{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,7]],"date-time":"2026-03-07T06:45:41Z","timestamp":1772865941105,"version":"3.50.1"},"reference-count":73,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2020,4,16]],"date-time":"2020-04-16T00:00:00Z","timestamp":1586995200000},"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>This article elucidates the magnetohydrodynamic 3D Maxwell nanofluid flow with heat absorption\/generation effects. The impact of the nonlinear thermal radiation with a chemical reaction is also an added feature of the presented model. The phenomenon of flow is supported by thermal and concentration stratified boundary conditions. The boundary layer set of non-linear PDEs (partial differential equation) are converted into ODEs (ordinary differential equation) with high nonlinearity via suitable transformations. The homotopy analysis technique is engaged to regulate the mathematical analysis. The obtained results for concentration, temperature and velocity profiles are analyzed graphically for various admissible parameters. A comparative statement with an already published article in limiting case is also added to corroborate our presented model. An excellent harmony in this regard is obtained. The impact of the Nusselt number for distinct parameters is also explored and discussed. It is found that the impacts of Brownian motion on the concentration and temperature distributions are opposite. It is also comprehended that the thermally stratified parameter decreases the fluid temperature.<\/jats:p>","DOI":"10.3390\/e22040453","type":"journal-article","created":{"date-parts":[[2020,4,21]],"date-time":"2020-04-21T03:23:06Z","timestamp":1587439386000},"page":"453","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":49,"title":["Effects of Chemical Species and Nonlinear Thermal Radiation with 3D Maxwell Nanofluid Flow with Double Stratification\u2014An Analytical Solution"],"prefix":"10.3390","volume":"22","author":[{"given":"Iskander","family":"Tlili","sequence":"first","affiliation":[{"name":"Department for Management of Science and Technology Development, Ton Duc Thang University, Ho Chi Minh City 758307, Vietnam"},{"name":"Faculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City 758307, Vietnam"}]},{"given":"Sania","family":"Naseer","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Bahria University, Islamabad 44000, Pakistan"}]},{"given":"Muhammad","family":"Ramzan","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Bahria University, Islamabad 44000, Pakistan"},{"name":"Department of Mechanical Engineering, Sejong University, Seoul 143-747, Korea"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1939-4842","authenticated-orcid":false,"given":"Seifedine","family":"Kadry","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, Faculty of Science, Beirut Arab University, Beirut 115020, Lebanon"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3318-9394","authenticated-orcid":false,"given":"Yunyoung","family":"Nam","sequence":"additional","affiliation":[{"name":"Department of Computer Science and Engineering, Soonchunhyang University, Asan 31538, Korea"}]}],"member":"1968","published-online":{"date-parts":[[2020,4,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"963","DOI":"10.1016\/j.cjph.2017.03.006","article-title":"A revised model for Darcy-Forchheimer flow of Maxwell nanofluid subject to convective boundary condition","volume":"55","author":"Muhammad","year":"2017","journal-title":"Chin. J. Phys."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"870","DOI":"10.1016\/j.molliq.2016.05.012","article-title":"Impact of double stratification and magnetic field in the mixed convective radiative flow of Maxwell nanofluid","volume":"220","author":"Hussain","year":"2016","journal-title":"J. Mol. Liq."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1172","DOI":"10.1016\/j.molliq.2016.10.082","article-title":"Stagnation-point heat and mass transfer of MHD Maxwell nanofluids over a stretching surface in the presence of thermophoresis","volume":"224","author":"Bai","year":"2016","journal-title":"J. Mol. Liq."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"166","DOI":"10.1016\/j.ijmecsci.2017.02.022","article-title":"Flow and heat transfer of magnetohydrodynamic three-dimensional Maxwell nanofluid over a permeable stretching\/shrinking surface with convective boundary conditions","volume":"124","author":"Jusoh","year":"2017","journal-title":"Int. J. Mech. Sci."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"801","DOI":"10.1016\/j.molliq.2016.10.055","article-title":"Thermally radiative stagnation point flow of Maxwell nanofluid due to unsteady convectively heated stretched surface","volume":"224","author":"Hayat","year":"2016","journal-title":"J. Mol. Liq."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1038\/s41598-019-43549-0","article-title":"MHD flow of Maxwell fluid with nanomaterials due to an exponentially stretching surface","volume":"9","author":"Farooq","year":"2019","journal-title":"Sci. Rep."},{"key":"ref_7","first-page":"3","article-title":"Soret and Dufour effects on three dimensional upper-convected Maxwell fluid with chemical reaction and non-linear radiative heat flux","volume":"15","author":"Ramzan","year":"2017","journal-title":"Int. J. Chem. React. Eng."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"015038","DOI":"10.1063\/1.5129569","article-title":"Entropy generation in MHD Maxwell nanofluid flow with variable thermal conductivity, thermal radiation, slip conditions, and heat source","volume":"10","author":"Aziz","year":"2020","journal-title":"AIP Adv."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1072","DOI":"10.1016\/j.rinp.2016.11.036","article-title":"Mixed convective flow of Maxwell nanofluid past a porous vertical stretched surface\u2014An optimal solution","volume":"6","author":"Ramzan","year":"2016","journal-title":"Result. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"415","DOI":"10.1016\/j.molliq.2017.01.061","article-title":"Influence of homogeneous-heterogeneous reactions on MHD 3D Maxwell fluid flow with Cattaneo-Christov heat flux and convective boundary condition","volume":"230","author":"Ramzan","year":"2017","journal-title":"J. Mol. Liq."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"100452","DOI":"10.1016\/j.csite.2019.100452","article-title":"Symmetry analysis on thermally magnetized fluid flow regime with heat source\/sink","volume":"14","author":"Rehman","year":"2019","journal-title":"Case Stud. Therm. Eng."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1038\/s41598-017-01358-3","article-title":"Heat transfer enhancement in free convection flow of CNTs Maxwell nanofluids with four different types of molecular liquids","volume":"7","author":"Aman","year":"2017","journal-title":"Sci. Rep."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2352","DOI":"10.1016\/j.cjche.2018.12.023","article-title":"Change in internal energy of thermal diffusion stagnation point Maxwell nanofluid flow along with solar radiation and thermal conductivity","volume":"27","author":"Khan","year":"2019","journal-title":"Chin. J. Chem. Eng."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"944","DOI":"10.1166\/jon.2018.1510","article-title":"Non-Linear thermal radiative marangoni boundary layer flow of gamma Al2O3 nanofluids past a stretching sheet","volume":"7","author":"Ganesh","year":"2018","journal-title":"J. Nanofluid."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"340","DOI":"10.1016\/j.csite.2018.04.019","article-title":"Magneto-Marangoni nano-boundary layer flow of water and ethylene glycol based \u03b3 Al2O3 nanofluids with non-linear thermal radiation effects","volume":"12","author":"Ganesh","year":"2018","journal-title":"Case Stud. Therm. Eng."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"100546","DOI":"10.1016\/j.csite.2019.100546","article-title":"Blasius and Sakiadis slip flow of H2O\u2013C2H6O2 (50: 50) based nanoliquid with different geometry of boehmite alumina nanoparticles","volume":"16","author":"Ganesh","year":"2019","journal-title":"Case Stud. Therm. Eng."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Abdelsalam, S., and Bhatti, M. (2020). Anomalous reactivity of thermo-bioconvective nanofluid towards oxytactic microorganisms. Appl. Math. Mech.","DOI":"10.1007\/s10483-020-2609-6"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"956","DOI":"10.1016\/j.powtec.2019.10.053","article-title":"Investigations of a new combined application of nanofluids in heat recovery and air purification","volume":"360","author":"Yang","year":"2020","journal-title":"Powder Techol."},{"key":"ref_19","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_20","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1038\/s41598-018-36459-0","article-title":"New insight into AuNP applications in tumour treatment and cosmetics through wavy annuli at the nanoscale","volume":"9","author":"Abdelsalam","year":"2019","journal-title":"Sci. Rep."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"7904","DOI":"10.1039\/C7RA13188G","article-title":"The study of non-Newtonian nanofluid with Hall and ion slip effects on peristaltically induced motion in a non-uniform channel","volume":"8","author":"Abdelsalam","year":"2018","journal-title":"RSC Adv."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"121206","DOI":"10.1016\/j.jclepro.2020.121206","article-title":"Acceleration of discharge process of clean energy storage unit with insertion of porous foam considering nanoparticle enhanced paraffin","volume":"261","author":"Sheikholeslami","year":"2020","journal-title":"J. Clean. Prod."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"974","DOI":"10.1016\/j.ijheatmasstransfer.2019.07.043","article-title":"Application of nano-refrigerant for boiling heat transfer enhancement employing an experimental study","volume":"141","author":"Sheikholeslami","year":"2019","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_24","first-page":"237","article-title":"Electromagnetic flow for two-layer immiscible fluids","volume":"22","author":"Elmaboud","year":"2019","journal-title":"Eng. Sci. Technol. Int. J."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"723","DOI":"10.1007\/s12668-019-00651-x","article-title":"Joint effect of magnetic field and heat transfer on particulate fluid suspension in a catheterized wavy tube","volume":"9","author":"Eldesoky","year":"2019","journal-title":"BioNanoScience"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"115301","DOI":"10.1088\/1402-4896\/ab207a","article-title":"Metachronal propulsion of a magnetised particle-fluid suspension in a ciliated channel with heat and mass transfer","volume":"94","author":"Abdelsalam","year":"2019","journal-title":"Phys. Scr."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"186","DOI":"10.1007\/s12668-018-0585-5","article-title":"Concurrent Development of Thermal Energy with Magnetic Field on a Particle-Fluid Suspension Through a Porous Conduit","volume":"9","author":"Eldesoky","year":"2019","journal-title":"BioNanoScience"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"450","DOI":"10.1016\/j.jksus.2018.07.007","article-title":"Heat transfer and second order slip effect on MHD flow of fractional Maxwell fluid in a porous medium","volume":"32","author":"Aman","year":"2020","journal-title":"J. King Saud Univ.-Sci."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"100413","DOI":"10.1016\/j.csite.2019.100413","article-title":"Numerical study of MHD effective Prandtl number boundary layer flow of \u03b3 Al2O3 nanofluids past a melting surface","volume":"13","author":"Ganesh","year":"2019","journal-title":"Case Stud. Therm. Eng."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"530","DOI":"10.1108\/MMMS-08-2017-0094","article-title":"The impact of impinging TiO2 nanoparticles in Prandtl nanofluid along with endoscopic and variable magnetic field effects on peristaltic blood flow","volume":"14","author":"Abdelsalam","year":"2018","journal-title":"Multidiscip. Model. Mat. Struct."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"398","DOI":"10.1016\/j.euromechflu.2017.02.002","article-title":"Combined effects of magnetic field and rheological properties on the peristaltic flow of a compressible fluid in a microfluidic channel","volume":"65","author":"Abdelsalam","year":"2017","journal-title":"Eur. J. Mech.-B\/Fluid."},{"key":"ref_32","first-page":"187","article-title":"Hall and transverse magnetic field effects on peristaltic flow of a Maxwell fluid through a porous medium","volume":"9","author":"Elkoumy","year":"2013","journal-title":"Global J. Pure Appl. Math"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"124702","DOI":"10.1088\/1674-1056\/22\/12\/124702","article-title":"Simultaneous effects of magnetic field and space porosity on compressible Maxwell fluid transport induced by a surface acoustic wave in a microchannel","volume":"22","author":"Mekheimer","year":"2013","journal-title":"Chin. Phys. B"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"433","DOI":"10.1016\/j.compfluid.2013.07.029","article-title":"The effect of double stratification on boundary-layer flow and heat transfer of nanofluid over a vertical plate","volume":"86","author":"Ibrahim","year":"2013","journal-title":"Comput. Fluids"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1016\/j.jmmm.2015.11.090","article-title":"Doubly stratified mixed convection flow of Maxwell nanofluid with heat generation\/absorption","volume":"404","author":"Abbasi","year":"2016","journal-title":"J. Magn. Magn. Mater."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"370","DOI":"10.1016\/j.joems.2013.02.006","article-title":"Effect of double stratification on MHD free convection in a micropolar fluid","volume":"21","author":"Srinivasacharya","year":"2013","journal-title":"J. Egypt. Math. Soc."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Hayat, T., Hussain, T., Shehzad, S.A., and Alsaedi, A. (2014). Thermal and concentration stratifications effects in radiative flow of Jeffrey fluid over a stretching sheet. PLoS ONE, 9.","DOI":"10.1371\/journal.pone.0107858"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"26","DOI":"10.1016\/j.imu.2017.05.008","article-title":"Cattaneo-Christov heat flux on UCM nanofluid flow across a melting surface with double stratification and exponential space dependent internal heat source","volume":"9","author":"Mahanthesh","year":"2017","journal-title":"Inform. Med. Unlocked"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"456","DOI":"10.1140\/epjp\/i2017-11748-5","article-title":"Double stratified radiative Jeffery magneto nanofluid flow along an inclined stretched cylinder with chemical reaction and slip condition","volume":"132","author":"Ramzan","year":"2017","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"12901","DOI":"10.1038\/s41598-017-13140-6","article-title":"Buoyancy effects on the radiative magneto Micropolar nanofluid flow with double stratification, activation energy and binary chemical reaction","volume":"7","author":"Ramzan","year":"2017","journal-title":"Sci. Rep."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"025702","DOI":"10.1088\/1402-4896\/ab4067","article-title":"Upshot of magnetic dipole on the flow of nanofluid along a stretched cylinder with gyrotactic microorganism in a stratified medium","volume":"95","author":"Alshomrani","year":"2019","journal-title":"Phys. Scr."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"3635","DOI":"10.1007\/s10973-019-08713-8","article-title":"Magnetohydrodynamics (MHD) stagnation point flow past a shrinking\/stretching surface with double stratification effect in a porous medium","volume":"139","author":"Arifin","year":"2020","journal-title":"J. Therm. Anal. Calorim."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"015701","DOI":"10.1088\/1402-4896\/ab3d06","article-title":"Thermally stratified Darcy Forchheimer nanofluid flow comprising carbon nanotubes with effects of Cattaneo-Christov heat flux and homogeneous-heterogeneous reactions","volume":"95","author":"Ramzan","year":"2019","journal-title":"Phys. Scr."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"723","DOI":"10.1088\/0253-6102\/67\/6\/723","article-title":"Effects of variable thermal conductivity and non-linear thermal radiation past an Eyring Powell nanofluid flow with chemical Reaction","volume":"67","author":"Ramzan","year":"2017","journal-title":"Commun. Theor. Phys."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1016\/j.ijmecsci.2017.06.009","article-title":"Radiative magnetohydrodynamic nanofluid flow due to gyrotactic microorganisms with chemical reaction and non-linear thermal radiation","volume":"130","author":"Ramzan","year":"2017","journal-title":"Int. J. Mech. Sci."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"222","DOI":"10.1016\/j.jmmm.2015.02.046","article-title":"Magnetohydrodynamic three-dimensional flow of viscoelastic nanofluid in the presence of nonlinear thermal radiation","volume":"385","author":"Hayat","year":"2015","journal-title":"J. Magn. Magn. Mater."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"387","DOI":"10.1007\/s40430-018-1297-z","article-title":"On three-dimensional MHD Oldroyd-B fluid flow with nonlinear thermal radiation and homogeneous\u2013heterogeneous reaction","volume":"40","author":"Lu","year":"2018","journal-title":"J. Braz. Soc. Mech. Sci. Eng."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"3127","DOI":"10.1007\/s10973-019-08324-3","article-title":"Hall current effect on unsteady rotational flow of carbon nanotubes with dust particles and nonlinear thermal radiation in Darcy\u2013Forchheimer porous media","volume":"138","author":"Bilal","year":"2019","journal-title":"J. Ther. Anal. Calorim"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"071","DOI":"10.1088\/0253-6102\/70\/1\/71","article-title":"Upshot of chemical species and nonlinear thermal radiation on Oldroyd-B nanofluid flow past a bi-directional stretched surface with heat generation\/absorption in a porous media","volume":"70","author":"Lu","year":"2018","journal-title":"Commun. Theor. Phys."},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Suleman, M., Ramzan, M., Zulfiqar, M., Bilal, M., Shafee, A., Chung, J.D., and Farooq, U. (2018). Entropy analysis of 3D non-Newtonian MHD nanofluid flow with nonlinear thermal radiation past over exponential stretched surface. Entropy, 20.","DOI":"10.3390\/e20120930"},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Lu, D., Ramzan, M., Ahmad, S., Shafee, A., and Suleman, M. (2018). Impact of nonlinear thermal radiation and entropy optimization coatings with hybrid nanoliquid flow past a curved stretched surface. Coatings, 8.","DOI":"10.3390\/coatings8120430"},{"key":"ref_52","first-page":"1","article-title":"Nonlinear radiation effect on MHD Carreau nanofluid flow over a radially stretching surface with zero mass flux at the surface","volume":"8","author":"Lu","year":"2018","journal-title":"Sci. Rep."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0065-2717(08)70172-2","article-title":"Second-law analysis in heat transfer and thermal design","volume":"Volume 15","author":"Bejan","year":"1982","journal-title":"Advances in Heat Transfer"},{"key":"ref_54","doi-asserted-by":"crossref","unstructured":"Lu, D., Ramzan, M., Mohammad, M., Howari, F., and Chung, J.D. (2019). A thin film flow of nanofluid comprising carbon nanotubes influenced by Cattaneo-Christov heat flux and entropy generation. Coatings, 9.","DOI":"10.3390\/coatings9050296"},{"key":"ref_55","doi-asserted-by":"crossref","unstructured":"Ramzan, M., Mohammad, M., Howari, F., and Chung, J.D. (2019). Entropy analysis of carbon nanotubes based nanofluid flow past a vertical cone with thermal radiation. Entropy, 21.","DOI":"10.3390\/e21070642"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"1863","DOI":"10.1016\/j.ijthermalsci.2009.02.013","article-title":"Similarity solutions of the entropy transport equation","volume":"48","author":"Weigand","year":"2009","journal-title":"Int. J. Therm. Sci."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"749","DOI":"10.1007\/s12648-018-1328-8","article-title":"Study of heat transfer and entropy generation in ferrofluid under low oscillating magnetic field","volume":"93","author":"Hassan","year":"2019","journal-title":"Indian J. Phys."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1016\/j.icheatmasstransfer.2010.12.016","article-title":"Numerical analysis of entropy generation in mixed convection flow with viscous dissipation effects in vertical channel","volume":"38","author":"Lai","year":"2011","journal-title":"Int. Commun. Heat Mass Transf."},{"key":"ref_59","doi-asserted-by":"crossref","unstructured":"Tlili, I., Ramzan, M., Kadry, S., Kim, H.W., and Nam, Y. (2020). Radiative MHD nanofluid flow over a moving thin needle with Entropy generation in a porous medium with dust particles and Hall current. Entropy, 22.","DOI":"10.3390\/e22030354"},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1038\/s41598-019-48645-9","article-title":"Magnetized suspended carbon nanotubes based nanofluid flow with bio-convection and entropy generation past a vertical cone","volume":"9","author":"Ramzan","year":"2019","journal-title":"Sci. Rep."},{"key":"ref_61","unstructured":"Liao, S.J. (1992). The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems. [Ph.D. Thesis, Shanghai Jiao Tong University]."},{"key":"ref_62","doi-asserted-by":"crossref","unstructured":"Liao, S. (2012). Homotopy Analysis Method in Nonlinear Differential Equations, Beijing Higher Education Press.","DOI":"10.1007\/978-3-642-25132-0"},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"759","DOI":"10.1016\/S0020-7462(98)00056-0","article-title":"An explicit, totally analytic approximation of Blasius\u2019 viscous flow problems","volume":"34","author":"Liao","year":"1999","journal-title":"Int. J. Non-Linear Mech."},{"key":"ref_64","unstructured":"Liao, S.J. (2003). Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman & Hall\/CRC Press."},{"key":"ref_65","first-page":"385","article-title":"New applications of the homotopy analysis method","volume":"63","author":"Elwakil","year":"2008","journal-title":"Z. Nat. A"},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"152","DOI":"10.1016\/j.matcom.2019.05.008","article-title":"Entropy analysis and nanofluid past a double stretchable spinning disk using Homotopy Analysis Method","volume":"171","author":"Renuka","year":"2020","journal-title":"Math. Comput. Simul."},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1140\/epjp\/i2019-12449-9","article-title":"Homotopy analysis method for the Sakiadis flow of a thixotropic fluid","volume":"134","author":"Ghiasi","year":"2019","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_68","unstructured":"Biswal, U., and Chakraverty, S. (2020). Investigation of Jeffery-Hamel fow for nanofluid in the presence of magnetic field by a new approach in the Optimal Homotopy analysis method. J. Appl. Comput. Mech."},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"11","DOI":"10.15406\/ipcse.2020.05.00118","article-title":"Homotopy analysis method to MHD-slip flow of an upper-convected maxwell viscoelastic nanofluid in a permeable channel embedded in a porous medium","volume":"5","author":"Sobamowo","year":"2020","journal-title":"Int. J. Petrochem. Sci. Eng."},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1007\/s40819-019-0765-1","article-title":"Non-similar solution of Eyring\u2013Powell fluid flow and heat transfer with convective boundary condition: Homotopy Analysis Method","volume":"6","author":"Ray","year":"2020","journal-title":"Int. J. Appl. Comput. Math."},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"1917","DOI":"10.1016\/j.aej.2017.03.039","article-title":"Three dimensional MHD upper-convected Maxwell nanofluid flow with nonlinear radiative heat flux","volume":"57","author":"Bilal","year":"2018","journal-title":"Alexandria Eng. J."},{"key":"ref_72","unstructured":"Rosseland, S. (2013). Astrophysik: Auf Atomtheoretischer Grundlage, Springer."},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"274","DOI":"10.1016\/j.ijthermalsci.2016.08.009","article-title":"An analytical solution for magnetohydrodynamic Oldroyd-B nanofluid flow induced by a stretching sheet with heat generation\/absorption","volume":"111","author":"Hayat","year":"2017","journal-title":"Int. J. Therm. Sci."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/4\/453\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T13:31:23Z","timestamp":1760362283000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/4\/453"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,4,16]]},"references-count":73,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2020,4]]}},"alternative-id":["e22040453"],"URL":"https:\/\/doi.org\/10.3390\/e22040453","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,4,16]]}}}