{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,9]],"date-time":"2026-03-09T15:31:58Z","timestamp":1773070318197,"version":"3.50.1"},"reference-count":50,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2020,4,22]],"date-time":"2020-04-22T00:00:00Z","timestamp":1587513600000},"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>A nonlocal fractional model of Brinkman type fluid (BTF) containing a hybrid nanostructure was examined. The magnetohydrodynamic (MHD) flow of the hybrid nanofluid was studied using the fractional calculus approach. Hybridized silver (Ag) and Titanium dioxide (TiO2) nanoparticles were dissolved in base fluid water (H2O) to form a hybrid nanofluid. The MHD free convection flow of the nanofluid (Ag-TiO2-H2O) was considered in a microchannel (flow with a bounded domain). The BTF model was generalized using a nonlocal Caputo-Fabrizio fractional operator (CFFO) without a singular kernel of order    \u03b1    with effective thermophysical properties. The governing equations of the model were subjected to physical initial and boundary conditions. The exact solutions for the nonlocal fractional model without a singular kernel were developed via the fractional Laplace transform technique. The fractional solutions were reduced to local solutions by limiting     \u03b1 \u2192 1    . To understand the rheological behavior of the fluid, the obtained solutions were numerically computed and plotted on various graphs. Finally, the influence of pertinent parameters was physically studied. It was found that the solutions were general, reliable, realistic and fixable. For the fractional parameter, the velocity and temperature profiles showed a decreasing trend for a constant time. By setting the values of the fractional parameter, excellent agreement between the theoretical and experimental results could be attained.<\/jats:p>","DOI":"10.3390\/sym12040663","type":"journal-article","created":{"date-parts":[[2020,4,23]],"date-time":"2020-04-23T02:10:52Z","timestamp":1587607852000},"page":"663","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":48,"title":["Symmetric MHD Channel Flow of Nonlocal Fractional Model of BTF Containing Hybrid Nanoparticles"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2248-9903","authenticated-orcid":false,"given":"Muhammad","family":"Saqib","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia JB, Johor Bahru 81310, Johor, Malaysia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sharidan","family":"Shafie","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia JB, Johor Bahru 81310, Johor, Malaysia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2056-9371","authenticated-orcid":false,"given":"Ilyas","family":"Khan","sequence":"additional","affiliation":[{"name":"Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh 72915, Vietnam"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yu-Ming","family":"Chu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Huzhou University, Huzhou 313000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5769-4320","authenticated-orcid":false,"given":"Kottakkaran Sooppy","family":"Nisar","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Arts and Sciences, Prince Sattam bin Abdulaziz University, Wadi Aldawaser 11991, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,4,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1347","DOI":"10.1016\/j.ijheatmasstransfer.2018.06.005","article-title":"Cu-Al2O3\/Water hybrid nanofluid through a permeable surface in the presence of nonlinear radiation and variable thermal conductivity via LSM","volume":"126","author":"Usman","year":"2018","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1231","DOI":"10.1080\/15376494.2016.1232454","article-title":"Shape effects of spherical and nonspherical nanoparticles in mixed convection flow over a vertical stretching permeable sheet","volume":"24","author":"Ellahi","year":"2017","journal-title":"Mech. Adv. Mater. Struct."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1007\/s12043-019-1781-8","article-title":"Peristaltic transport of Jeffrey fluid in a rectangular duct through a porous medium under the effect of partial slip: An application to upgrade industrial sieves\/filters","volume":"93","author":"Ellahi","year":"2019","journal-title":"Pramana"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1615\/JPorMedia.v20.i5.70","article-title":"Study of peristaltic flow of nanofluid with entropy generation in a porous medium","volume":"20","author":"Ellahi","year":"2017","journal-title":"J. Porous Media"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1016\/j.molliq.2016.12.073","article-title":"On boundary layer nano-ferroliquid flow under the influence of low oscillating stretchable rotating disk","volume":"229","author":"Ellahi","year":"2017","journal-title":"J. Mol. Liq."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Ellahi, R., Zeeshan, A., Hussain, F., and Abbas, T. (2019). Two-phase couette flow of couple stress fluid with temperature dependent viscosity thermally affected by magnetized moving surface. Symmetry, 11.","DOI":"10.3390\/sym11050647"},{"key":"ref_7","unstructured":"Routbort, J.L., Singh, D., and Chen, G. (2006). Heavy Vehicle Systems Optimization Merit Review and Peer Evaluation, Argonne National Laboratory. Annual Report."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"280","DOI":"10.1115\/1.2825978","article-title":"Measuring thermal conductivity of fluids containing oxide nanoparticles","volume":"121","author":"Lee","year":"1999","journal-title":"J. Heat Transf."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Mahian, O., Kolsi, L., Amani, M., Estell\u00e9, P., Ahmadi, G., Kleinstreuer, C., Marshall, J.S., Siavashi, M., Taylor, R.A., and Niazmand, H. (2018). Recent advances in modeling and simulation of nanofluid flows-part I: Fundamental and theory. Phys. Rep., 1\u201348.","DOI":"10.1016\/j.physrep.2018.11.004"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Mahian, O., Kolsi, L., Amani, M., Estell\u00e9, P., Ahmadi, G., Kleinstreuer, C., Marshall, J.S., Taylor, R.A., Abu-Nada, E., and Rashidi, S. (2018). Recent advances in modeling and simulation of nanofluid flows-part II: Applications. Phys. Rep.","DOI":"10.1016\/j.physrep.2018.11.003"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1016\/j.icheatmasstransfer.2018.10.001","article-title":"Rheological behaviour of functionalized graphene nanoplatelet nanofluids based on water and propylene glycol: Water mixtures","volume":"99","author":"Vallejo","year":"2018","journal-title":"Int. Commun. Heat Mass Transf."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Vallejo, J.P., \u017by\u0142a, G., Fern\u00e1ndez-Seara, J., and Lugo, L. (2019). Influence of Six Carbon-Based Nanomaterials on the Rheological Properties of Nanofluids. Nanomaterials, 9.","DOI":"10.3390\/nano9020146"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1016\/j.physe.2018.07.023","article-title":"Ethylene glycol based silicon nitride nanofluids: An experimental study on their thermophysical, electrical and optical properties","volume":"104","author":"Fal","year":"2018","journal-title":"Phys. E Low Dimens. Syst. Nanostruct."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Fal, J., Mahian, O., and \u017by\u0142a, G. (2018). Nanofluids in the Service of High Voltage Transformers: Breakdown Properties of Transformer Oils with Nanoparticles, a Review. Energies, 11.","DOI":"10.3390\/en11112942"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"155","DOI":"10.12693\/APhysPolA.132.155","article-title":"Effect of Temperature and Mass Concentration of SiO2 Nanoparticles on Electrical Conductivity of Ethylene Glycol","volume":"132","author":"Fal","year":"2017","journal-title":"Acta Phys. Pol. A"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1134","DOI":"10.1016\/j.rser.2017.03.113","article-title":"A state of the art review on viscosity of nanofluids","volume":"76","author":"Murshed","year":"2017","journal-title":"Renew. Sustain. Energy Rev."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"931","DOI":"10.1016\/j.rser.2018.07.006","article-title":"Current trends in surface tension and wetting behavior of nanofluids","volume":"94","author":"Cabaleiro","year":"2018","journal-title":"Renew. Sustain. Energy Rev."},{"key":"ref_18","first-page":"140","article-title":"MHD Natural Convection in a Triangular Cavity filled with a Cu-Al2O3\/Water Hybrid Nanofluid with Localized Heating from Below and Internal Heat Generation","volume":"7","author":"Rashad","year":"2018","journal-title":"J. Heat Transf."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"154","DOI":"10.1016\/j.ijmecsci.2018.11.019","article-title":"Natural convection of a Magnetizable hybrid nanofluid inside a porous enclosure subjected to two variable magnetic fields","volume":"151","author":"Izadi","year":"2019","journal-title":"Int. J. Mech. Sci."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1016\/j.jtice.2017.12.029","article-title":"Heat transfer and entropy generation optimization for flow of a non-Newtonian hybrid nanofluid containing coated CNT\/Fe3O4 nanoparticles in a concentric annulus","volume":"84","author":"Shahsavar","year":"2018","journal-title":"J. Taiwan Inst. Chem. Eng."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Farooq, U., Afridi, M., Qasim, M., and Lu, D. (2018). Transpiration and Viscous Dissipation Effects on Entropy Generation in Hybrid Nanofluid Flow over a Nonlinear Radially Stretching Disk. Entropy, 20.","DOI":"10.3390\/e20090668"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"529","DOI":"10.1111\/j.1365-246X.1967.tb02303.x","article-title":"Linear models of dissipation whose Q is almost frequency independent\u2014II","volume":"13","author":"Caputo","year":"1967","journal-title":"Geophys. J. Int."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Saqib, M., Khan, I., and Shafie, S. (2019). New Direction of Atangana\u2014Baleanu Fractional Derivative with Mittag-Leffler Kernel for Non-Newtonian Channel Flow. Fractional Derivatives with Mittag-Leffler Kernel, Springer.","DOI":"10.1007\/978-3-030-11662-0_15"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0370-1573(00)00070-3","article-title":"The random walk\u2019s guide to anomalous diffusion: A fractional dynamics approach","volume":"339","author":"Metzler","year":"2000","journal-title":"Phys. Rep."},{"key":"ref_25","first-page":"1","article-title":"A new definition of fractional derivative without singular kernel","volume":"1","author":"Caputo","year":"2015","journal-title":"Prog. Fract. Differ. Appl."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1140\/epjp\/i2016-16377-x","article-title":"Application of Caputo-Fabrizio derivatives to MHD free convection flow of generalized Walters\u2019-B fluid model","volume":"131","author":"Ali","year":"2016","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1865","DOI":"10.1007\/s00521-016-2815-5","article-title":"A modern approach of Caputo\u2014Fabrizio time-fractional derivative to MHD free convection flow of generalized second-grade fluid in a porous medium","volume":"30","author":"Sheikh","year":"2018","journal-title":"Neural Comput. Appl."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2789","DOI":"10.1016\/j.aej.2016.07.022","article-title":"Flow over an infinite plate of a viscous fluid with non-integer order derivative without singular kernel","volume":"55","author":"Zafar","year":"2016","journal-title":"Alex. Eng. J."},{"key":"ref_29","first-page":"270","article-title":"Derivatives with non-singular kernels from the Caputo\u2014Fabrizio definition and beyond: Appraising analysis with emphasis on diffusion models","volume":"1","author":"Hristov","year":"2017","journal-title":"Front. Fract. Calc."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"5","DOI":"10.22436\/jnsa.009.05.46","article-title":"On the new fractional derivative and application to nonlinear Baggs and Freedman model","volume":"9","author":"Atangana","year":"2016","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"763","DOI":"10.2298\/TSCI160111018A","article-title":"New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model","volume":"20","author":"Atangana","year":"2016","journal-title":"Therm. Sci."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"3043","DOI":"10.1007\/s11012-017-0652-y","article-title":"On the notion of fractional derivative and applications to the hysteresis phenomena","volume":"52","author":"Caputo","year":"2017","journal-title":"Meccanica"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"135","DOI":"10.3389\/fphy.2018.00135","article-title":"Linear viscoelastic responses: The Prony decomposition naturally leads into the Caputo-Fabrizio fractional operator","volume":"6","author":"Hristov","year":"2018","journal-title":"Front. Phys."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1007\/s00231-007-0251-x","article-title":"Magnetic field effect on heat transfer and fluid flow characteristics of blood flow in multi-stenosis arteries","volume":"44","author":"Tashtoush","year":"2008","journal-title":"Heat Mass Transf."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1472","DOI":"10.1002\/num.22200","article-title":"Engine oil based generalized brinkman-type nano-liquid with molybdenum disulphide nanoparticles of spherical shape: Atangana-Baleanu fractional model","volume":"34","author":"Jan","year":"2018","journal-title":"Numer. Methods Partial Differ. Equ."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"630","DOI":"10.1016\/j.euromechflu.2009.05.006","article-title":"Natural convection cooling of a localised heat source at the bottom of a nanofluid-filled enclosure","volume":"28","author":"Aminossadati","year":"2009","journal-title":"Eur. J. Mech. B Fluids"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Brinkman, H.C. (1952). The viscosity of concentrated suspensions and solutions. J. Chem. Phys., 20.","DOI":"10.1063\/1.1700493"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1016\/j.ijheatmasstransfer.2013.09.006","article-title":"Modeling the natural convective flow of micropolar nanofluids","volume":"68","author":"Bourantas","year":"2014","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"425","DOI":"10.3367\/UFNr.0135.198111d.0425","article-title":"Maxwell a treatise on electricity and magnetism","volume":"135","author":"Levin","year":"1981","journal-title":"Uspekhi Fiz. Nauk"},{"key":"ref_40","unstructured":"Maxwell, J.C. (1954). Electricity and Magnetism, Dover."},{"key":"ref_41","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_42","first-page":"16","article-title":"Recent Advancment in Thermophysical Properties of Nanofluids and Hybrid Nanofluids: An Overview","volume":"3","author":"Saqib","year":"2019","journal-title":"City Univ. Int. J. Comput. Anal."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1054","DOI":"10.1016\/j.ijheatmasstransfer.2017.06.135","article-title":"Entropy generation analysis in MHD mixed convection of hybrid nanofluid in an open cavity with a horizontal channel containing an adiabatic obstacle","volume":"114","author":"Hussain","year":"2017","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1186\/s13662-019-1988-5","article-title":"Application of fractional differential equations to heat transfer in hybrid nanofluid: Modeling and solution via integral transforms","volume":"2019","author":"Saqib","year":"2019","journal-title":"Adv. Differ. Equ."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"1663","DOI":"10.1061\/(ASCE)0733-9399(1993)119:8(1663)","article-title":"Dynamic analysis of generalized viscoelastic fluids","volume":"119","author":"Makris","year":"1993","journal-title":"J. Eng. Mech."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"082001","DOI":"10.1063\/1.4996034","article-title":"Free convection flow of some fractional nanofluids over a moving vertical plate with uniform heat flux and heat source","volume":"29","author":"Azhar","year":"2017","journal-title":"Phys. Fluids"},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Safaei, M.R., Ahmadi, G., Goodarzi, M.S., Safdari Shadloo, M., Goshayeshi, H.R., and Dahari, M. (2016). Heat Transfer and Pressure Drop in Fully Developed Turbulent Flows of Graphene Nanoplatelets\u2014Silver\/Water Nanofluids. Fluids, 1.","DOI":"10.3390\/fluids1030020"},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Abdollahzadeh Jamalabadi, M.Y., Ghasemi, M., Alamian, R., Wongwises, S., Afrand, M., and Shadloo, M.S. (2019). Modeling of Subcooled Flow Boiling with Nanoparticles under the Influence of a Magnetic Field. Symmetry, 11.","DOI":"10.3390\/sym11101275"},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Irandoost Shahrestani, M., Maleki, A., Safdari Shadloo, M., and Tlili, I. (2020). Numerical Investigation of Forced Convective Heat Transfer and Performance Evaluation Criterion of Al2O3\/Water Nanofluid Flow inside an Axisymmetric Microchannel. Symmetry, 12.","DOI":"10.3390\/sym12010120"},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Ellahi, R., Hussain, F., Abbas, S.A., Sarafraz, M.M., Goodarzi, M., and Safdari Shadloo, M. (2020). Study of two-phase newtonian nanofluid flow hybrid with hafnium particles under the effects of slip. Inventions, 5.","DOI":"10.3390\/inventions5010006"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/4\/663\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T14:26:48Z","timestamp":1760365608000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/4\/663"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,4,22]]},"references-count":50,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2020,4]]}},"alternative-id":["sym12040663"],"URL":"https:\/\/doi.org\/10.3390\/sym12040663","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,4,22]]}}}