{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,19]],"date-time":"2026-06-19T01:39:26Z","timestamp":1781833166956,"version":"3.54.5"},"reference-count":38,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2020,4,14]],"date-time":"2020-04-14T00:00:00Z","timestamp":1586822400000},"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 study mainly concerns with the examination of heat transfer rate, mass and motile micro-organisms for convective second grade nanofluid flow. The considered model comprises of both nanoparticles as well as gyrotactic micro-organisms. Microorganisms stabilize the suspension of nanoparticles by bio-convective flow which is generated by the combined effects of nanoparticles and buoyancy forces. The Brownian motion and thermophoretic mechanisms along with Newtonian heating are also considered. Appropriately modified transformations are invoked to get a non-linear system of differential equations. The resulting problems are solved using a numerical scheme. Velocity field, thermal and solute distributions and motile micro-organism density are discussed graphically. Wall-drag (skin-friction) coefficient, Nusselt, Sherwood and motile micro-organisms are numerically examined for various parameters. The outcomes indicate that for a larger Rayleigh number, the bio-convection restricts the upward movement of nanoparticles that are involved in nanofluid for the given buoyancy effect. Furthermore, larger buoyancy is instigated which certainly opposes the fluid flow and affects the concentration. For a larger values of fluid parameter, the fluid viscosity faces a decline and certainly less restriction is faced by the fluid. In both assisting and opposing cases, we notice a certain rise in fluid motion. Thermal layer receives enhancement for larger values of Brownian diffusion parameter. The random motion for stronger Brownian impact suddenly raises which improves the heat convection and consequently thermal distribution receives enhancement. Thermal distribution receives enhancement for a larger Lewis number whereas the decline is noticed in concentration distribution. The larger Rayleigh number results in a strong buoyancy force that effectively increases the fluid temperature. This also increases the concentration difference, thus more nanoparticles transport between surface and micro-organisms. Furthermore, for larger (Nb), the thermal state of fluid receives enhancement while a decline in motile density is observed. Numerical results show that mass flux is an enhancing function of both the (Le) and (Nb).<\/jats:p>","DOI":"10.3390\/sym12040621","type":"journal-article","created":{"date-parts":[[2020,4,15]],"date-time":"2020-04-15T04:01:46Z","timestamp":1586923306000},"page":"621","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":98,"title":["Second Grade Bioconvective Nanofluid Flow with Buoyancy Effect and Chemical Reaction"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7186-7216","authenticated-orcid":false,"given":"Anum","family":"Shafiq","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China"},{"name":"International Institute for Symmetry Analysis and Mathematical Modeling, Department of Mathematical Sciences, North-West University, Ma keng Campus, Private Bag X 2046, Mmabatho 2735, South Africa"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5880-9553","authenticated-orcid":false,"given":"Ghulam","family":"Rasool","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1986-4859","authenticated-orcid":false,"given":"Chaudry Masood","family":"Khalique","sequence":"additional","affiliation":[{"name":"International Institute for Symmetry Analysis and Mathematical Modeling, Department of Mathematical Sciences, North-West University, Ma keng Campus, Private Bag X 2046, Mmabatho 2735, South Africa"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Sohail","family":"Aslam","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Preston University, Islamabad 44000, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2020,4,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1241","DOI":"10.1007\/s11671-010-9638-6","article-title":"Nanofluids Research: Key Issue","volume":"5","author":"Wang","year":"2010","journal-title":"Nanoscale Res. Lett."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1421","DOI":"10.1016\/j.icheatmasstransfer.2010.08.015","article-title":"The onset of nanofluid bioconvection in a suspension containing both nanoparticles and gyrotactic micro-organisms","volume":"37","author":"Kuznetsov","year":"2010","journal-title":"Int. Commun. Heat Mass Trans."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"055505","DOI":"10.1088\/0169-5983\/43\/5\/055505","article-title":"Nanofluid bio-thermal convection: Simultaneous effects of gyrotactic and oxytactic micro-organisms","volume":"43","author":"Kuznetsov","year":"2011","journal-title":"Fluid Dyn. Res."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1016\/j.ijthermalsci.2012.01.011","article-title":"Free convection boundary layer flow past a horizontal flat plate embedded in porous medium filled by nanofluid containing gyrotactic micro-organisms","volume":"56","author":"Aziz","year":"2012","journal-title":"Int. J. Therm. Sci."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"647","DOI":"10.1016\/j.ijheatmasstransfer.2013.03.012","article-title":"Mixed convection flow over a solid sphere embedded in a porous medium filled by a nanofluid containing gyrotactic micro-organisms","volume":"62","author":"Tham","year":"2013","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/j.euromechflu.2014.02.005","article-title":"Fully developed mixed convection flow in a horizontal channel filled by a nanofluid containing both nanoparticles and gyrotactic micro-organisms","volume":"46","author":"Xu","year":"2014","journal-title":"Eur. J. Mech. B Fluids"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1007\/s11242-012-0120-z","article-title":"Free convection of Non-Newtonian nanofluids in porous media with gyrotactic micro-organisms","volume":"97","author":"Khan","year":"2013","journal-title":"Transp. Porous Med."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"88","DOI":"10.1016\/j.compfluid.2014.02.026","article-title":"Hydromagnetic bioconvection of nanofluid over a permeable vertical plate due to gyrotactic micro-organisms","volume":"95","author":"Mutuku","year":"2014","journal-title":"Comput. Fluids"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"214","DOI":"10.1108\/HFF-07-2013-0242","article-title":"Bioconvection peristaltic flow in an asymmetric channel filled by nanofluid containing gyrotactic micro-organism: Bio nano engineering model","volume":"25","author":"Akbar","year":"2015","journal-title":"Int. J. Numer. Meth. Heat Fluid Flow"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1063\/1.4903574","article-title":"Analytical modelling of free convection of non-Newtonian nanofluids flow in porous media with gyrotactic micro-organisms using OHAM","volume":"1635","author":"Mabood","year":"2014","journal-title":"AIP Conf. Proc."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"105221","DOI":"10.1088\/1402-4896\/ab18c8","article-title":"Magnetohydrodynamic Darcy Forchheimer nanofluid flow over non-linear stretching sheet","volume":"94","author":"Rasool","year":"2019","journal-title":"Phys. Scr."},{"key":"ref_12","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_13","doi-asserted-by":"crossref","unstructured":"Lund, L.A., Omar, Z., Khan, I., Raza, J., Bakouri, M., and Tlili, I. (2019). Stability analysis of Darcy-Forchheimer flow of Casson type Nanofluid over an exponential sheet: Investigation of critical points. Symmetry, 11.","DOI":"10.3390\/sym11030412"},{"key":"ref_14","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_15","doi-asserted-by":"crossref","first-page":"1283","DOI":"10.1007\/s11771-019-4087-6","article-title":"Multiple solutions of Cu-C6 H9 NaO7 and Ag-C6 H9 NaO7 nanofluids flow over non-linear shrinking surface","volume":"26","author":"Lund","year":"2019","journal-title":"J. Centr. South Univ."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"274","DOI":"10.1002\/htj.21612","article-title":"Marangoni convective nano-fluid flow over an electromagnetic actuator in the presence of first order chemical reaction","volume":"49","author":"Rasool","year":"2019","journal-title":"Heat Transf.-Asian Res."},{"key":"ref_17","unstructured":"Rasool, G., Shafiq, A., and Durur, H. (2019). Darcy-Forchheimer relation in Magnetohydrodynamic Jeffrey nanofluid flow over stretching surface. Discr. Contin. Dyn. Syst.-Ser. S, accepted."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Sohail, M., Naz, R., and Abdelsalam, S.I. (2019). On the onset of entropy generation for a nanofluid with thermal radiation and gyrotactic micro-organisms through 3D flows. Phys. Scr., accepted.","DOI":"10.1088\/1402-4896\/ab3c3f"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Shafiq, A., Rasool, G., and Khalique, C.M. (2020). Significance of thermal slip and convective boundary conditions in three dimensional rotating Darcy-Forchheimer nanofluid flow. Symmetry, accepted.","DOI":"10.3390\/sym12050741"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Sohail, M., and Naz, R. (2020). Modified heat and mass transmission models in the magnetohydrodynamic flow of Sutterby nanofluid in stretching cylinder. Phys. A Stat. Mech. Its Appl., 124088.","DOI":"10.1016\/j.physa.2019.124088"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"e01479","DOI":"10.1016\/j.heliyon.2019.e01479","article-title":"Characteristics of chemical reaction and convective boundary conditions in Powell-Eyring nanofluid flow along a radiative Riga plate","volume":"5","author":"Rasool","year":"2019","journal-title":"Heliyon"},{"key":"ref_22","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\u2014Forchheimer Williamson Nanofluid Flow Over Non-Linearly Stretching Surface. Entropy, 22.","DOI":"10.3390\/e22010018"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Chamkha, A.J., Ismael, M., Kasaeipoor, A., and Armaghani, T. (2016). Entropy Generation and Natural Convection of CuO-Water Nanofluid in C-Shaped Cavity under Magnetic Field. Entropy, 18.","DOI":"10.3390\/e18020050"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"94","DOI":"10.1108\/09615530010306939","article-title":"Similarity Solutions for Hydromagnetic Mixed Convection Heat and Mass Transfer for Hiemenz Flow Through Porous Media","volume":"10","author":"Chamkha","year":"2000","journal-title":"Int. J. Numer. Methods Heat Fluid Flow"},{"key":"ref_25","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-Forchheimer nanofluid flow bounded by non-linearly stretching surface. Symmetry, accepted.","DOI":"10.3390\/sym12040652"},{"key":"ref_26","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_27","doi-asserted-by":"crossref","first-page":"392","DOI":"10.1016\/0142-727X(94)90053-1","article-title":"Natural convection boundary-layer flow on a vertical surface with Newtonian heating","volume":"15","author":"Merkin","year":"1992","journal-title":"Int. J. Heat Fluid Flow"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1515\/zna-2014-0280","article-title":"Boundary-layer flow of Walters\u2019B fluid with Newtonian heating","volume":"70","author":"Hayat","year":"2015","journal-title":"Zeitschrift f\u00fcr Naturforschung A"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"555","DOI":"10.1590\/0104-6632.20150322s00001918","article-title":"Newtonian heating, thermal diffusion and duffsion thermo effects in an axisymmetric flow of Jeffrey fluid over a stretching surface","volume":"32","author":"Awais","year":"2015","journal-title":"Braz. J. Chem. Eng."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1017\/jmech.2014.93","article-title":"MHD flow of a Jeffrey fluid with Newtonian heating","volume":"31","author":"Farooq","year":"2015","journal-title":"J. Mech."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"68","DOI":"10.1108\/HFF-03-2014-0070","article-title":"Three dimensional flow of an Oldroyd-B fluid with Newtonian Heating","volume":"25","author":"Ramzan","year":"2015","journal-title":"Int. J. Numer. Meth. Heat Fluid Flow"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1007\/s10483-015-1895-9","article-title":"Newtonian heating in stagnation point flow of Burgers fluid","volume":"36","author":"Hayat","year":"2015","journal-title":"Appl. Math. Mech."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0735-1933(03)00196-9","article-title":"Effect of small particles on the stability of bioconvection in a suspension of gyrotactic micro-organisms in a layer of finite depth","volume":"31","author":"Kuznetsov","year":"2014","journal-title":"Int. Commun. Heat Mass Transf."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"629","DOI":"10.1016\/S0735-1933(04)00050-8","article-title":"Effect of small solid particles on the development of bioconvection plumes","volume":"31","author":"Geng","year":"2004","journal-title":"Int. Commun. Heat Mass Transf."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"328","DOI":"10.1108\/09615530510590597","article-title":"The interaction of bioconvection caused by gyrotactic micro-organisms and settling of small solid particles","volume":"15","author":"Kuznetsov","year":"2005","journal-title":"Int. J. Numer. Methods Heat Fluid Flow"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"2048","DOI":"10.1016\/j.cjph.2017.08.005","article-title":"Bioconvection nanofluid slip flow past a wavy surface with applications in nano-biofuel cells","volume":"55","author":"Uddin","year":"2017","journal-title":"Chin. J. Phys."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1186\/1556-276X-6-100","article-title":"Nanofluid bioconvection in water-based suspensions containing nanoparticles and oxytactic micro-organisms: Oscillatory instability","volume":"6","author":"Kuznetsov","year":"2011","journal-title":"Nanoscale Res. Lett."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"111231","DOI":"10.1016\/j.molliq.2019.111231","article-title":"Analysis on the bioconvection flow of modified second-grade nanofluid containing gyrotactic micro-organisms and nanoparticles","volume":"291","author":"Waqas","year":"2019","journal-title":"J. Mol. Liq."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/4\/621\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T13:30:44Z","timestamp":1760362244000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/4\/621"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,4,14]]},"references-count":38,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2020,4]]}},"alternative-id":["sym12040621"],"URL":"https:\/\/doi.org\/10.3390\/sym12040621","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,4,14]]}}}