{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,11]],"date-time":"2026-05-11T14:06:43Z","timestamp":1778508403329,"version":"3.51.4"},"reference-count":141,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2020,5,3]],"date-time":"2020-05-03T00:00:00Z","timestamp":1588464000000},"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>In the present era, nanofluids are one of the most important and hot issue for scientists, physicists, and mathematicians. Nanofluids have many important and updated characteristics compared to conventional fluids. The thermal conductivity, thermal expansion, and the heat transfer rate of conventional fluids are not up to the mark for industrial and experimental uses. To overcome these deficiencies, nanoparticles have been dispersed into base fluids to make them more efficient. The heat transfer characteristics through symmetry trapezoidal-corrugated channels can be enhanced using nanofluids. In the present article, a literature survey has been presented for different models of nanofluids and their solutions\u2014particularly, exact solutions. The models for hybrid nanofluids were also mentioned in the present study. Furthermore, some important and most used models for the viscosity, density, coefficient of thermal expansion, coefficient of mass expansion, heat capacitance, electrical conductivity, and thermal conductivity are also presented in tabular form. Moreover, some future suggestions are also provided in this article.<\/jats:p>","DOI":"10.3390\/sym12050725","type":"journal-article","created":{"date-parts":[[2020,5,5]],"date-time":"2020-05-05T06:41:20Z","timestamp":1588660880000},"page":"725","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":24,"title":["A Comprehensive Review on Theoretical Aspects of Nanofluids: Exact Solutions and Analysis"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2080-1695","authenticated-orcid":false,"given":"Nadeem Ahmad","family":"Sheikh","sequence":"first","affiliation":[{"name":"Fundamental and Applied Science Department, Universiti Teknologi PETRONAS, Perak 32610, Malaysia"},{"name":"Department of Mathematics, City University of Science and Information Technology, Peshawar 25000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dennis Ling","family":"Chuan Ching","sequence":"additional","affiliation":[{"name":"Fundamental and Applied Science Department, Universiti Teknologi PETRONAS, Perak 32610, Malaysia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ilyas","family":"Khan","sequence":"additional","affiliation":[{"name":"Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City 72915, Vietnam"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,5,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"055705","DOI":"10.1088\/0957-4484\/21\/5\/055705","article-title":"Enhanced thermal conductivities of nanofluids containing graphene oxide nanosheets","volume":"21","author":"Yu","year":"2009","journal-title":"Nanotechnology"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1016\/j.molliq.2017.04.011","article-title":"Impact of nonlinear radiation on 3D magnetohydrodynamic flow of methanol and kerosene based ferrofluids with temperature dependent viscosity","volume":"236","author":"Reddy","year":"2017","journal-title":"J. Mol. Liq."},{"key":"ref_3","unstructured":"Choi, S.U., and Eastman, J.A. (1995, January 1). Enhancing thermal conductivity of fluids with nanoparticles. Proceedings of the International Mechanical Engineering Congress and Exhibition, San Francisco, CA, USA."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1016\/j.powtec.2009.07.025","article-title":"A review on development of nanofluid preparation and characterization","volume":"196","author":"Li","year":"2009","journal-title":"Powder Technol."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1260\/1759-3093.1.4.269","article-title":"A review of thermal conductivity data, mechanisms and models for nanofluids","volume":"1","author":"Lee","year":"2010","journal-title":"Int. J. Micro-Nano Scale Transp."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"4051","DOI":"10.1016\/j.ijheatmasstransfer.2011.04.014","article-title":"A review of nanofluid stability properties and characterization in stationary conditions","volume":"54","author":"Ghadimi","year":"2011","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"334","DOI":"10.1186\/1556-276X-6-334","article-title":"Review of thermo-physical properties, wetting and heat transfer characteristics of nanofluids and their applicability in industrial quench heat treatment","volume":"6","author":"Ramesh","year":"2011","journal-title":"Nanoscale Res. Lett."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"4410","DOI":"10.1016\/j.ijheatmasstransfer.2011.04.048","article-title":"A critical synthesis of thermophysical characteristics of nanofluids","volume":"54","author":"Khanafer","year":"2011","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"040801","DOI":"10.1115\/1.4002633","article-title":"Review of heat conduction in nanofluids","volume":"133","author":"Fan","year":"2011","journal-title":"J. Heat Transf."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"4063","DOI":"10.1016\/j.ijheatmasstransfer.2012.03.048","article-title":"A review and analysis on influence of temperature and concentration of nanofluids on thermophysical properties, heat transfer and pumping power","volume":"55","author":"Vajjha","year":"2012","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"512","DOI":"10.1016\/j.rser.2005.01.010","article-title":"Critical review of heat transfer characteristics of nanofluids","volume":"11","author":"Trisaksri","year":"2007","journal-title":"Renew. Sustain. Energy Rev."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"797","DOI":"10.1016\/j.rser.2005.06.005","article-title":"A critical review of convective heat transfer of nanofluids","volume":"11","author":"Daungthongsuk","year":"2007","journal-title":"Renew. Sustain. Energy Rev."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"3187","DOI":"10.1016\/j.ijheatmasstransfer.2009.02.006","article-title":"Review of convective heat transfer enhancement with nanofluids","volume":"52","author":"Pramuanjaroenkij","year":"2009","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"629","DOI":"10.1016\/j.rser.2009.10.004","article-title":"Enhancement of heat transfer using nanofluids\u2014An overview","volume":"14","author":"Godson","year":"2010","journal-title":"Renew. Sustain. Energy Rev."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"3271","DOI":"10.1016\/j.rser.2011.04.025","article-title":"A critical review on convective heat transfer correlations of nanofluids","volume":"15","author":"Sarkar","year":"2011","journal-title":"Renew. Sustain. Energy Rev."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1016\/j.molliq.2016.12.024","article-title":"Magnetic field influence on nanofluid thermal radiation in a cavity with tilted elliptic inner cylinder","volume":"229","author":"Sheikholeslami","year":"2017","journal-title":"J. Mol. Liq."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"799","DOI":"10.1016\/j.ijheatmasstransfer.2015.05.110","article-title":"Three dimensional mesoscopic simulation of magnetic field effect on natural convection of nanofluid","volume":"89","author":"Sheikholeslami","year":"2015","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1016\/j.jtice.2016.05.014","article-title":"Nanofluid convective heat transfer using semi analytical and numerical approaches: A review","volume":"65","author":"Sheikholeslami","year":"2016","journal-title":"J. Taiwan Inst. Chem. Eng."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2477","DOI":"10.1016\/j.ijheatmasstransfer.2010.01.032","article-title":"Boundary-layer flow of a nanofluid past a stretching sheet","volume":"53","author":"Khan","year":"2010","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1002\/htj.21375","article-title":"Convective heat transfer of Titania nanofluids of different base fluids in cylindrical annulus with discrete heat source","volume":"48","year":"2019","journal-title":"Heat Transf. Asian Res."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"737","DOI":"10.1108\/MMMS-07-2018-0133","article-title":"Magnetohydrodynamic flow of molybdenum disulfide nanofluid in a channel with shape effects","volume":"15","author":"Raza","year":"2019","journal-title":"Multidiscip. Modeling Mater. Struct."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"51","DOI":"10.4028\/www.scientific.net\/DDF.387.51","article-title":"MHD slip flow of Cu-Kerosene nanofluid in a channel with stretching walls using 3-stage Lobatto IIIA formula","volume":"387","author":"Reza","year":"2018","journal-title":"Defect Diffus. Forum"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"146","DOI":"10.1016\/j.cis.2015.08.014","article-title":"Review on thermal properties of nanofluids: Recent developments","volume":"225","author":"Angayarkanni","year":"2015","journal-title":"Adv. Colloid Interface Sci."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1203","DOI":"10.1016\/j.ijheatmasstransfer.2017.08.108","article-title":"Simulation of nanofluid heat transfer in presence of magnetic field: A review","volume":"115","author":"Sheikholeslami","year":"2017","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1080\/08916159808946559","article-title":"Hydrodynamic and heat transfer study of dispersed fluids with submicron metallic oxide particles","volume":"11","author":"Pak","year":"1998","journal-title":"Exp. Heat Transf. Int. J."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Eastman, J. (1999). Novel Thermal Properties of Nanostructured Materials, Argonne National Lab.","DOI":"10.4028\/www.scientific.net\/MSF.312-314.629"},{"key":"ref_27","first-page":"433","article-title":"Nanofluid flow and heat transfer over a stretching porous cylinder considering thermal radiation","volume":"39","author":"Sheikholeslami","year":"2015","journal-title":"Iran. J. Sci. Technol. Trans. Sci."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"412","DOI":"10.1016\/j.molliq.2016.08.068","article-title":"MHD flow of water-based Brinkman type nanofluid over a vertical plate embedded in a porous medium with variable surface velocity, temperature and concentration","volume":"223","author":"Ali","year":"2016","journal-title":"J. Mol. Liq."},{"key":"ref_29","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_30","unstructured":"Einstein, A. (1956). Investigations on the Theory of the Brownian Movement, Courier Corporation."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1017\/S0022112077001062","article-title":"The effect of Brownian motion on the bulk stress in a suspension of spherical particles","volume":"83","author":"Batchelor","year":"1977","journal-title":"J. Fluid Mech."},{"key":"ref_32","unstructured":"D\u00e1valos-Orozco, L.A., and Del Castillo, L.F. (2005). Hydrodynamic Behavior of Suspensions of Polar Particles, CRC Press."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"571","DOI":"10.1063\/1.1700493","article-title":"The viscosity of concentrated suspensions and solutions","volume":"20","author":"Brinkman","year":"1952","journal-title":"J. Chem. Phys."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1492","DOI":"10.1016\/j.ijheatfluidflow.2007.02.004","article-title":"Temperature and particle-size dependent viscosity data for water-based nanofluids\u2013hysteresis phenomenon","volume":"28","author":"Nguyen","year":"2007","journal-title":"Int. J. Heat Fluid Flow"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"679","DOI":"10.1016\/j.ijheatfluidflow.2009.02.003","article-title":"Effects of variable viscosity and thermal conductivity of Al2O3\u2013water nanofluid on heat transfer enhancement in natural convection","volume":"30","year":"2009","journal-title":"Int. J. Heat Fluid Flow"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"243112","DOI":"10.1063\/1.2824393","article-title":"Particle concentration and tube size dependence of viscosities of Al2O3-water nanofluids flowing through micro-and minitubes","volume":"91","author":"Jang","year":"2007","journal-title":"Appl. Phys. Lett."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"577","DOI":"10.1007\/s11051-004-3170-5","article-title":"A new thermal conductivity model for nanofluids","volume":"6","author":"Koo","year":"2004","journal-title":"J. Nanopart. Res."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"530","DOI":"10.1016\/j.ijheatfluidflow.2005.02.004","article-title":"Heat transfer enhancement by using nanofluids in forced convection flows","volume":"26","author":"Maiga","year":"2005","journal-title":"Int. J. Heat Fluid Flow"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1016\/j.ijthermalsci.2007.01.033","article-title":"Viscosity data for Al2O3\u2013water nanofluid\u2014hysteresis: Is heat transfer enhancement using nanofluids reliable?","volume":"47","author":"Nguyen","year":"2008","journal-title":"Int. J. Therm. Sci."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"055501","DOI":"10.1088\/0022-3727\/42\/5\/055501","article-title":"A new model for calculating the effective viscosity of nanofluids","volume":"42","author":"Masoumi","year":"2009","journal-title":"J. Phys. D Appl. Phys."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1486","DOI":"10.1016\/j.ijthermalsci.2009.01.008","article-title":"Experimental investigation of nanofluids in confined laminar radial flows","volume":"48","author":"Gherasim","year":"2009","journal-title":"Int. J. Therm. Sci."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"657","DOI":"10.1007\/s41779-018-0275-3","article-title":"The unsteady flow of generalized hybrid nanofluids: Applications in cementitious materials","volume":"55","author":"Gohar","year":"2018","journal-title":"J. Aust. Ceram. Soc."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1016\/j.powtec.2013.02.006","article-title":"Investigation of squeezing unsteady nanofluid flow using ADM","volume":"239","author":"Sheikholeslami","year":"2013","journal-title":"Powder Technol."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1021\/i160003a005","article-title":"Thermal conductivity of heterogeneous two-component systems","volume":"1","author":"Hamilton","year":"1962","journal-title":"Ind. Eng. Chem. Fundam."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"4316","DOI":"10.1063\/1.1756684","article-title":"Role of Brownian motion in the enhanced thermal conductivity of nanofluids","volume":"84","author":"Jang","year":"2004","journal-title":"Appl. Phys. Lett."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"2665","DOI":"10.1016\/S0017-9310(03)00016-4","article-title":"A fractal model for predicting the effective thermal conductivity of liquid with suspension of nanoparticles","volume":"46","author":"Wang","year":"2003","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"153107","DOI":"10.1063\/1.2093936","article-title":"Empirical correlation finding the role of temperature and particle size for nanofluid (Al2O3) thermal conductivity enhancement","volume":"87","author":"Chon","year":"2005","journal-title":"Appl. Phys. Lett."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"1038","DOI":"10.1002\/aic.690490420","article-title":"Aggregation structure and thermal conductivity of nanofluids","volume":"49","author":"Xuan","year":"2003","journal-title":"AIChE J."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"819","DOI":"10.1016\/0017-9310(91)90128-2","article-title":"Forced convection heat transfer in microencapsulated phase change material slurries: Flow in circular ducts","volume":"34","author":"Charunyakorn","year":"1991","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1146\/annurev.matsci.34.052803.090621","article-title":"Thermal transport in nanofluids","volume":"34","author":"Eastman","year":"2004","journal-title":"Annu. Rev. Mater. Res."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1023\/A:1024438603801","article-title":"The role of interfacial layers in the enhanced thermal conductivity of nanofluids: A renovated Maxwell model","volume":"5","author":"Yu","year":"2003","journal-title":"J. Nanopart. Res."},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Patel, H.E., Anoop, K., Sundararajan, T., and Das, S.K. (2006, January 13\u201318). A micro-convection model for thermal conductivity of nanofluids. Proceedings of the International Heat Transfer Conference 13, Sydney, Australia.","DOI":"10.1615\/IHTC13.p8.240"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1016\/j.ijthermalsci.2008.03.009","article-title":"New temperature dependent thermal conductivity data for water-based nanofluids","volume":"48","author":"Mintsa","year":"2009","journal-title":"Int. J. Therm. Sci."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"17541","DOI":"10.1039\/c1jm12671g","article-title":"Effect of carbon nanotube surface modification on thermal properties of copper\u2013CNT composites","volume":"21","author":"Firkowska","year":"2011","journal-title":"J. Mater. Chem."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"021914","DOI":"10.1063\/1.2430914","article-title":"Effects of anisotropy, aspect ratio, and nonstraightness of carbon nanotubes on thermal conductivity of carbon nanotube composites","volume":"90","author":"Deng","year":"2007","journal-title":"Appl. Phys. Lett."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"302","DOI":"10.1016\/j.physb.2005.07.024","article-title":"Model for thermal conductivity of carbon nanotube-based composites","volume":"368","author":"Xue","year":"2005","journal-title":"Phys. B Condens. Matter"},{"key":"ref_57","first-page":"834","article-title":"Flow of a generalized Oldroyd-B fluid due to a constantly accelerating plate","volume":"201","author":"Vieru","year":"2008","journal-title":"Appl. Math. Comput."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1016\/j.apnum.2018.03.016","article-title":"A time\u2013space spectral tau method for the time fractional cable equation and its inverse problem","volume":"130","author":"Yang","year":"2018","journal-title":"Appl. Numer. Math."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"789","DOI":"10.1016\/j.rinp.2017.01.025","article-title":"Comparison and analysis of the Atangana\u2013Baleanu and Caputo\u2013Fabrizio fractional derivatives for generalized Casson fluid model with heat generation and chemical reaction","volume":"7","author":"Sheikh","year":"2017","journal-title":"Results Phys."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1016\/j.jmmm.2016.09.125","article-title":"Magnetic field effect on blood flow of Casson fluid in axisymmetric cylindrical tube: A fractional model","volume":"423","author":"Ali","year":"2017","journal-title":"J. Magn. Magn. Mater."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"795","DOI":"10.1155\/S0161171200002817","article-title":"MHD flow of an elastico-viscous fluid under periodic body acceleration","volume":"23","author":"Elbarbary","year":"2000","journal-title":"Int. J. Math. Math. Sci."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"515","DOI":"10.1016\/j.ijnonlinmec.2004.07.016","article-title":"Stokes\u2019 first problem for a second grade fluid in a porous half-space with heated boundary","volume":"40","author":"Tan","year":"2005","journal-title":"Int. J. Non-Linear Mech."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"023101","DOI":"10.1063\/1.1850409","article-title":"Stokes\u2019 first problem for an Oldroyd-B fluid in a porous half space","volume":"17","author":"Tan","year":"2005","journal-title":"Phys. Fluids"},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"1371","DOI":"10.1016\/j.ijnonlinmec.2003.12.003","article-title":"On a new exact solution to Stokes\u2019 first problem for Maxwell fluids","volume":"39","author":"Jordan","year":"2004","journal-title":"Int. J. Non-Linear Mech."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"423","DOI":"10.1016\/S0020-7462(01)00062-2","article-title":"A new exact solution for the flow of a Maxwell fluid past an infinite plate","volume":"38","author":"Fetecau","year":"2003","journal-title":"Int. J. Non-Linear Mech."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"1367","DOI":"10.1080\/00986441003626078","article-title":"Exact Solutions for the Unsteady Flow of a Burgers\u2019 Fluid between Two Sidewalls Perpendicular to the Plate","volume":"197","author":"Khan","year":"2010","journal-title":"Chem. Eng. Commun."},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"584","DOI":"10.1016\/j.nonrwa.2008.10.055","article-title":"New exact solutions corresponding to the second problem of Stokes for second grade fluids","volume":"11","author":"Nazar","year":"2010","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1016\/j.rinp.2011.04.001","article-title":"New exact solution for Rayleigh\u2013Stokes problem of Maxwell fluid in a porous medium and rotating frame","volume":"1","author":"Salah","year":"2011","journal-title":"Results Phys."},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"3096","DOI":"10.1016\/j.camwa.2008.07.003","article-title":"Some exact solutions for the helical flow of a generalized Oldroyd-B fluid in a circular cylinder","volume":"56","author":"Fetecau","year":"2008","journal-title":"Comput. Math. Appl."},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"682","DOI":"10.1016\/j.mcm.2009.10.040","article-title":"Exact solutions for some oscillating motions of a fractional Burgers\u2019 fluid","volume":"51","author":"Khan","year":"2010","journal-title":"Math. Comput. Model."},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"777","DOI":"10.1007\/s10409-009-0277-z","article-title":"Exact solution for the rotational flow of a generalized second grade fluid in a circular cylinder","volume":"25","author":"Siddique","year":"2009","journal-title":"Acta Mech. Sin."},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"573","DOI":"10.1007\/s11012-007-9081-7","article-title":"Starting solutions for the oscillating motion of a Maxwell fluid in cylindrical domains","volume":"42","author":"Vieru","year":"2007","journal-title":"Meccanica"},{"key":"ref_73","first-page":"459","article-title":"Flow of a viscoelastic fluid with the fractional Maxwell model between two side walls perpendicular to a plate","volume":"200","author":"Vieru","year":"2008","journal-title":"Appl. Math. Comput."},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"109454","DOI":"10.1016\/j.chaos.2019.109454","article-title":"Hydromagnetic flow over a moving plate of second grade fluids with time fractional derivatives having non-singular kernel","volume":"130","author":"Fetecau","year":"2020","journal-title":"Chaos Solitons Fractals"},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1186\/s13662-016-0775-9","article-title":"Solutions with Wright functions for time fractional convection flow near a heated vertical plate","volume":"2016","author":"Shakeel","year":"2016","journal-title":"Adv. Differ. Equ."},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"534","DOI":"10.1016\/j.rinp.2018.09.023","article-title":"Exact solution of non-Newtonian fluid motion between side walls","volume":"11","author":"Asif","year":"2018","journal-title":"Results Phys."},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"1849","DOI":"10.1016\/j.aej.2017.03.017","article-title":"Exact solutions for free convection flow of generalized Jeffrey fluid: A Caputo-Fabrizio fractional model","volume":"57","author":"Saqib","year":"2018","journal-title":"Alex. Eng. J."},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"310","DOI":"10.1140\/epjp\/i2016-16310-5","article-title":"Solutions with special functions for time fractional free convection flow of Brinkman-type fluid","volume":"131","author":"Ali","year":"2016","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_79","doi-asserted-by":"crossref","unstructured":"Ali, F., Khan, I., and Shafie, S. (2014). Closed form solutions for unsteady free convection flow of a second grade fluid over an oscillating vertical plate. PLoS ONE, 9.","DOI":"10.1371\/journal.pone.0085099"},{"key":"ref_80","first-page":"635","article-title":"Exact Solutions for Unsteady Magnetohydrodynamic oscillatory flow of a maxwell fluid in a porous medium","volume":"68","author":"Khan","year":"2013","journal-title":"Z. Nat. A"},{"key":"ref_81","first-page":"377","article-title":"A note on new exact solutions for some unsteady flows of Brinkman-type fluids over a plane wall","volume":"67","author":"Ali","year":"2012","journal-title":"Z. Nat. A"},{"key":"ref_82","doi-asserted-by":"crossref","first-page":"2565","DOI":"10.1007\/s13369-017-2521-3","article-title":"Solutions with Wright function for time fractional free convection flow of Casson fluid","volume":"42","author":"Ali","year":"2017","journal-title":"Arab. J. Sci. Eng."},{"key":"ref_83","doi-asserted-by":"crossref","first-page":"1865","DOI":"10.1007\/s00521-016-2815-5","article-title":"A modern approach of Caputo\u2013Fabrizio 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_84","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1140\/epjp\/i2018-11953-8","article-title":"A comparative mathematical analysis of RL and RC electrical circuits via Atangana-Baleanu and Caputo-Fabrizio fractional derivatives","volume":"133","author":"Abro","year":"2018","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_85","first-page":"18","article-title":"Solutions of Stokes Second Problem for Casson Fluid over an Infinite Plate: A Comparison of the Laplace and Fourier Transforms","volume":"2","author":"Ali","year":"2018","journal-title":"City Univ. Int. J. Comput. Anal."},{"key":"ref_86","doi-asserted-by":"crossref","first-page":"2197","DOI":"10.1007\/s10973-018-7302-z","article-title":"Dual thermal analysis of magnetohydrodynamic flow of nanofluids via modern approaches of Caputo\u2013Fabrizio and Atangana\u2013Baleanu fractional derivatives embedded in porous medium","volume":"135","author":"Abro","year":"2018","journal-title":"J. Therm. Anal. Calorim."},{"key":"ref_87","doi-asserted-by":"crossref","unstructured":"Sheikh, N.A., Ching, D.L.C., Khan, I., Kumar, D., and Nisar, K.S. (2019). A new model of fractional Casson fluid based on generalized Fick\u2019s and Fourier\u2019s laws together with heat and mass transfer. Alex. Eng. J.","DOI":"10.1016\/j.aej.2019.12.023"},{"key":"ref_88","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1051\/mmnp\/2018007","article-title":"Application of Atangana-Baleanu fractional derivative to convection flow of MHD Maxwell fluid in a porous medium over a vertical plate","volume":"13","author":"Abro","year":"2018","journal-title":"Math. Model. Nat. Phenom."},{"key":"ref_89","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1016\/j.ijheatmasstransfer.2012.12.009","article-title":"Heat and mass transfer of unsteady natural convection flow of some nanofluids past a vertical infinite flat plate with radiation effect","volume":"59","author":"Turkyilmazoglu","year":"2013","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_90","first-page":"2323","article-title":"Unsteady hydromagnetic radiative flow of a nanofluid past a flat plate with ramped temperature","volume":"975","author":"Nandkeolyar","year":"2013","journal-title":"J. Orissa Math. Soc."},{"key":"ref_91","doi-asserted-by":"crossref","unstructured":"Khalid, A., Khan, I., and Shafie, S. (2015). Exact solutions for free convection flow of nanofluids with ramped wall temperature. Eur. Phys. J. Plus, 130.","DOI":"10.1140\/epjp\/i2015-15057-9"},{"key":"ref_92","doi-asserted-by":"crossref","first-page":"987","DOI":"10.1016\/j.icheatmasstransfer.2010.06.004","article-title":"Mixed convection boundary layer flow from a vertical flat plate embedded in a porous medium filled with nanofluids","volume":"37","author":"Ahmad","year":"2010","journal-title":"Int. Commun. Heat Mass Transf."},{"key":"ref_93","doi-asserted-by":"crossref","unstructured":"Gul, A., Khan, I., Shafie, S., Khalid, A., and Khan, A. (2015). Heat Transfer in MHD Mixed Convection Flow of a Ferrofluid along a Vertical Channel. PLoS ONE, 10.","DOI":"10.1371\/journal.pone.0141213"},{"key":"ref_94","first-page":"244","article-title":"Mixed convective magnetohydrodynamic flow in a vertical channel filled with nanofluids","volume":"18","author":"Das","year":"2015","journal-title":"Eng. Sci. Technol. Int. J."},{"key":"ref_95","doi-asserted-by":"crossref","first-page":"1175","DOI":"10.1016\/j.molliq.2016.06.105","article-title":"Heat transfer in ferrofluid with cylindrical shape nanoparticles past a vertical plate with ramped wall temperature embedded in a porous medium","volume":"221","author":"Khalid","year":"2016","journal-title":"J. Mol. Liq."},{"key":"ref_96","first-page":"1","article-title":"Molybdenum disulfide nanoparticles suspended in water-based nanofluids with mixed convection and flow inside a channel filled with saturated porous medium","volume":"1775","author":"Shafie","year":"2016","journal-title":"AIP Conf. Proc."},{"key":"ref_97","doi-asserted-by":"crossref","unstructured":"Ali, F., Gohar, M., Khan, I., Sheikh, N.A., Jan, S.A.A., and Saqib, M. (2018). Magnetite Molybdenum Disulphide Nanofluid of Grade Two: A Generalized Model with Caputo-Fabrizio Derivative. Microfluid. Nanofluidics.","DOI":"10.5772\/intechopen.72863"},{"key":"ref_98","first-page":"1","article-title":"A new definition of fractional derivative without singular kernel","volume":"1","author":"Caputo","year":"2015","journal-title":"Progr. Fract. Differ. Appl."},{"key":"ref_99","doi-asserted-by":"crossref","first-page":"138","DOI":"10.1016\/j.molliq.2016.06.098","article-title":"The impact silver nanoparticles on MHD free convection flow of Jeffrey fluid over an oscillating vertical plate embedded in a porous medium","volume":"222","author":"Khan","year":"2016","journal-title":"J. Mol. Liq."},{"key":"ref_100","first-page":"893","article-title":"Magnetohydrodynamic flow of brinkman-type engine oil based MoS2-nanofluid in a rotating disk with hall effect","volume":"4","author":"Ali","year":"2017","journal-title":"Int. J. Heat Technol."},{"key":"ref_101","doi-asserted-by":"crossref","unstructured":"Aman, S., Khan, I., Ismail, Z., Salleh, M.Z., Alshomrani, A.S., and Alghamdi, M.S. (2017). Magnetic field effect on Poiseuille flow and heat transfer of carbon nanotubes along a vertical channel filled with Casson fluid. AIP Adv., 7.","DOI":"10.1063\/1.4975219"},{"key":"ref_102","doi-asserted-by":"crossref","first-page":"864","DOI":"10.1016\/j.applthermaleng.2016.08.129","article-title":"Velocity, mass and temperature analysis of gravity-driven convection nanofluid flow past an oscillating vertical plate in the presence of magnetic field in a porous medium","volume":"110","author":"Kataria","year":"2017","journal-title":"Appl. Therm. Eng."},{"key":"ref_103","doi-asserted-by":"crossref","first-page":"2445","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_104","doi-asserted-by":"crossref","first-page":"442","DOI":"10.1016\/j.molliq.2017.03.009","article-title":"Shape effects of MoS 2 nanoparticles on MHD slip flow of molybdenum disulphide nanofluid in a porous medium","volume":"233","author":"Khan","year":"2017","journal-title":"J. Mol. Liq."},{"key":"ref_105","doi-asserted-by":"crossref","unstructured":"Sheikh, N.A., Ali, F., Khan, I., Gohar, M., and Saqib, M. (2017). On the applications of nanofluids to enhance the performance of solar collectors: A comparative analysis of Atangana-Baleanu and Caputo-Fabrizio fractional models. Eur. Phys. J. Plus, 132.","DOI":"10.1140\/epjp\/i2017-11809-9"},{"key":"ref_106","doi-asserted-by":"crossref","first-page":"1502","DOI":"10.1002\/num.22195","article-title":"Numerical approximation of Riemann-Liouville definition of fractional derivative: From Riemann-Liouville to Atangana-Baleanu","volume":"34","author":"Atangana","year":"2018","journal-title":"Numer. Methods Partial Differ. Equ."},{"key":"ref_107","unstructured":"Ali, F., Gohar, M., Khan, I., Sheikh, N.A., and Jan, S.A.A. (2018). Thermal Radiation and Magnetic Field Effects on Different Channel Flows of CNTs Brinkman-Type Nanofluids with Water, Kerosene and Engine-oil. City Univ. Int. J. Comput. Anal., 2."},{"key":"ref_108","doi-asserted-by":"crossref","first-page":"482","DOI":"10.1016\/j.molliq.2016.12.040","article-title":"Convection heat transfer in micropolar nanofluids with oxide nanoparticles in water, kerosene and engine oil","volume":"229","author":"Hussanan","year":"2017","journal-title":"J. Mol. Liq."},{"key":"ref_109","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1016\/j.chaos.2018.09.007","article-title":"Application of Atangana\u2013Baleanu fractional derivative to MHD channel flow of CMC-based-CNT\u2019s nanofluid through a porous medium","volume":"116","author":"Saqib","year":"2018","journal-title":"Chaos Solitons Fractals"},{"key":"ref_110","doi-asserted-by":"crossref","first-page":"619","DOI":"10.18280\/ijht.360227","article-title":"Natural convection in polyethylene glycol based molybdenum disulfide nanofluid with thermal radiation, chemical reaction and ramped wall temperature","volume":"36","author":"Ali","year":"2018","journal-title":"Int. J. Heat Technol."},{"key":"ref_111","doi-asserted-by":"crossref","first-page":"789","DOI":"10.1007\/s00521-016-2688-7","article-title":"Impacts of gold nanoparticles on MHD mixed convection Poiseuille flow of nanofluid passing through a porous medium in the presence of thermal radiation, thermal diffusion and chemical reaction","volume":"30","author":"Aman","year":"2018","journal-title":"Neural. Comput. Appl."},{"key":"ref_112","doi-asserted-by":"crossref","first-page":"515","DOI":"10.1016\/j.ijheatmasstransfer.2013.03.004","article-title":"Entropy generation in steady MHD flow due to a rotating porous disk in a nanofluid","volume":"62","author":"Rashidi","year":"2013","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_113","doi-asserted-by":"crossref","first-page":"514","DOI":"10.1016\/j.ijheatmasstransfer.2013.06.010","article-title":"A review of entropy generation in nanofluid flow","volume":"65","author":"Mahian","year":"2013","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_114","doi-asserted-by":"crossref","first-page":"677","DOI":"10.1016\/j.molliq.2018.12.109","article-title":"Entropy generation in flow of Carreau nanofluid","volume":"278","author":"Khan","year":"2019","journal-title":"J. Mol. Liq."},{"key":"ref_115","doi-asserted-by":"crossref","first-page":"947","DOI":"10.1016\/j.rinp.2018.03.012","article-title":"Entropy generation in a mixed convection Poiseulle flow of molybdenum disulphide Jeffrey nanofluid","volume":"9","author":"Gul","year":"2018","journal-title":"Results Phys."},{"key":"ref_116","doi-asserted-by":"crossref","first-page":"8645","DOI":"10.1038\/s41598-018-26994-1","article-title":"MHD Flow of Sodium Alginate-Based Casson Type Nanofluid Passing Through A Porous Medium With Newtonian Heating","volume":"8","author":"Khan","year":"2018","journal-title":"Sci. Rep."},{"key":"ref_117","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1016\/j.chaos.2018.08.020","article-title":"A theoretical study on the performance of a solar collector using CeO2 and Al2O3 water based nanofluids with inclined plate: Atangana\u2013Baleanu fractional model","volume":"115","author":"Sheikh","year":"2018","journal-title":"Chaos Solitons Fractals"},{"key":"ref_118","doi-asserted-by":"crossref","first-page":"1352","DOI":"10.1016\/j.rinp.2018.04.007","article-title":"A new Caputo time fractional model for heat transfer enhancement of water based graphene nanofluid: An application to solar energy","volume":"9","author":"Aman","year":"2018","journal-title":"Results Phys."},{"key":"ref_119","first-page":"131","article-title":"Effect of MHD and porosity on exact solutions and flow of a hybrid Casson-nanofluid","volume":"44","author":"Aman","year":"2018","journal-title":"J. Adv. Res. Fluid Mech. Therm. Sci."},{"key":"ref_120","first-page":"769","article-title":"Channel flow of fractionalized H2O-based CNTs nanofluids with Newtonian heating","volume":"13","author":"Khan","year":"2018","journal-title":"Discret. Contin. Dyn. Syst. S"},{"key":"ref_121","doi-asserted-by":"crossref","unstructured":"Tzou, D.Y. (2014). Macro-to Microscale Heat Transfer: The Lagging Behavior, John Wiley & Sons.","DOI":"10.1002\/9781118818275"},{"key":"ref_122","doi-asserted-by":"crossref","first-page":"15285","DOI":"10.1038\/s41598-018-33547-z","article-title":"Effects of Different Shaped Nanoparticles on the Performance of Engine-Oil and Kerosene-Oil: A generalized Brinkman-Type Fluid model with Non-Singular Kernel","volume":"8","author":"Ali","year":"2018","journal-title":"Sci. Rep."},{"key":"ref_123","first-page":"1243","article-title":"Effect of thermal radiation on magneto-nanofluids free convective flow over an ac-celerated moving ramped temperature plate","volume":"25","author":"Hussain","year":"2017","journal-title":"Sci. Iran."},{"key":"ref_124","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_125","doi-asserted-by":"crossref","first-page":"531","DOI":"10.1007\/s13369-018-3342-8","article-title":"Entropy Generation in Different Types of Fractionalized Nanofluids","volume":"44","author":"Saqib","year":"2018","journal-title":"Arab. J. Sci. Eng."},{"key":"ref_126","doi-asserted-by":"crossref","first-page":"523","DOI":"10.1007\/s10973-018-7054-9","article-title":"Convection in ethylene glycol-based molybdenum disulfide nanofluid","volume":"135","author":"Saqib","year":"2018","journal-title":"J. Therm. Anal. Calorim."},{"key":"ref_127","doi-asserted-by":"crossref","unstructured":"Ali, F., Khan, I., Sheikh, N.A., and Gohar, M. (2019). Exact solutions for the Atangana-Baleanu time-fractional model of a Brinkman-type nanofluid in a rotating frame: Applications in solar collectors. Eur. Phys. J. Plus, 134.","DOI":"10.1140\/epjp\/i2019-12455-y"},{"key":"ref_128","doi-asserted-by":"crossref","first-page":"318","DOI":"10.1039\/C6RA25625B","article-title":"Enhanced thermal conductivity of nanofluid-based ethylene glycol containing Cu nanoparticles decorated on a Gr\u2013MWCNT hybrid material","volume":"7","author":"Anh","year":"2017","journal-title":"RSC Adv."},{"key":"ref_129","doi-asserted-by":"crossref","first-page":"388","DOI":"10.1016\/j.molliq.2018.12.104","article-title":"Variable magnetic forces impact on magnetizable hybrid nanofluid heat transfer through a circular cavity","volume":"277","author":"Sheikholeslami","year":"2019","journal-title":"J. Mol. Liq."},{"key":"ref_130","doi-asserted-by":"crossref","first-page":"105093","DOI":"10.1016\/j.cmpb.2019.105093","article-title":"Magneto rotating flow of hybrid nanofluid with entropy generation","volume":"183","author":"Khan","year":"2019","journal-title":"Comput. Methods Programs Biomed."},{"key":"ref_131","doi-asserted-by":"crossref","first-page":"671","DOI":"10.1615\/JPorMedia.v20.i8.10","article-title":"Unsteady MHD flow of second-grade fluid over an oscillating vertical plate with isothermal temperature in a porous medium with heat and mass transfer by using the Laplace transform technique","volume":"20","author":"Ali","year":"2017","journal-title":"J. Porous Media"},{"key":"ref_132","first-page":"374","article-title":"Analysis of time dependent third grade fluid in wire coating","volume":"8","author":"Gul","year":"2017","journal-title":"J. Nonlinear Sci. Lett. A"},{"key":"ref_133","doi-asserted-by":"crossref","unstructured":"Sheikh, N.A., Ali, F., Khan, I., Saqib, M., and Khan, A. (2017). MHD flow of micropolar fluid over an oscillating vertical plate embedded in porous media with constant temperature and concentration. Math. Probl. Eng., 2017.","DOI":"10.1155\/2017\/9402964"},{"key":"ref_134","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1007\/s00216-001-1200-z","article-title":"Electron transfer of hemoglobin at electrodes modified with colloidal clay nanoparticles","volume":"372","author":"Lei","year":"2002","journal-title":"Anal. Bioanal. Chem."},{"key":"ref_135","doi-asserted-by":"crossref","first-page":"2121","DOI":"10.1021\/nn800511k","article-title":"Comparison of the mechanism of toxicity of zinc oxide and cerium oxide nanoparticles based on dissolution and oxidative stress properties","volume":"2","author":"Xia","year":"2008","journal-title":"ACS Nano"},{"key":"ref_136","doi-asserted-by":"crossref","first-page":"4806","DOI":"10.1021\/nn1006368","article-title":"Improved synthesis of graphene oxide","volume":"4","author":"Marcano","year":"2010","journal-title":"ACS Nano"},{"key":"ref_137","doi-asserted-by":"crossref","first-page":"3303","DOI":"10.1039\/b407904c","article-title":"Biosynthesis of zirconia nanoparticles using the fungus Fusarium oxysporum","volume":"14","author":"Bansal","year":"2004","journal-title":"J. Mater. Chem."},{"key":"ref_138","doi-asserted-by":"crossref","first-page":"314","DOI":"10.1016\/j.jcis.2010.06.069","article-title":"Fabrication, characterization, and application of greigite nanoparticles for cancer hyperthermia","volume":"363","author":"Chang","year":"2011","journal-title":"J. Colloid Interface Sci."},{"key":"ref_139","doi-asserted-by":"crossref","unstructured":"Yang, X.J., Srivastava, H.M., and Machado, J. (2015). A new fractional derivative without singular kernel: Application to the modelling of the steady heat flow. arXiv.","DOI":"10.2298\/TSCI151224222Y"},{"key":"ref_140","doi-asserted-by":"crossref","first-page":"118","DOI":"10.3389\/fphy.2018.00118","article-title":"Sinc-fractional operator on Shannon wavelet space","volume":"6","author":"Cattani","year":"2018","journal-title":"Front. Phys."},{"key":"ref_141","doi-asserted-by":"crossref","first-page":"3567","DOI":"10.1140\/epjst\/e2018-00020-2","article-title":"A new fractional derivative involving the normalized sinc function without singular kernel","volume":"226","author":"Yang","year":"2017","journal-title":"Eur. Phys. J. Spec. Top."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/5\/725\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T14:09:53Z","timestamp":1760364593000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/5\/725"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,5,3]]},"references-count":141,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2020,5]]}},"alternative-id":["sym12050725"],"URL":"https:\/\/doi.org\/10.3390\/sym12050725","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,5,3]]}}}