{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,6]],"date-time":"2026-01-06T19:26:42Z","timestamp":1767727602254,"version":"build-2238731810"},"reference-count":40,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2016,4,18]],"date-time":"2016-04-18T00:00:00Z","timestamp":1460937600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["www.mdpi.com"],"crossmark-restriction":true},"short-container-title":["Entropy"],"abstract":"<jats:p>Numerical study of the slip effects and radiative heat transfer on a steady state fully developed Williamson flow of an incompressible Newtonian fluid; between parallel vertical walls of a microchannel with isothermal walls in a porous medium is performed. The slip effects are considered at both boundary conditions. Radiative highly absorbing medium is modeled by the Rosseland approximation. The non-dimensional governing Navier\u2013Stokes and energy coupled partial differential equations formed a boundary problem are solved numerically using the fourth order Runge\u2013Kutta algorithm by means of a shooting method. Numerical outcomes for the skin friction coefficient, the rate of heat transfer represented by the local Nusselt number were presented even as the velocity and temperature profiles illustrated graphically and analyzed. The effects of the temperature number, Grashof number, thermal radiation parameter, Reynolds number, velocity slip length, Darcy number, and temperature jump, on the flow field and temperature field and their effects on the boundaries are presented and discussed.<\/jats:p>","DOI":"10.3390\/e18040147","type":"journal-article","created":{"date-parts":[[2016,4,18]],"date-time":"2016-04-18T10:37:17Z","timestamp":1460975837000},"page":"147","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":17,"title":["Numerical Simulation of Williamson Combined Natural and Forced Convective Fluid Flow between Parallel Vertical Walls with Slip Effects and Radiative Heat Transfer in a Porous Medium"],"prefix":"10.3390","volume":"18","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4423-1025","authenticated-orcid":false,"given":"Mohammad","family":"Abdollahzadeh Jamalabadi","sequence":"first","affiliation":[{"name":"Department of Mechanical, Robotics and Energy Engineering, Dongguk University, Seoul 04620, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Payam","family":"Hooshmand","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, Sanandaj Branch, Islamic Azad University, Sanandaj 6616935391, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Navid","family":"Bagheri","sequence":"additional","affiliation":[{"name":"Department of Energy Engineering, Graduate School of the Environment and Energy, Science and Research Branch, Islamic Azad University, Tehran 1477893855, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9701-9394","authenticated-orcid":false,"given":"HamidReza","family":"KhakRah","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, College of Engineering, Shiraz Branch, Islamic Azad University, Shiraz 71987-74731, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Majid","family":"Dousti","sequence":"additional","affiliation":[{"name":"Faculty of Engineering, Zanjan University, Zanjan 38791-45371, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2016,4,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1115\/1.3679915","article-title":"On Combined Free and Forced Convection in Channels","volume":"82","author":"Tao","year":"1960","journal-title":"J. Heat Transf."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"372","DOI":"10.1137\/0139031","article-title":"Combined Natural- and Forced-Convection between Parallel Vertical Walls","volume":"39","author":"Beckett","year":"1980","journal-title":"SIAM J. Appl. Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"485","DOI":"10.1115\/1.3246958","article-title":"Theory of Fully Developed, Combined Convection Including Flow Reversal","volume":"108","author":"Aung","year":"1986","journal-title":"J. Heat Transf."},{"key":"ref_4","first-page":"249","article-title":"Analysis of Fully Developed Opposing Mixed Convection between Inclined Parallel Plates","volume":"23","author":"Lavine","year":"1988","journal-title":"Heat Mass Transf."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"375","DOI":"10.2514\/3.190","article-title":"Flow Reversal and Heat Transfer of Fully Developed Mixed Convection in Vertical Channels","volume":"4","author":"Cheng","year":"1990","journal-title":"J. Thermophys. Heat Transf."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"507","DOI":"10.1115\/1.2910593","article-title":"Analysis of Laminar Fully Developed Mixed Convection in a Vertical Channel with Opposing Buoyancy","volume":"113","author":"Hamadah","year":"1991","journal-title":"J. Heat Transf."},{"key":"ref_7","first-page":"97","article-title":"Radiation effects on mixed convection flow and viscous heating in a vertical channel partially filled with a porous medium","volume":"14","author":"Chauhan","year":"2011","journal-title":"Tamkang J. Sci. Eng."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"843","DOI":"10.1615\/JPorMedia.v18.i9.20","article-title":"Effects of Micro and Macro Scale Viscous Dissipations with Heat Generation and Local Thermal Non-Equilibrium on Thermal Developing Forced Convection in Saturated Porous Media","volume":"18","year":"2015","journal-title":"J. Porous Media"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"65","DOI":"10.2514\/1.9124","article-title":"Investigation of liquid flow in microchannels","volume":"18","author":"Liu","year":"2004","journal-title":"AIAA J. Thermo. Phys. Heat Transf."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1023","DOI":"10.1007\/s00231-011-0951-0","article-title":"Experimental study on convective heat transfer coefficient around a vertical hexagonal rod bundle","volume":"48","author":"Makhmalbaf","year":"2012","journal-title":"Heat Mass Transf."},{"key":"ref_11","unstructured":"Makhmalbaf, M., Liu, T., and Merati, P. (2015, January 22\u201324). Experimental Simulation of Buoyancy-Driven Vortical Flow in Jupiter Great Red Spot. Proceedings of the 68th Annual Meeting of the APS Division of Fluid Dynamics, Boston, MA, USA."},{"key":"ref_12","first-page":"356","article-title":"Biofidelity Corridors for Whole-Body Pedestrian Impact with a Generic Buck","volume":"49","author":"Forman","year":"2015","journal-title":"IRCOBI Conf."},{"key":"ref_13","first-page":"1","article-title":"Effect of fuel inject angle on non-premixed combustion of air\/methane mixturesin in vertical cylinder","volume":"1","year":"2015","journal-title":"Int. J. Multidiscip. Res. Dev."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"100606","DOI":"10.1063\/1.1896985","article-title":"Characterization of surface roughness effects on pressure drop in microchannels","volume":"10","author":"Kandlikar","year":"2005","journal-title":"Phys. Fluids"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"132","DOI":"10.1016\/j.euromechflu.2015.08.013","article-title":"Mixed convection flow of an electrically conducting fluid in a vertical channel using Robin boundary conditions with heat source\/sink","volume":"55","author":"Umavathia","year":"2016","journal-title":"Eur. J. Mech. B\/Fluids"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1016\/j.rinp.2014.12.002","article-title":"Numerical Solutions of Williamson Fluid with Pressure Dependent Viscosity","volume":"5","author":"Zehra","year":"2015","journal-title":"Results Phys."},{"key":"ref_17","first-page":"27","article-title":"MHD mixed convection visco-elastic slip flow through a porous medium in a vertical porous channel with thermal radiation","volume":"35","author":"Singh","year":"2013","journal-title":"Kragujev. J. Sci."},{"key":"ref_18","first-page":"125","article-title":"Thermal radiation, joule heating, and viscous dissipation effects on mhd forced convection flow with uniform surface temperature","volume":"2","author":"Park","year":"2014","journal-title":"Open J. Fluid Dyn."},{"key":"ref_19","first-page":"281","article-title":"Analytical Study of Magnetohydrodynamic Propulsion Stability","volume":"3","year":"2014","journal-title":"J. Mar. Sci. Appl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2667","DOI":"10.1109\/TMAG.2009.2018954","article-title":"Flow Analysis of Non-Newtonian Blood in a Magnetohydrodynamic Pump","volume":"6","author":"Shahidian","year":"2009","journal-title":"IEEE Trans. Magn."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Takagi, D., and Huppert, H.E. (2008). Viscous gravity currents inside confining channels and fractures. Phys. Fluids, 20.","DOI":"10.1063\/1.2883991"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Longo, S., di Federico, V., and Chiapponi, L. (2015). Propagation of viscous gravity currents inside confining boundaries: The effects of fluid rheology and channel geometry. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci., 471.","DOI":"10.1098\/rspa.2015.0070"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"061002","DOI":"10.1115\/1.4029125","article-title":"Control of Magnetic Bearing with Material Saturation Nonlinearity","volume":"137","author":"Gerami","year":"2015","journal-title":"J. Dyn. Syst. Meas. Control"},{"key":"ref_24","unstructured":"Gerami, A., Allaire, P., and Fittro, R. (2014, January 11\u201314). Nonlinear Modeling and Control of a Magnetic Bearing with Material Saturation. Proceedings of the 14th International Conference on Magnetic Bearings, Linz, Austria."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Dousti, S., Dimond, T.W., Allaire, P.E., and Wood, H.E. (2013, January 15\u201321). Time Transient Analysis of Horizontal Rigid Rotor Supported with O-Ring Sealed Squeeze Film Damper. Proceedings of the ASME 2013 International Mechanical Engineering Congress and Exposition, San Diego, CA, USA.","DOI":"10.1115\/IMECE2013-63907"},{"key":"ref_26","first-page":"80","article-title":"A Numerical CFD Analysis on Supply Groove Effects in High Pressure, Open End Squeeze Film Dampers","volume":"1","author":"Dousti","year":"2016","journal-title":"Int. J. Eng. Innov. Res."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Asfer, M., and Panigrahi, P.K. (2015). Boundary Slip of Liquids. Encyclopedia of Microfluidics and Nanofluidics, Springer-Verlag US.","DOI":"10.1007\/978-1-4614-5491-5_124"},{"key":"ref_28","first-page":"51","article-title":"Governing Equations and Slip Models, Microflows and Nanoflows","volume":"29","author":"Karniadakis","year":"2005","journal-title":"Fundam. Simul."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1063\/1.3006031","article-title":"Molecular effects on boundary condition in micro\/nanoliquid flows","volume":"20","author":"Ulmanella","year":"2008","journal-title":"Phys. Fluids"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"685","DOI":"10.1039\/b616490k","article-title":"Flow Boundary Conditions from Nano- to Micro-Scales","volume":"3","author":"Bocquet","year":"2007","journal-title":"Soft Matter"},{"key":"ref_31","unstructured":"Karniadakis, G., Beskok, A., and Aluru, N. (2005). Microflows and Nanoflows: Fundamentals and Simulation, Springer."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1016\/j.jksus.2014.12.002","article-title":"Fully developed MHD natural convection flow in a vertical annular microchannel: An exact solution","volume":"27","author":"Jha","year":"2015","journal-title":"J. King Saud Univ. Sci."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1108","DOI":"10.1021\/ie50239a035","article-title":"The Flow of Pseudoplastic Materials","volume":"21","author":"Williamson","year":"1929","journal-title":"Ind. Eng. Chem."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"515","DOI":"10.1007\/s10652-014-9369-9","article-title":"Non-Newtonian Power-Law Gravity Currents Propagating in Confining Boundaries","volume":"15","author":"Longo","year":"2015","journal-title":"Environ. Fluid Mech."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1016\/j.jvolgeores.2010.06.011","article-title":"Initial advance of long lava flows in open channels","volume":"195","author":"Takagi","year":"2010","journal-title":"J. Volcanol. Geotherm. Res."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1017\/S0022112003003951","article-title":"Dipole solutions for viscous gravity currents: Theory and experiments","volume":"483","author":"King","year":"2003","journal-title":"J. Fluid Mech."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"534","DOI":"10.1017\/jfm.2015.405","article-title":"A dipole solution for power-law gravity currents in porous formations","volume":"778","author":"Longo","year":"2015","journal-title":"J. Fluid Mech."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"558","DOI":"10.1017\/jfm.2012.630","article-title":"Fluid drainage from the edge of a porous reservoir","volume":"718","author":"Zheng","year":"2013","journal-title":"J. Fluid Mech."},{"key":"ref_39","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":"15","author":"Bejan","year":"1982","journal-title":"Adv. Heat Transf."},{"key":"ref_40","unstructured":"Bejan, A. (1996). Entropy Generation Minimization, CRC."}],"updated-by":[{"DOI":"10.3390\/e19110593","type":"correction","label":"Correction","source":"publisher","updated":{"date-parts":[[2016,4,18]],"date-time":"2016-04-18T00:00:00Z","timestamp":1460937600000}}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/18\/4\/147\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,8,3]],"date-time":"2025-08-03T23:09:09Z","timestamp":1754262549000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/18\/4\/147"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,4,18]]},"references-count":40,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2016,4]]}},"alternative-id":["e18040147"],"URL":"https:\/\/doi.org\/10.3390\/e18040147","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,4,18]]}}}