{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,26]],"date-time":"2026-02-26T20:24:54Z","timestamp":1772137494535,"version":"3.50.1"},"reference-count":36,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2019,12,27]],"date-time":"2019-12-27T00:00:00Z","timestamp":1577404800000},"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>The aim of this current contribution is to examine the rheological significance of Maxwell fluid configured between two isothermal stretching disks. The energy equation is also extended by evaluating the heat source and sink features. The governing partial differential equations (PDEs) are converted into the ordinary differential equations (ODEs) by using appropriate variables. An analytically-based technique is adopted to compute the series solution of the dimensionless flow problem. The convergence of this series solution is carefully ensured. The physical interpretation of important physical parameters like the Hartmann number, Prandtl number, Archimedes number, Eckert number, heat source\/sink parameter and the activation energy parameter are presented for velocity, pressure and temperature profiles. The numerical values of different involved parameters for skin friction coefficient and local Nusselt number are expressed in tabular and graphical forms. Moreover, the significance of an important parameter, namely Frank-Kamenetskii, is presented both in tabular and graphical form. This particular study reveals that both axial and radial velocity components decrease by increasing the Frank\u2013Kamenetskii number and stretching the ratio parameter. The pressure distribution is enhanced with an increasing Frank\u2013Kamenetskii number and stretching ratio parameter. It is also observed that thetemperature distribution increases with the increasing Hartmann number, Eckert number and Archimedes number.<\/jats:p>","DOI":"10.3390\/sym12010062","type":"journal-article","created":{"date-parts":[[2019,12,27]],"date-time":"2019-12-27T11:42:47Z","timestamp":1577446967000},"page":"62","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":28,"title":["A Theoretical Analysis for Mixed Convection Flow of Maxwell Fluid between Two Infinite Isothermal Stretching Disks with Heat Source\/Sink"],"prefix":"10.3390","volume":"12","author":[{"given":"Nargis","family":"Khan","sequence":"first","affiliation":[{"name":"Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur 63100, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7167-3822","authenticated-orcid":false,"given":"Hossam A.","family":"Nabwey","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia"},{"name":"Department of Basic Engineering Science, Faculty of Engineering, Menoufia University, Shebin El-Kom 32511, Egypt"}]},{"given":"Muhammad Sadiq","family":"Hashmi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, The Govt. Sadiq College Women University, Bahawalpur 63100, Pakistan"}]},{"given":"Sami Ullah","family":"Khan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, COMSATS University Islamabad, Sahiwal 57000, Pakistan"}]},{"given":"Iskander","family":"Tlili","sequence":"additional","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"}]}],"member":"1968","published-online":{"date-parts":[[2019,12,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"692516","DOI":"10.1155\/2013\/692516","article-title":"Effects of exothermic\/endothermic chemical reactions with Arrhenius activation energy on MHD free convection and mass transfer flow in presence of thermal radiation","volume":"2013","author":"Maleque","year":"2013","journal-title":"J. Thermodyn."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"216","DOI":"10.1016\/j.icheatmasstransfer.2017.11.001","article-title":"Activation energy impact in nonlinear radiative stagnation point flow of Cross nanofluid","volume":"91","author":"Khan","year":"2018","journal-title":"Int. Commun. Heat Mass Transf."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"627","DOI":"10.1016\/j.rinp.2016.09.006","article-title":"Boundary layer flow of Maxwell fluid in rotating frame with binary chemical reaction and activation energy","volume":"6","author":"Shafique","year":"2016","journal-title":"Results Phys."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Awad, F.G., Motsa, S., and Khumalo, M. (2014). Heat and Mass Transfer in Unsteady Rotating Fluid Flow with Binary Chemical Reaction and Activation Energy. PLoS ONE, 9.","DOI":"10.1371\/journal.pone.0107622"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"486","DOI":"10.1016\/j.energy.2017.05.004","article-title":"To promote radiation electrical MHD activation energy thermal extrusion manufacturing system efficiency by using Carreau-Nanofluid with parameters control method","volume":"130","author":"Hsiao","year":"2017","journal-title":"Energy"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1023\/A:1006541819777","article-title":"Convective flows on reactive surfaces in porous media","volume":"47","author":"Merkin","year":"1998","journal-title":"Transp. Porous Media"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1016\/S0017-9310(97)00114-2","article-title":"Free convection driven by an exothermic on a vertical surface embedded in porous media","volume":"41","author":"Minto","year":"1998","journal-title":"Int. J. Heat Mass Trans. Transf."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"492","DOI":"10.1016\/B978-0-444-89851-7.50048-0","article-title":"Mixed convection heat transfer of horizontal channel flow over a heated block","volume":"2","author":"Chou","year":"1992","journal-title":"Transp. Phenom. Heat Mass Trans."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"115207","DOI":"10.1088\/1402-4896\/ab1eb6","article-title":"MHD stagnation point flow of viscous nanofluid over a curved surface","volume":"94","author":"Nadeem","year":"2019","journal-title":"Phys. Scr."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Ahmed, Z., Al-Qahtani, A., Nadeem, S., and Saleem, S. (2019). Computational Study of MHD Nanofluid Flow Possessing Micro-Rotational Inertia over a Curved Surface with Variable Thermophysical Properties. Processes, 7.","DOI":"10.3390\/pr7060387"},{"key":"ref_11","first-page":"66","article-title":"Existence and stability of heat and fluid flow in the presence of nanoparticles along a curved surface by mean of dual nature solution","volume":"353","author":"Khan","year":"2019","journal-title":"Appl. Math. Comput."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"4751","DOI":"10.1039\/C8RA09698H","article-title":"Numerical simulation of oscillatory oblique stagnation point flow of a magneto micropolar nanofluid","volume":"9","author":"Sadiq","year":"2019","journal-title":"RSC Adv."},{"key":"ref_13","unstructured":"Harris, J. (1977). Rheology and Non-Newtonian Flow, Longman Inc."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"396","DOI":"10.1016\/j.physleta.2006.04.117","article-title":"Series Solution for the Upper-Convected Maxwell Fluid over a Porous Stretching Plate","volume":"358","author":"Hayat","year":"2006","journal-title":"Phys. Lett. A"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"779","DOI":"10.1016\/j.cnsns.2007.12.003","article-title":"The Influence of Thermal Radiation on MHD Flow of Maxwellian Fluid above Stretching Sheet","volume":"14","author":"Aliakbar","year":"2009","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"840","DOI":"10.1016\/j.chaos.2007.01.067","article-title":"MHD Stagnation-point Flow of an Upper-Convected Maxwell Fluid over a Stretching Surface","volume":"39","author":"Hayat","year":"2009","journal-title":"ChaosSolitons Fractals"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1425","DOI":"10.1007\/s11012-011-9526-x","article-title":"MHD Flow and Heat Transfer of a UCM Fluid over a Stretching Surface with Variable Thermophysical Properties","volume":"47","author":"Prasad","year":"2012","journal-title":"Meccanica"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1007\/s40010-017-0470-6","article-title":"Heat transfer characteristics in oscillatory hydromagnetic channel flow of Maxwell fluid using Cattaneo-Christov model","volume":"89","author":"Khan","year":"2019","journal-title":"Proc. Natl. Acad. Sci. India Sect. A Phys. Sci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"22","DOI":"10.1016\/j.cjph.2019.02.010","article-title":"MHD swirling flow and heat transfer in Maxwell fluid driven by two coaxially rotating disks with variable thermal conductivity","volume":"60","author":"Ahmed","year":"2019","journal-title":"Chin. J. Phys."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1093\/qjmam\/47.3.405","article-title":"Free convection boundary layer driven by an exothermic surface reaction","volume":"47","author":"Merkin","year":"1994","journal-title":"Q. J. Mech. Appl. Math."},{"key":"ref_21","first-page":"1513","article-title":"Analytical solutions of a coupled nonlinear system arising in a flow between stretching disks","volume":"216","author":"Gorder","year":"2010","journal-title":"Appl. Math. Comput."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1090","DOI":"10.1016\/j.ijheatmasstransfer.2015.09.001","article-title":"Heat and mass transfer on MHD mixed convection axisymmetric chemically reactive flow of Maxwell fluid driven by exothermal and isothermal stretching disks","volume":"92","author":"Khan","year":"2016","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1017\/S0022112003004865","article-title":"On the analytic solution of magnetohydrodynamic flow of non-Newtonian fluids over a stretching sheet","volume":"488","author":"Liao","year":"2003","journal-title":"J. Fluid Mech."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"4097","DOI":"10.1016\/j.cnsns.2012.01.030","article-title":"Solution of the Thomas\u2013Fermi equation with a convergent approach","volume":"17","author":"Turkyilmazoglu","year":"2012","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_25","first-page":"21","article-title":"ParametrizedAdomian decomposition method with optimum convergence","volume":"24","author":"Turkyilmazoglu","year":"2017","journal-title":"ACM Trans. Modeling Comput. Simul."},{"key":"ref_26","first-page":"320","article-title":"Analytical and Numerical Study of Diffusion of Chemically Reactive Species in Eyring-Powell Fluid over an Oscillatory Stretching Surface","volume":"49","author":"Khan","year":"2017","journal-title":"Bulg. Chem. Commun."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1021","DOI":"10.1007\/s10973-018-7482-6","article-title":"Thermally radiated squeezed flow of magneto-nanofluid between two parallel disks with chemical reaction","volume":"135","author":"Ullah","year":"2019","journal-title":"J. Therm. Anal. Calorim."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1197","DOI":"10.1615\/JPorMedia.2018029010","article-title":"Darcy-Forchheimer MHD Couple Stress liquid flow by oscillatory stretched sheet with thermophoresis and heat generation\/absorption","volume":"21","author":"Khan","year":"2018","journal-title":"J. Porous Media"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1002\/apj.1954","article-title":"Aggregation effects on water base Al2O3\u2014Nano fluid over permeable wedge in mixed convection","volume":"11","author":"Ellahi","year":"2016","journal-title":"Asia-Pac. J. Chem. Eng."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"643","DOI":"10.1177\/0954408918787122","article-title":"Analysis of natural convective flow of non-Newtonian fluid under the effects of nanoparticles of different materials","volume":"233","author":"Hassan","year":"2019","journal-title":"J. Process Mech. Eng."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1061","DOI":"10.1615\/HeatTransRes.2018028397","article-title":"Effects of chemical reaction on third grade magnetohydrodynamics fluid flow under the influence of heat and mass transfer with variable reactive index","volume":"50","author":"Ambreen","year":"2019","journal-title":"Heat Transf. Res."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1108\/HFF-08-2017-0301","article-title":"Darcy\u2013Forchheimer flow of nanofluid due to a curved stretching surface","volume":"29","author":"Sajjad","year":"2019","journal-title":"Int. J. Numer. Methods Heat Fluid Flow"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1139\/cjp-2018-0159","article-title":"A comparitive study of magnetic and non-magnetic particles in nanofluidpropigating over over a wedge","volume":"97","author":"Hassan","year":"2019","journal-title":"Can. J. Phys."},{"key":"ref_34","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":"Hussain","year":"2019","journal-title":"Pramana J. Phys."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"2015","DOI":"10.2298\/TSCI150524129E","article-title":"A study of heat transfer in power law nanofluid","volume":"20","author":"Ellahi","year":"2016","journal-title":"Therm. Sci. J."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"871","DOI":"10.1177\/0954408916646139","article-title":"Shape effect of nanosize particles in unsteady mixed convection flow of nanofluid over disk with entropy generation","volume":"231","author":"Zeeshan","year":"2017","journal-title":"Proc. Inst. Mech. Eng. Part E J. Process Mech. Eng."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/1\/62\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:46:16Z","timestamp":1760190376000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/1\/62"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,12,27]]},"references-count":36,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2020,1]]}},"alternative-id":["sym12010062"],"URL":"https:\/\/doi.org\/10.3390\/sym12010062","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,12,27]]}}}