{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,14]],"date-time":"2026-01-14T20:30:00Z","timestamp":1768422600332,"version":"3.49.0"},"reference-count":29,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2021,6,25]],"date-time":"2021-06-25T00:00:00Z","timestamp":1624579200000},"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 present paper studies the flow and heat transfer of the hybrid nanofluids flows induced by a permeable power-law stretching\/shrinking surface modulated orthogonal surface shear. The governing partial differential equations were converted into non-linear ordinary differential equations by using proper similarity transformations. These equations were then solved applying a numerical technique, namely bvp4c solver in MATLAB. Results of the flow field, temperature distribution, reduced skin friction coefficient and reduced Nusselt number were deduced. It was found that increasing mass flux parameter slows down the velocity and, hence, decreases the temperature. Furthermore, on enlarging the stretching parameter, the velocity and temperature increases and decreases, respectively. In addition, that the radiation parameter can effectively control the thermal boundary layer. Finally, the temperature decreases when the values of the temperature parameter increases. We apply similarity transformation in order to transform the governing model into a system of ODEs (ordinary differential equations). Numerical solutions for particular values of involved parameters are in very good agreement with previous calculations. The most important and interesting result of this paper is that for both the cases of shrinking and stretching sheet flows exhibit dual solutions in some intervals of the shrinking and stretching parameter. In spite of numerous published papers on the flow and heat transfer over a permeable stretching\/shrinking surface in nanofluids and hybrid nanofluids, none of the researchers studied the present problem. Therefore, we believe that the results of the present paper are new, and have many industrial applications.<\/jats:p>","DOI":"10.3390\/e23070813","type":"journal-article","created":{"date-parts":[[2021,6,25]],"date-time":"2021-06-25T11:07:40Z","timestamp":1624619260000},"page":"813","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":16,"title":["Hybrid Nanofluids Flows Determined by a Permeable Power-Law Stretching\/Shrinking Sheet Modulated by Orthogonal Surface Shear"],"prefix":"10.3390","volume":"23","author":[{"given":"Natalia C.","family":"Ro\u015fca","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Mathematics and Computer Science, Babe\u015f-Bolyai University, 400084 Cluj-Napoca, Romania"}]},{"given":"Ioan","family":"Pop","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Mathematics and Computer Science, Babe\u015f-Bolyai University, 400084 Cluj-Napoca, Romania"}]}],"member":"1968","published-online":{"date-parts":[[2021,6,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1560","DOI":"10.4028\/www.scientific.net\/AMR.328-330.1560","article-title":"Synthesis, characterization of Al2O3-Cu nanocomposite powder and water-based nanofluids","volume":"328\u2013330","author":"Suresh","year":"2011","journal-title":"Adv. Mater. Res."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1016\/j.ijheatmasstransfer.2018.04.059","article-title":"Hybrid nanofluids for heat transfer applications\u2014A state-of-the-art review","volume":"125","author":"Huminic","year":"2018","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2018.11.004","article-title":"Recent advances in modeling and simulation of nanofluid flows-Part I: Fundamentals and theory","volume":"790","author":"Mahian","year":"2019","journal-title":"Physic Rep."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2018.11.003","article-title":"Recent advances in modeling and simulation of nanofluid flows-Part II: Applications","volume":"791","author":"Mahian","year":"2019","journal-title":"Physic Rep."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"569","DOI":"10.1016\/j.jmmm.2014.09.013","article-title":"MHD boundary layer flow and heat transfer of nanofluids over a nonlinear stretching sheet: A numerical study","volume":"374","author":"Mabood","year":"2015","journal-title":"J. Magn. Magn. Mater."},{"key":"ref_6","first-page":"419","article-title":"Heat transfer enhancement of Cu-Al2O3\/water hybrid nanofluid flow over a stretching sheet","volume":"36","author":"Devi","year":"2017","journal-title":"J. Niger. Math. Soc."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"147059","DOI":"10.1155\/2014\/147059","article-title":"Augmentation of the heat transfer performance of a sinusoidal corrugated enclosure by employing hybrid nanofluid","volume":"6","author":"Takabi","year":"2014","journal-title":"Adv. Mech. Engng."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1787","DOI":"10.1016\/j.aej.2020.04.048","article-title":"Flow and heat transfer of hybrid nanofluid over a permeable shrinking cylinder with Joule heating: A comparative analysis","volume":"59","author":"Arifin","year":"2020","journal-title":"Alex. Eng. J."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2465","DOI":"10.1016\/j.cjph.2018.06.013","article-title":"Dual solutions for mixed convective stagnation-point flow of an aqueous silica\u2013alumina hybrid nanofluid","volume":"56","author":"Rostami","year":"2018","journal-title":"Chin. J. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"507","DOI":"10.1007\/s10483-020-2584-7","article-title":"MHD flow and heat transfer of a hybrid nanofluid past a permeable stretching\/shrinking wedge","volume":"41","author":"Waini","year":"2020","journal-title":"Appl. Math. Mech.-Engl. Ed."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"216","DOI":"10.1108\/HFF-02-2020-0086","article-title":"Hybrid nanofluid flow towards a stagnation point on an exponentially stretching\/shrinking vertical sheet with buoyancy effects","volume":"31","author":"Waini","year":"2021","journal-title":"Int. J. Numer. Methods Heat Fluid Flow"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1016\/j.cjph.2020.03.032","article-title":"Three-dimensional hybrid nanofluid flow and heat transfer past a permeable stretching\/shrinking sheet with velocity slip and convective condition","volume":"66","author":"Arifin","year":"2020","journal-title":"Chin. J. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Zainal, N.A., Nazar, R., Naganthran, K., and Pop, I. (2020). Unsteady three-dimensional MHD non-axisymmetric Homann stagnation point flow of a hybrid nanofluid with stability analysis. Mathematics, 8.","DOI":"10.3390\/math8050784"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Rahman, M.M., Saghir, Z., and Pop, I. (2021). Free convective heat transfer efficiency in Al2O3\u2013Cu\/water hybrid nanofluid inside a recto-trapezoidal enclosure. Int. J. Numer. Methods Heat Fluid Flow.","DOI":"10.1615\/TFEC2021.nma.036792"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1038\/s41598-021-87956-8","article-title":"A sensitivity study on carbon nanotubes significance in Darcy\u2013Forchheimer flow towards a rotating disk by response surface methodology","volume":"11","author":"Shafiq","year":"2021","journal-title":"Sci. Rep."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1140\/epjp\/s13360-021-01394-z","article-title":"A study of dual stratification on stagnation point Walters\u2019 B nanofluid flow via radiative Riga plate: A statistical approach","volume":"136","author":"Shafiq","year":"2021","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"4533","DOI":"10.1016\/j.aej.2020.08.007","article-title":"Numerical investigation and sensitivity analysis on bioconvective tangent hyperbolic nanofluid flow towards stretching surface by response surface methodology","volume":"59","author":"Anum","year":"2020","journal-title":"Alex. Eng. J."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"283","DOI":"10.3389\/fphy.2021.670930","article-title":"Nonlinear radiative treatment of hydromagnetic non-Newtonian fluid flow induced Kehinde nonlinear convective boundary driven curved sheet with dissipations and chemical reaction effects","volume":"9","author":"Sanni","year":"2021","journal-title":"Front. Phys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"633","DOI":"10.1007\/s13204-013-0242-9","article-title":"Fluid flow and heat transfer of carbon nanotubes along a flat plate with Navier slip boundary","volume":"4","author":"Khan","year":"2013","journal-title":"Appl. Nanosci."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"395","DOI":"10.2514\/1.6186","article-title":"Fluid flow and heat transfer from a cylinder between parallel planes","volume":"18","author":"Khan","year":"2004","journal-title":"J. Thermophys. Heat Transf."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"083601","DOI":"10.1063\/1.5046331","article-title":"Numerical study of unsteady hydromagnetic radiating fluid flow past a slippery stretching sheet embedded in a porous medium","volume":"30","author":"Makinde","year":"2018","journal-title":"Phys. Fluids"},{"key":"ref_22","unstructured":"Fisher, E.G. (1976). Extrusion of Plastics, Wiley."},{"key":"ref_23","first-page":"375","article-title":"Similarity solutions of the boundary-layer euations for a stretching wall","volume":"2","author":"Banks","year":"1983","journal-title":"J. M\u00e9cnique Th\u00e9orique Appl."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1016\/j.crme.2016.10.016","article-title":"Flows induced by power-law stretching surface motion modulated by transverse or orthogonal surface shear","volume":"345","author":"Weidman","year":"2017","journal-title":"Comptes Rendus Mec."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1326","DOI":"10.1016\/j.ijheatfluidflow.2008.04.009","article-title":"Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids","volume":"29","author":"Oztop","year":"2008","journal-title":"Int. J. Heat Fluid Flow"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1623","DOI":"10.1007\/s11012-012-9541-6","article-title":"Oblique stagnation-point flow and heat towards a shrinking sheet with thermal radiation","volume":"47","author":"Mahapatra","year":"2012","journal-title":"Meccanica"},{"key":"ref_27","first-page":"609","article-title":"Flow over a nonlinearly stretching sheet","volume":"181","author":"Vajravelu","year":"2006","journal-title":"Appl. Math. Comput."},{"key":"ref_28","first-page":"864","article-title":"A nonlinearly stretching sheet","volume":"184","author":"Cortell","year":"2007","journal-title":"Appl. Math. Comput."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Shampine, L.F., Gladwell, I., and Thompson, S. (2003). Solving ODEs with MATLAB, Cambridge University Press.","DOI":"10.1017\/CBO9780511615542"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/7\/813\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:24:25Z","timestamp":1760163865000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/7\/813"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,6,25]]},"references-count":29,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2021,7]]}},"alternative-id":["e23070813"],"URL":"https:\/\/doi.org\/10.3390\/e23070813","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,6,25]]}}}