy Generation","volume":"2022","author":"Ali","year":"2022","journal-title":"J. Nanomater."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"El-Zahar, E.R., Mahdy, A.E.N., Rashad, A.M., Saad, W., and Seddek, L.F. (2021). Unsteady MHD mixed convection flow of Non-Newtonian Casson hybrid nanofluid in the stagnation zone of sphere spinning impulsively. Fluids, 6.","DOI":"10.3390\/fluids6060197"},{"key":"ref_18","first-page":"1781","article-title":"Wasserbewegungdurch Boden","volume":"45","author":"Forchheimer","year":"1901","journal-title":"Z. VDI"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"219","DOI":"10.3389\/fphy.2019.00219","article-title":"A numerical simulation for Darcy-Forchheimer flow of nanofluid by a rotating disk with partial slip effects","volume":"7","author":"Ullah","year":"2020","journal-title":"Front. Phys."},{"key":"ref_20","first-page":"1","article-title":"Darcy-Forchheimer flow of Cu-water nanofluid over a vertical sheet owing to solar radiation","volume":"95","author":"Sarkar","year":"2021","journal-title":"J. Phys."},{"key":"ref_21","first-page":"1756","article-title":"Thermal analysis for radiative flow of Darcy-Forchheimer nanomaterials subject to entropy generation","volume":"9","author":"Khan","year":"2022","journal-title":"J. Comput. Des. Eng."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"3399","DOI":"10.1007\/s00542-019-04340-3","article-title":"Darcy-Forchheimer flow and heat transfer augmentation of a viscoelastic fluid over an incessant moving needle in the presence of viscous dissipation","volume":"25","author":"Khan","year":"2019","journal-title":"Microsyst. Technol."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1685","DOI":"10.1007\/s10483-020-2680-8","article-title":"Darcy-Forchheimer flow with nonlinear mixed convection","volume":"41","author":"Hayat","year":"2020","journal-title":"J. Appl. Math. Mech."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"2644","DOI":"10.1016\/j.rinp.2017.07.030","article-title":"Numerical analysis for Darcy-Forchheimer flow in presence of homogeneous-heterogeneous reactions","volume":"7","author":"Khan","year":"2017","journal-title":"Results Phys."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"7387","DOI":"10.1016\/j.jmrt.2020.04.074","article-title":"Flow and thermal analysis on Darcy-Forchheimer flow of copper-water nanofluid due to a rotating disk: A static and dynamic approach","volume":"9","author":"Nayak","year":"2020","journal-title":"J. Mater. Res. Technol."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Mahdy, A., El-Zahar, E.R., Rashad, A.M., Saad, W., and Al-Juaydi, H.S. (2021). The magneto-natural convection flow of a micropolar hybrid nanofluid over a vertical plate saturated in a porous medium. Fluids, 6.","DOI":"10.3390\/fluids6060202"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"821","DOI":"10.1016\/j.aej.2020.10.011","article-title":"Impact of heated obstacle position on magneto-hybrid nanofluid flow in a lid-driven porous cavity with Cattaneo-Christov heat flux pattern","volume":"60","author":"Jakeer","year":"2021","journal-title":"Alex. Eng. J."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"El-Zahar, E.R., Rashad, A.M., and Al-Juaydi, H.S. (2022). Studying massive suction impact on magneto-flow of a hybridized Casson nanofluid on a porous continuous moving or fixed surface. Symmetry, 14.","DOI":"10.3390\/sym14030627"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1016\/j.jppr.2017.01.003","article-title":"Double diffusive magnetohydrodynamic heat and mass transfer of nanofluids over a nonlinear stretching\/shrinking sheet with viscous-Ohmic dissipation and thermal radiation","volume":"6","author":"Pal","year":"2017","journal-title":"Propuls. Power Res."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1016\/j.aej.2015.03.003","article-title":"Magnetohydrodynamic mixed convective slip flow over an inclined porous plate with viscous dissipation and Joule heating","volume":"54","author":"Das","year":"2015","journal-title":"Alex. Eng. J."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"353","DOI":"10.1016\/j.asej.2015.08.017","article-title":"Dissipation effect on MHD mixed convection flow over a stretching sheet through porous medium with non-uniform heat source\/sink","volume":"8","author":"Bhukta","year":"2017","journal-title":"Ain Shams Eng. J."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"901","DOI":"10.1016\/j.jmrt.2021.06.103","article-title":"A flow study of Carreau fluid near the boundary layer region of paraboloid surface with viscous dissipation and variable fluid properties","volume":"14","author":"Salahuddin","year":"2021","journal-title":"J. Mater. Res. Technol."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1861","DOI":"10.1016\/j.aej.2020.11.034","article-title":"Numerical analysis of higher order chemical reaction on electrically MHD nanofluid under influence of viscous dissipation","volume":"60","author":"Gopal","year":"2021","journal-title":"Alex. Eng. J."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"3502","DOI":"10.1016\/j.rinp.2017.08.026","article-title":"Effects of viscous dissipation on MHD tangent hyperbolic fluid over a nonlinear stretching sheet with convective boundary conditions","volume":"7","author":"Hussain","year":"2017","journal-title":"Results Phys."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"464","DOI":"10.1016\/j.jart.2017.05.007","article-title":"Double stratification effects on unsteady electrical MHD mixed convection flow of nanofluid with viscous dissipation and Joule heating","volume":"15","author":"Daniel","year":"2017","journal-title":"J. Appl. Res. Technol."},{"key":"ref_36","first-page":"168","article-title":"Effect of heat radiation in a Walter\u2019s liquid B fluid over a stretching sheet with non-uniform heat source\/sink and elastic deformation","volume":"26","author":"Hakeem","year":"2014","journal-title":"J. King Saud Univ. Eng. Sci."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"517","DOI":"10.1016\/j.asej.2016.02.008","article-title":"Momentum and heat transfer behaviour of Jeffrey, Maxwell and Oldroyd-B nanofluids past a stretching surface with non-uniform heat source\/sink","volume":"9","author":"Sandeep","year":"2018","journal-title":"Ain Shams Eng. J."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"100435","DOI":"10.1016\/j.rineng.2022.100435","article-title":"The effects of MHD radiating and non-uniform heat source\/sink with heating on the momentum and heat transfer of Eyring-Powell fluid over a stretching","volume":"14","author":"Manvi","year":"2022","journal-title":"Results Eng."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"207","DOI":"10.18869\/acadpub.jafm.67.221.20343","article-title":"Effects of radiation on Darcy-Forchheimer convective flow over a stretching sheet in a micropolar fluid with non-uniform heat source\/sink","volume":"8","author":"Pal","year":"2015","journal-title":"J. Appl. Fluid Mech."},{"key":"ref_40","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_41","doi-asserted-by":"crossref","first-page":"1009","DOI":"10.1016\/j.aej.2017.01.026","article-title":"Combined influence of viscous dissipation and non-uniform heat source\/sink on MHD non-Newtonian fluid flow with Cattaneo-Christov heat flux","volume":"57","author":"Ramandevi","year":"2018","journal-title":"Alex. Eng. J."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1016\/j.aej.2021.04.089","article-title":"Unsteady mixed convection flow of magneto-Williamson nanofluid due to stretched cylinder with significant non-uniform heat source\/sink features","volume":"61","author":"Song","year":"2022","journal-title":"Alex. Eng. J."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"2135","DOI":"10.1007\/s10973-020-09603-0","article-title":"Entropy generation in electrical magnetohydrodynamic flow of Al2O3-Cu\/H2O hybrid nanofluid with non-uniform heat flux","volume":"143","author":"Mumraiz","year":"2021","journal-title":"J. Therm. Anal. Calorim."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"487","DOI":"10.1016\/j.icheatmasstransfer.2010.12.042","article-title":"Analytical solution of natural convection flow of a nanofluid over a linearly stretching sheet in the presence of magnetic field","volume":"38","author":"Hamad","year":"2011","journal-title":"Int. Commun. Heat Mass Transf."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"7587","DOI":"10.1016\/j.ijheatmasstransfer.2012.07.065","article-title":"Hydromagnetic nanofluid flow due to a stretching or shrinking sheet with viscous dissipation and chemical reaction effects","volume":"55","author":"Kameswaran","year":"2012","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1166\/jon.2014.1109","article-title":"Magnetohydrodynamic nanofluid flow over a stretching sheet with thermal radiation, viscous dissipation, chemical reaction and ohmic effects","volume":"3","author":"Shankar","year":"2014","journal-title":"J. Nanofluids"},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Eswaramoorthi, S., Loganathan, K., Faisal, M., Botmart, T., and Shah, N.A. (2022). Analytical and numerical investigation of Darcy-Forchheimer flow of a nonlinear-radiative non-Newtonian fluid over a Riga plate with entropy optimization. Ain Shams Eng. J., 101887.","DOI":"10.1016\/j.asej.2022.101887"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"2490524","DOI":"10.1155\/2021\/2490524","article-title":"Mixed Convection and Thermally Radiative Flow of MHD Williamson Nanofluid with Arrhenius Activation Energy and Cattaneo\u2013Christov Heat-Mass Flux","volume":"2021","author":"Eswaramoorthi","year":"2021","journal-title":"J. Math."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1016\/j.jppr.2022.07.004","article-title":"Entropy generation for mixed convection flow in vertical annulus with two regions hydromagnetic viscous and Cu-Ag water hybrid nanofluid through porous zone: A comparative numerical study","volume":"11","author":"Rahim","year":"2022","journal-title":"Propuls. Power Res."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/1\/16\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:45:31Z","timestamp":1760147131000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/1\/16"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,12,21]]},"references-count":49,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,1]]}},"alternative-id":["sym15010016"],"URL":"https:\/\/doi.org\/10.3390\/sym15010016","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,12,21]]}}}