{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,11]],"date-time":"2026-04-11T01:06:48Z","timestamp":1775869608218,"version":"3.50.1"},"reference-count":53,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2023,3,26]],"date-time":"2023-03-26T00:00:00Z","timestamp":1679788800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Princess Nourah bint Abdulrahman University","award":["PNURSP2023R41"],"award-info":[{"award-number":["PNURSP2023R41"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In the present study, the effects of reduced gravity and solar radiation on the magnetohydrodynamics (MHD) fluid flow and heat transfer past a solid and stationary sphere embedded in a porous medium are investigated. A model describing the considered configuration is put in dimensionless form using appropriate dimensionless variables and then transformed to primitive form for a smooth algorithm on a computing tool. A primitive form of the model is solved by employing the finite difference method. Solutions for variables of interest, such as velocity distribution and temperature field, along with their gradients, are depicted in graphs and tables. The main goal of the paper is to study the physical impact of reduced gravity on heat transfer and fluid flow around a sphere surface inserted in a porous medium in the presence of an applied magnetic field and solar radiation. The effects of the governing parameters, which are the reduced gravity parameter, magnetic field parameter, radiation parameter, porous medium parameter, and the Prandtl number, are discussed and physically interpreted. The displayed solutions indicate that velocity rises with the reduced gravity and solar radiation parameters but decreases with augmenting the Prandtl number, magnetic field parameter, and porous medium parameter. It is deduced from the presented results that the temperature becomes lower by increasing the values of the reduced gravity parameter and the Prandtl number, but, on the other hand, it becomes higher by increasing the values of the magnetic field, the porous medium, and the radiation parameters at all the considered positions of the surface of the sphere. A comparison between the present and already published results is performed to check the validity of the proposed numerical model.<\/jats:p>","DOI":"10.3390\/sym15040806","type":"journal-article","created":{"date-parts":[[2023,3,27]],"date-time":"2023-03-27T04:31:21Z","timestamp":1679891481000},"page":"806","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":29,"title":["The Effects of Reduced Gravity and Radiative Heat Transfer on the Magnetohydrodynamic Flow Past a Non-Rotating Stationary Sphere Surrounded by a Porous Medium"],"prefix":"10.3390","volume":"15","author":[{"given":"Amir","family":"Abbas","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, University of Gujrat, Sub-Campus, Mandi Bahauddin 50400, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6542-0490","authenticated-orcid":false,"given":"Ioannis E.","family":"Sarris","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, University of West Attica, 12244 Athens, Greece"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7016-9181","authenticated-orcid":false,"given":"Muhammad","family":"Ashraf","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, University of Sargodha, Sargodha 40100, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9372-251X","authenticated-orcid":false,"given":"Kaouther","family":"Ghachem","sequence":"additional","affiliation":[{"name":"Department of Industrial Engineering and Systems, College of Engineering, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9888-389X","authenticated-orcid":false,"given":"Nidhal","family":"Hnaien","sequence":"additional","affiliation":[{"name":"Research Laboratory of Metrology and Energy Systems, National Engineering School, University of Monastir, LR18ES21, Monastir 5000, Tunisia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6819-3695","authenticated-orcid":false,"given":"Badr M.","family":"Alshammari","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering, College of Engineering, University of Ha\u2019il, P.O. Box 2440, Ha\u2019il City 81451, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2023,3,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1017\/S0022112095004265","article-title":"Boundary-layer analysis of the thermal bar","volume":"303","author":"Kay","year":"1995","journal-title":"J. Fluid Mech."},{"key":"ref_2","unstructured":"Ostrach, S. (1952). An Analysis of Laminar Free-Convection Flow and Heat Transfer about a Flat Plate Parallel to the Direction of the Generating Body Force, National Aeronautics and Space Administration Cleveland Oh Lewis Research Center."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1007\/BF02383574","article-title":"Free convection on a heated vertical plate: The solution for small Prandtl number","volume":"23","author":"Merkin","year":"1989","journal-title":"J. Eng. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2319","DOI":"10.1016\/0017-9310(87)90224-9","article-title":"Natural convection heat transfer in a square enclosure containing water near its density maximum","volume":"30","author":"Lin","year":"1987","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1115\/1.3250628","article-title":"Convection near the temperature of maximum density for high Rayleigh number, low aspect ratio, rectangular cavities","volume":"111","author":"Ivey","year":"1989","journal-title":"ASME J. Heat Transfer."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"769","DOI":"10.1017\/S0022112080001395","article-title":"Free convection from a heated sphere at large Grashof number","volume":"100","author":"Potter","year":"1980","journal-title":"J. Fluid Mech."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1016\/0045-7930(86)90022-8","article-title":"The heat transfer from a sphere in free convective flow","volume":"14","author":"Riley","year":"1986","journal-title":"Comput. Fluids"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"976","DOI":"10.1139\/cjp-2016-0593","article-title":"Periodic momentum and thermal boundary layer mixed convection flow around the surface of a sphere in the presence of viscous dissipation","volume":"95","author":"Ashraf","year":"2017","journal-title":"Can. J. Phys."},{"key":"ref_9","first-page":"701","article-title":"Numerical simulation of the effect of transient shear stress and the rate of heat transfer around different positions of sphere in the presence of viscous dissipation","volume":"140","year":"2018","journal-title":"J. Heat Transf."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1127","DOI":"10.1002\/htj.21424","article-title":"Natural convection boundary layer flow of nanofluids around different stations of the sphere and into the plume above the sphere","volume":"48","author":"Ashraf","year":"2019","journal-title":"Heat Transf.\u2014Asian Res."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Abbas, A., Ashraf, M., Chu, Y., Zia, S., Khan, I., and Nisar, K.S. (2020). Computational Study of the Coupled Mechanism of Thermophoretic Transportation and Mixed Convection Flow around the Surface of a Sphere. Molecules, 25.","DOI":"10.3390\/molecules25112694"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"4089","DOI":"10.2298\/TSCI190518137A","article-title":"Combined effects of variable viscosity and thermophoretic transportation on mixed convection flow around the surface of a sphere","volume":"24","author":"Abbas","year":"2020","journal-title":"Therm. Sci."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"085005","DOI":"10.1063\/5.0018674","article-title":"Numerical simulation of the combined effects of thermophoretic motion and variable thermal conductivity on free convection heat transfer","volume":"10","author":"Ashraf","year":"2020","journal-title":"AIP Adv."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"3243","DOI":"10.1016\/j.aej.2021.01.038","article-title":"Combined effects of thermal radiation and thermophoretic motion on mixed convection boundary layer flow","volume":"60","author":"Abbas","year":"2021","journal-title":"Alex. Eng. J."},{"key":"ref_15","first-page":"1809","article-title":"Computational analysis of the effect of nano particle material motion on mixed convection flow in the presence of heat generation and absorption","volume":"65","author":"Ashraf","year":"2020","journal-title":"Comput. Mater. Contin."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"7349","DOI":"10.1002\/htj.22232","article-title":"Computations of mixed convection slip flow around the surface of a sphere: Effects of thermophoretic transportation and viscous dissipation","volume":"50","author":"Ashraf","year":"2021","journal-title":"Heat Transf."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"12907","DOI":"10.1038\/s41598-021-92409-3","article-title":"Mixed convection flow along a curved surface in the presence of exothermic catalytic chemical reaction","volume":"11","author":"Ahmad","year":"2021","journal-title":"Sci. Rep."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"101640","DOI":"10.1016\/j.csite.2021.101640","article-title":"Combined effects of variable density and thermal radiation on MHD Sakiadis flow","volume":"28","author":"Abbas","year":"2021","journal-title":"Case Stud. Therm. Eng."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"498","DOI":"10.1016\/j.rinp.2017.01.005","article-title":"Magnetohydrodynamic flow of Casson fluid over a stretching cylinder","volume":"7","author":"Tamoor","year":"2017","journal-title":"Results Phys."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1615\/InterJFluidMechRes.2020028543","article-title":"Influence of chemical reaction on magnetohydrodynamic flow over an exponential stretching sheet: A numerical study","volume":"47","author":"Pattnaik","year":"2020","journal-title":"Int. J. Fluid Mech. Res."},{"key":"ref_21","first-page":"68","article-title":"MHD flow over exponential radiating stretching sheet using homotopy analysis method","volume":"29","author":"Mabood","year":"2017","journal-title":"J. King Saud Univ.-Eng. Sci."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"124231","DOI":"10.1016\/j.physa.2020.124231","article-title":"Numerical modeling and analysis of bioconvection on MHD flow due to an upper paraboloid surface of revolution","volume":"553","author":"Khan","year":"2020","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"2589760","DOI":"10.1155\/2020\/2589760","article-title":"Magnetohydrodynamics Free Convection Flow of Incompressible Fluids over Corrugated Vibrating Bottom Surface with Hall Currents and Heat and Mass Transfers","volume":"2020","author":"Bulinda","year":"2020","journal-title":"J. Appl. Math."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"102818","DOI":"10.1016\/j.rinp.2019.102818","article-title":"MHD natural convection of Sodium Alginate Cassonnanofluid over a solid sphere","volume":"16","author":"Alwawi","year":"2020","journal-title":"Results Phys."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"34","DOI":"10.11648\/j.acm.20170601.12","article-title":"Double-Diffusion MHD free convective flow along a sphere in the presence of a homogeneous chemical reaction and Soret and Dufour effects","volume":"6","author":"Chamkha","year":"2017","journal-title":"Appl. Comput. Math."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"126","DOI":"10.26637\/MJM0S20\/0024","article-title":"Pulsatile flow through a circular pipe with porous medium under the influence of time varying pressure gradient: Effects of with and without visco-elastic fluid","volume":"S","author":"Chitra","year":"2020","journal-title":"Malaya J. Mat."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Abbas, A., Jeelani, M.B., and Alharthi, N.H. (2022). Magnetohydrodynamic Effects on Third-Grade Fluid Flow and Heat Transfer with Darcy\u2013Forchheimer Law over an Inclined Exponentially Stretching Sheet Embedded in a Porous Medium. Magnetochemistry, 8.","DOI":"10.3390\/magnetochemistry8060061"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Abbas, A., Jeelani, M.B., and Alharthi, N.H. (2022). Darcy\u2013Forchheimer Relation Influence on MHD Dissipative Third-Grade Fluid Flow and Heat Transfer in Porous Medium with Joule Heating Effects: A Numerical Approach. Processes, 10.","DOI":"10.3390\/pr10050906"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Abbas, A., Shafqat, R., Jeelani, M.B., and Alharthi, N.H. (2022). Convective Heat and Mass Transfer in Third-Grade Fluid with Darcy\u2013Forchheimer Relation in the Presence of Thermal-Diffusion and Diffusion-Thermo Effects over an Exponentially Inclined Stretching Sheet Surrounded by a Porous Medium: A CFD Study. Processes, 10.","DOI":"10.3390\/pr10040776"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Abbas, A., Shafqat, R., Jeelani, M.B., and Alharthi, N.H. (2022). Significance of Chemical Reaction and Lorentz Force on Third-Grade Fluid Flow and Heat Transfer with Darcy\u2013Forchheimer Law over an Inclined Exponentially Stretching Sheet Embedded in a Porous Medium. Symmetry, 14.","DOI":"10.3390\/sym14040779"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Abbas, A., Jeelani, M.B., Alnahdi, A.S., and Ilyas, A. (2022). MHD Williamson Nanofluid Fluid Flow and Heat Transfer Past a Non-Linear Stretching Sheet Implanted in a Porous Medium: Effects of Heat Generation and Viscous Dissipation. Processes, 10.","DOI":"10.3390\/pr10061221"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"106559","DOI":"10.1016\/j.icheatmasstransfer.2022.106559","article-title":"Convection analysis of the radiative nanofluid flow through porous media over a stretching surface with inclined magnetic field","volume":"140","author":"Hussain","year":"2023","journal-title":"Int. Commun. Heat Mass Transf."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"20999","DOI":"10.1016\/j.ijhydene.2022.04.201","article-title":"Numerical study on methane\/air combustion characteristics in a heat-recirculating micro combustor embedded with porous media","volume":"47","author":"Yan","year":"2022","journal-title":"Int. J. Hydrog. Energy"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"678","DOI":"10.1080\/16583655.2018.1499171","article-title":"The combined effects of anisotropic porous medium and stably stratified fluid on free convective flow through an annulus","volume":"12","author":"Jha","year":"2018","journal-title":"J. Taibah Univ. Sci."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1","DOI":"10.21608\/bfemu.2021.146283","article-title":"Enhancement of Mixed Convection in a Channel with Discrete Heat Sources by Using a Highly Conducting Porous Medium.(Dept. M)","volume":"25","year":"2021","journal-title":"MEJ. Mansoura Eng. J."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Sparrow, E.M., and Gregg, J.L. (1959). Details of Exact Low Prandtl Number Boundary-Layer Solutions for Forced and For Free Convection, NTRS. No. NASA-MEMO-2-27-59E.","DOI":"10.1063\/1.1705928"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1007\/BF02328616","article-title":"Radiation effect on mixed convection along a vertical plate with uniform surface temperature","volume":"31","author":"Hossain","year":"1996","journal-title":"Heat Mass Transf."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"54","DOI":"10.1115\/1.1991863","article-title":"Magneto hydrodynamics-mixed convection from radiate vertical isothermal surface embedded in a saturated porous media","volume":"73","author":"Damseh","year":"2006","journal-title":"J. Appl. Mech."},{"key":"ref_39","first-page":"47","article-title":"Influence of thermal radiation on free convection inside a porous enclosure","volume":"12","author":"Zahmatkesh","year":"2007","journal-title":"Emir. J. Eng. Res."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1007\/s11012-008-9156-0","article-title":"Radiation effects on combined convection over a vertical flat plate embedded in a porous medium of variable porosity","volume":"44","author":"Pal","year":"2009","journal-title":"Meccanica"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1800","DOI":"10.1016\/j.ijthermalsci.2009.01.019","article-title":"Influence of chemical reaction and thermal radiation on the heat and mass transfer in MHD micropolar flow over a vertical moving porous plate in a porous medium with heat generation","volume":"48","author":"Mohamed","year":"2009","journal-title":"Int. J. Therm. Sci"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"2945","DOI":"10.1016\/j.ijheatmasstransfer.2012.01.051","article-title":"Effects of thermal radiation on micropolar fluid flow and heat transfer over a porous shrinking sheet","volume":"55","author":"Bhattacharyya","year":"2012","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"498","DOI":"10.3846\/13926292.2012.706653","article-title":"MHD mixed convection stagnation-point flow of a micropolar fluid in a porous medium towards a heated stretching sheet with thermal radiation","volume":"17","author":"Pal","year":"2012","journal-title":"Math. Model. Anal."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1016\/j.ijthermalsci.2012.08.014","article-title":"On mixed convection\u2013radiation interaction about an inclined plate through a porous medium","volume":"64","author":"Moradi","year":"2013","journal-title":"Int. J. Therm. Sci."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"501","DOI":"10.1016\/j.energy.2013.07.070","article-title":"Numerical investigation of MHD effects on Al2O3\u2013water nanofluid flow and heat transfer in a semi-annulus enclosure using LBM","volume":"60","author":"Sheikholeslami","year":"2013","journal-title":"Energy"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"194","DOI":"10.1016\/j.jfranklin.2012.07.005","article-title":"Radiation effects on the thermal boundary layer flow of a micropolar fluid towards a permeable stretching sheet","volume":"350","author":"Hussain","year":"2013","journal-title":"J. Frankl. Inst."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"485","DOI":"10.1016\/j.asej.2012.10.007","article-title":"Slip effects on MHD boundary layer flow over an exponentially stretching sheet with suction\/blowing and thermal radiation","volume":"4","author":"Mukhopadhyay","year":"2013","journal-title":"Ain Shams Eng. J."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"155","DOI":"10.2514\/1.T4113","article-title":"Thermally stratified radiative flow of third grade fluid over a stretching surface","volume":"28","author":"Hayat","year":"2014","journal-title":"J. Thermopphys. Heat Transf."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"1277","DOI":"10.1016\/j.asej.2014.05.004","article-title":"Effect of radiation on transient MHD flow of micropolar fluid between porous vertical channel with boundary conditions of the third kind","volume":"5","author":"Prakash","year":"2014","journal-title":"Ain Shams Eng. J."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"489","DOI":"10.2298\/TSCI110712085U","article-title":"Influence of thermal radiation and heat generation\/absorption on MHD heat transfer flow of a micropolar fluid past a wedge considering hall and ion slip currents","volume":"18","author":"Uddin","year":"2014","journal-title":"Therm. Sci."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"1340","DOI":"10.1088\/0256-307X\/25\/4\/048","article-title":"Nnumerical tackling on Sakiadis flow with thermal radiation","volume":"25","author":"Cortell","year":"2008","journal-title":"Chin. Phys. Lett."},{"key":"ref_52","unstructured":"Siegel, R., and Howell, J.R. (1992). Thermal Radiation: Heat Transfer, Hemisphere. [3rd ed.]."},{"key":"ref_53","unstructured":"Sparrow, E.M., and Cess, R.D. (1978). Radiation Heat Transfer, Hemisphere."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/4\/806\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:03:26Z","timestamp":1760123006000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/4\/806"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,3,26]]},"references-count":53,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2023,4]]}},"alternative-id":["sym15040806"],"URL":"https:\/\/doi.org\/10.3390\/sym15040806","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,3,26]]}}}