{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,12]],"date-time":"2026-06-12T21:34:43Z","timestamp":1781300083628,"version":"3.54.1"},"reference-count":45,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2020,8,26]],"date-time":"2020-08-26T00:00:00Z","timestamp":1598400000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This research article aims to investigate the consequences of binary chemical reaction, thermal radiation, and Soret\u2013Dufour effects on a steady incompressible Darcy\u2013Forchheimer flow of nanofluids. Stretching surface is assumed to drive the fluid along positive horizontal direction. Brownian motion, and the Thermophoresis are accounted in particular. The governing highly nonlinear system of problems which are advanced version of Navier\u2013Stokes equations are transformed into ordinary differential equations (ODEs) using appropriately adjusted transformations invoking symmetric property of the independent variables. The numerical approach using RK45 in connection with shooting technique is adopted to solve the final equations. Graphical approach is used to interpret the results and the values of important physical quantities are given in tabular data form. Velocity field, temperature distribution and concentration distribution are graphically analyzed for variation in respective fluid parameters. Furthermore, density graphs and stream lines are sketched for the present model. The outputs indicate a rise of temperature field in connection with thermal radiation parameter. A clear decline is noticed in velocity field for elevated values of Forchheimer number and porosity factor. The Dufour effect anticipates a rising factor for temperature distribution and the same is noticed for concentration distribution in lieu of Soret effect. Thermal radiation and binary chemical reaction has strong impact on heat transport mechanism. The results for physical quantities such as skin friction, heat and mass flux rates are given in tabular data form in last section of this study.<\/jats:p>","DOI":"10.3390\/sym12091421","type":"journal-article","created":{"date-parts":[[2020,8,26]],"date-time":"2020-08-26T09:05:37Z","timestamp":1598432737000},"page":"1421","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":86,"title":["Consequences of Soret\u2013Dufour Effects, Thermal Radiation, and Binary Chemical Reaction on Darcy Forchheimer Flow of Nanofluids"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5880-9553","authenticated-orcid":false,"given":"Ghulam","family":"Rasool","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7186-7216","authenticated-orcid":false,"given":"Anum","family":"Shafiq","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0286-7244","authenticated-orcid":false,"given":"Dumitru","family":"Baleanu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Cankaya University, 06530 Ankara, Turkey"},{"name":"Institute of Space Sciences, 077125 Magurele, Romania"},{"name":"Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40250, Taiwan"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2020,8,26]]},"reference":[{"key":"ref_1","unstructured":"Choi, U.S., and Eastman, J.A. (1995, January 12\u201317). Enhancing thermal conductivity of fluids with nanoparticles. Proceedings of the ASME International Mechanical Engineering Congress and Exposition (IMECE), San Francisco, CA, USA."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"110526","DOI":"10.1016\/j.solmat.2020.110526","article-title":"MXene based new class of silicone oil nanofluids for the performance improvement of concentrated photovoltaic thermal collector","volume":"211","author":"Aslfattahi","year":"2020","journal-title":"Sol. Energy Mater. Sol. Cells"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"112014","DOI":"10.1016\/j.molliq.2019.112014","article-title":"Nanofluid: Potential evaluation in automotive radiator","volume":"297","author":"Abbas","year":"2020","journal-title":"J. Mol. Liq."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"112787","DOI":"10.1016\/j.molliq.2020.112787","article-title":"Comprehensive study on nanofluid and ionanofluid for heat transfer enhancement: A review on current and future perspective","volume":"305","author":"Bakthavatchalam","year":"2020","journal-title":"J. Mol. Liq."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"113310","DOI":"10.1016\/j.molliq.2020.113310","article-title":"Investigation the nanofluid flow through a nanochannel to study the effect of nanoparticles on the condensation phenomena","volume":"311","author":"Ghahremanian","year":"2020","journal-title":"J. Mol. Liq."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"106347","DOI":"10.1016\/j.ijthermalsci.2020.106347","article-title":"Bernardo HerreraNanofluids stability effect on a thermosyphon thermal performance","volume":"153","author":"Cacua","year":"2020","journal-title":"Int. J. Therm. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Lund, L.A., Omar, Z., Khan, I., Sherif, E.-S.M., and Abdo, H.S. (2020). Stability Analysis of the Magnetized Casson Nanofluid Propagating through an Exponentially Shrinking\/Stretching Plate: Dual Solutions. Symmetry, 12.","DOI":"10.3390\/sym12071162"},{"key":"ref_8","first-page":"159","article-title":"Darcy-Forchheimer relation in Casson type MHD nanofluid flow over non-linear stretching surface","volume":"9","author":"Rasool","year":"2020","journal-title":"Prop. Power Res."},{"key":"ref_9","first-page":"2134","article-title":"Numerical analysis of the onset of longitudinal convective rolls in a porous medium saturated by an electrically conducting nanofluid in the presence of an external magnetic field","volume":"7","author":"Wakif","year":"2017","journal-title":"Res. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"085704","DOI":"10.1088\/1402-4896\/ab1307","article-title":"Active and passive controls of 3D nanofluid flow by a convectively heated nonlinear stretching surface","volume":"94","author":"Hayat","year":"2019","journal-title":"Phy. Scr."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Rasool, G., and Wakif, A. (2020). Numerical spectral examination of EMHD mixed convective flow of second-grade nanofluid towards a vertical Riga plate using an advanced version of the revised Buongiorno\u2019s nanofluid model. J. Therm. Anal. Calorim., accepted.","DOI":"10.1007\/s10973-020-09865-8"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"511","DOI":"10.1080\/10407780120202","article-title":"Coupled heat and mass transfer by natural convection about a truncated cone in the presence of magnetic field and radiation effects","volume":"39","author":"Chamkha","year":"2001","journal-title":"Numer. Heat Trans. Part A"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"122009","DOI":"10.1063\/1.4993866","article-title":"Heat source location and natural convection in a c-shaped enclosure saturated by a nanofluid","volume":"29","author":"Mohebbi","year":"2017","journal-title":"Phys. Fluids"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Rasool, G., Shafiq, A., Khan, I., Baleanu, D., Nisar, K.S., and Shahzadi, G. (2020). Entropy generation and consequences of MHD in Darcy\u2013Forchheimer nanofluid flow bounded by non-linearly stretching surface. Symmetry, 12.","DOI":"10.3390\/sym12040652"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Lund, L.A., Omar, Z., Raza, J., Khan, I., and Sherif, E.-S.M. (2020). Effects of Stefan Blowing and Slip Conditions on Unsteady MHD Casson Nanofluid Flow Over an Unsteady Shrinking Sheet: Dual Solutions. Symmetry, 12.","DOI":"10.3390\/sym12030487"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"045206","DOI":"10.1088\/1402-4896\/ab3c3f","article-title":"On the onset of entropy generation for a nanofluid with thermal radiation and gyrotactic microorganisms through 3D flows","volume":"95","author":"Sohail","year":"2020","journal-title":"Phys. Scr."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Rasool, G., Zhang, T., Chamkha, A.J., Shafiq, A., Tlili, I., and Shahzadi, G. (2020). Entropy generation and consequences of binary chemical reaction on MHD Darcy\u2013Forchheimer Williamson nanofluid flow over non-linearly stretching surface. Entropy, 22.","DOI":"10.3390\/e22010018"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2846","DOI":"10.1016\/j.physleta.2018.06.015","article-title":"An optimal analysis for Darcy\u2013Forchheimer 3D flow of nanofluid with convective condition and homogeneous\u2013heterogeneous reactions","volume":"382","author":"Hayat","year":"2018","journal-title":"Phys. Lett. A"},{"key":"ref_19","first-page":"1632","article-title":"Darcy-Forchheimer relation in Magnetohydrodynamic Jeffrey nanofluid flow over stretching surface","volume":"30","author":"Rasool","year":"2020","journal-title":"Discret. Contin. Dyn. Syst.-S"},{"key":"ref_20","first-page":"1438","article-title":"A semi-analytical analysis of electro-thermo-hydrodynamic stability in dielectric nanofluids using Buongiorno\u2019s mathematical model together with more realistic boundary conditions","volume":"9","author":"Wakif","year":"2018","journal-title":"Res. Phys."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1207","DOI":"10.1016\/j.apt.2016.04.005","article-title":"Soret and Dufour effects on MHD convective flow of Al2O3-Water and TiO2-Water Nanofluids Past a Stretching Sheet in Porous Media with Heat Generation\/Absorption","volume":"27","author":"Reddy","year":"2016","journal-title":"Adv. Powder Technol."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"125212","DOI":"10.1088\/1402-4896\/ab3990","article-title":"Second grade nanofluidic flow past a convectively heated vertical Riga plate","volume":"94","author":"Rasool","year":"2019","journal-title":"Phys. Scr."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1007\/s40819-018-0513-y","article-title":"Numerical Analysis of the Unsteady Natural Convection MHD Couette Nanofluid Flow in the Presence of Thermal Radiation Using Single and Two-Phase Nanofluid Models for Cu\u2013Water Nanofluids","volume":"4","author":"Wakif","year":"2018","journal-title":"Int. J. Appl. Comput. Math."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"738","DOI":"10.1016\/j.ijheatmasstransfer.2019.04.037","article-title":"Natural convective flow and heat transfer of nano-encapsulated phase change materials (NEPCMs) in a cavity","volume":"138","author":"Ghalambaz","year":"2019","journal-title":"Int. J. Heat Mass Trans."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Rasool, G., and Zhang, T. (2019). Darcy-Forchheimer nanofluidic flow manifested with Cattaneo-Christov theory of heat and mass flux over non-linearly stretching surface. PLoS ONE, 14.","DOI":"10.1371\/journal.pone.0221302"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"629","DOI":"10.1016\/j.molliq.2018.06.017","article-title":"Entropy generation in Darcy-Forchheimer bidirectional flow of water-based carbon nanotubes with convective boundary conditions","volume":"265","author":"Hayat","year":"2018","journal-title":"J. Mol. Liq."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"105251","DOI":"10.1088\/1402-4896\/ab18c8","article-title":"Magnetohydrodynamic Darcy\u2013Forchheimer nanofluid flow over a nonlinear stretching sheet","volume":"94","author":"Rasool","year":"2019","journal-title":"Phys. Scr."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Lund, L.A., Omar, Z., Raza, J., and Khan, I. (2020). Magnetohydrodynamic flow of Cu\u2013Fe3O4\/H2O hybrid nanofluid with effect of viscous dissipation: Dual similarity solutions. J. Therm. Anal. Calorim.","DOI":"10.1007\/s10973-020-09602-1"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"124088","DOI":"10.1016\/j.physa.2019.124088","article-title":"Modified heat and mass transmission models in the magnetohydrodynamic flow of Sutterby nanofluid in stretching cylinder","volume":"549","author":"Sohail","year":"2020","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_30","unstructured":"Darcy, H. (1856). Les Fontaines Publiques de la Ville de Dijon, Victor Dalmont."},{"key":"ref_31","first-page":"1782","article-title":"Wasserbewegung durch boden","volume":"45","author":"Forchheimer","year":"1901","journal-title":"Z. Ver. Deutsch. Ing."},{"key":"ref_32","unstructured":"Muskat, M. (1946). The Flow of Homogeneous Fluids Through Porous Media, Edwards."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Muhammad, T., Rafique, K., Asma, M., and Alghamdi, M. (2020). Darcy\u2013Forchheimer flow over an exponentially stretching curved surface with Cattaneo\u2013Christov double diffusion. Physica A, accepted.","DOI":"10.1016\/j.physa.2019.123968"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"115231","DOI":"10.1016\/j.applthermaleng.2020.115231","article-title":"Analysis of hydro-thermal and entropy generation characteristics of nanofluid in an aluminium foam heat sink by employing Darcy-Forchheimer-Brinkman model coupled with multiphase Eulerian model","volume":"173","author":"Ambreen","year":"2020","journal-title":"App. Therm. Eng."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"124024","DOI":"10.1016\/j.physa.2019.124024","article-title":"Significance of Arrhenius activation energy in Darcy\u2013Forchheimer 3D rotating flow of nanofluid with radiative heat transfer","volume":"550","author":"Ullah","year":"2020","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_36","unstructured":"Huda, N.U., Hamid, A., and Khan, M. (2020). Impact of Cattaneo-Christov model on Darcy\u2013Forchheimer flow of ethylene glycol basefluid over a moving needle. J. Mater. Res. Technol., accepted."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"035102","DOI":"10.1063\/1.5019218","article-title":"Darcy-Forchheimer flow of Maxwell nanofluid flow with nonlinear thermal radiation and activation energy","volume":"8","author":"Sajid","year":"2018","journal-title":"AIP Adv."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"119758","DOI":"10.1016\/j.ijheatmasstransfer.2020.119758","article-title":"Physical modeling of simultaneous heat and mass transfer: Species interdiffusion, Soret effect and Dufour effect","volume":"15","author":"Jiang","year":"2020","journal-title":"Int. J. Heat Mass Trans."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"713","DOI":"10.1016\/j.ijheatmasstransfer.2019.05.026","article-title":"Multiple-relaxation-time lattice Boltzmann model for double-diffusive convection with Dufour and Soret effects","volume":"13","author":"Liu","year":"2019","journal-title":"Int. J. Heat Mass Trans."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"809","DOI":"10.1016\/j.ijheatmasstransfer.2019.03.132","article-title":"Investigation of mixed convection flow of Carreau nanofluid over a wedge in the presence of Soret and Dufour effects","volume":"137","author":"Sardar","year":"2019","journal-title":"Int. J. Heat Mass Trans."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"435","DOI":"10.1016\/j.molliq.2018.04.095","article-title":"Impacts of binary chemical reaction with activation energy on unsteady flow of magneto-Williamson nanofluid","volume":"262","author":"Hamid","year":"2018","journal-title":"J. Mol. Liq."},{"key":"ref_42","first-page":"149","article-title":"Activation energy and binary chemical reaction effects in mixed convective nanofluid flow with convective boundary conditions","volume":"6","author":"Dhlamini","year":"2019","journal-title":"J. Comput. Des. Eng."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"188","DOI":"10.1016\/j.jmrt.2019.10.044","article-title":"Activation energy on MHD flow of titanium alloy (Ti6Al4V) nanoparticle along with a cross flow and stream wise direction with binary chemical reaction and non-linear radiation: Dual Solutions","volume":"9","author":"Khan","year":"2020","journal-title":"J. Mater. Res. Technol."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1942","DOI":"10.1016\/j.cnsns.2010.08.033","article-title":"Effects of Soret Dufour, chemical reaction and thermal radiation on MHD non-Darcy unsteady mixed convective heat and mass transfer over a stretching sheet","volume":"16","author":"Pal","year":"2011","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"150","DOI":"10.1016\/j.ijheatmasstransfer.2014.06.052","article-title":"Three dimensional MHD flow and heat transfer over a stretching\/shrinking surface in a viscoelastic fluid with various physical effects","volume":"78","author":"Turkyilmazoglu","year":"2014","journal-title":"Int. J. Heat Mass Trans."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/9\/1421\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:06:57Z","timestamp":1760177217000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/9\/1421"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,8,26]]},"references-count":45,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2020,9]]}},"alternative-id":["sym12091421"],"URL":"https:\/\/doi.org\/10.3390\/sym12091421","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,8,26]]}}}