{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,24]],"date-time":"2026-01-24T10:25:53Z","timestamp":1769250353307,"version":"3.49.0"},"reference-count":30,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2023,9,4]],"date-time":"2023-09-04T00:00:00Z","timestamp":1693785600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"King Saud University, Riyadh, Saudi Arabia","award":["RSPD2023R909"],"award-info":[{"award-number":["RSPD2023R909"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The two-field thermal conditions of local thermal nonequilibrium (LTNE) were used to investigate linear stability of thermal convection in a liquid-saturated, porous layer via the extended Brinkman\u2013Darcy model for different non-uniform basic thermal gradients. The critical values were numerically computed by the Galerkin method for rigid isothermal boundaries. The impact of LTNE and different forms of non-uniform basic temperature gradients on the onset of porous convection was examined. The porosity modified conductivity ratio has no influence on system stability at a small inter-phase heat transport coefficient limit. However, for higher values of the inter-phase heat transport coefficient, an increase in the porosity modified conductivity ratio hastens the onset of convection. An increase in the Darcy number delays the convective motions. The results for different basic temperature profiles are symmetric qualitatively. In addition, the possibility of control of convection by a basic temperature profile was studied in detail.<\/jats:p>","DOI":"10.3390\/sym15091695","type":"journal-article","created":{"date-parts":[[2023,9,4]],"date-time":"2023-09-04T10:14:11Z","timestamp":1693822451000},"page":"1695","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["The Onset of Darcy\u2013Brinkman Convection in a Porous Layer with Mutual Impact of Thermal Non-Equilibrium and Non-Uniform Temperature Gradients"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9645-7157","authenticated-orcid":false,"given":"Suma","family":"Shyabal","sequence":"first","affiliation":[{"name":"Department of Mathematics, School of Applied Sciences, REVA University, Bengaluru 560064, India"}]},{"given":"B. N.","family":"Hanumagowda","sequence":"additional","affiliation":[{"name":"Department of Mathematics, School of Applied Sciences, REVA University, Bengaluru 560064, India"}]},{"given":"M.","family":"Ravisha","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Dr. G. Shankar Government Women\u2019s First Grade College, Post Graduate Study Centre, Ajjarakadu, Udupi 576101, India"}]},{"given":"A. L.","family":"Mamatha","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Government First Grade College and Centre for PG Studies, Thenkanidiyoor 576106, India"}]},{"given":"N.","family":"Shivaraju","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Government First Grade College for Women, Davanagere 577004, India"}]},{"given":"Soumya","family":"D. O.","sequence":"additional","affiliation":[{"name":"Department of Mathematics, PES Institute of Technology & Management, Shivamogga 577201, India"}]},{"given":"Shalan","family":"Alkarni","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Sciences, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia"}]},{"given":"Nehad Ali","family":"Shah","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, Sejong University, Seoul 05006, Republic of Korea"}]}],"member":"1968","published-online":{"date-parts":[[2023,9,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Nield, D.A., and Bejan, A. (2017). Convection in Porous Media, Springer. [5th ed.].","DOI":"10.1007\/978-3-319-49562-0"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"2221","DOI":"10.1016\/S0017-9310(01)00331-3","article-title":"Onset of Darcy\u2013B\u00e9nard convection using a thermal nonequilibrium model","volume":"45","author":"Banu","year":"2002","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1587","DOI":"10.1016\/j.ijthermalsci.2008.01.002","article-title":"The onset of a Darcy\u2013Brinkman convection using a thermal nonequilibrium model","volume":"47","author":"Postelnicu","year":"2008","journal-title":"Part II Int. J. Thermal Sci."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"961","DOI":"10.1002\/er.928","article-title":"The onset of Darcy-Brinkman convection in a porous medium using a thermal non-equilibrium model. Part 1: Stress-free boundaries","volume":"27","author":"Postelnicu","year":"2003","journal-title":"Int. J. Energy Res."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1155","DOI":"10.1016\/j.ijheatmasstransfer.2004.09.027","article-title":"The onset of Lapwood\u2013Brinkman convection using a thermal non-equilibrium model","volume":"48","author":"Malashetty","year":"2005","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_6","first-page":"409","article-title":"Global non-linear stability in porous convection with a thermal non-equilibrium model","volume":"462","author":"Straughan","year":"2006","journal-title":"Proc. R. Soc. London"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1615\/SpecialTopicsRevPorousMedia.2023045844","article-title":"Impact of diffusion-thermo and thermal-diffusion on the flow of Walters-B fluid over a sheet saturated in a porous medium using local thermal non-equilibrium condition","volume":"14","author":"Kumar","year":"2023","journal-title":"Spec. Top. Amp. Rev. Porous Media Int. J."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"613","DOI":"10.1615\/JPorMedia.v13.i7.30","article-title":"Effects of variable viscosity and density maximum on the onset of Darcy\u2013B\u00e9nard convection using a thermal non-equilibrium model","volume":"13","author":"Shivakumara","year":"2010","journal-title":"J. Porous Med."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1610","DOI":"10.1007\/s42452-019-1651-5","article-title":"Effects of Hall current and permeability on the stability of peristaltic flow","volume":"1","author":"Mekheimer","year":"2019","journal-title":"SN Appl. Sci."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2475","DOI":"10.1007\/s12648-021-02216-9","article-title":"Assessment of the local thermal non-equilibrium condition for nanofluid flow through porous media: A comparative analysis","volume":"96","author":"Prasannakumara","year":"2022","journal-title":"Indian J. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"102654","DOI":"10.1016\/j.csite.2022.102654","article-title":"Hydrothermal and mass aspects of MHD non-Darcian convective flows of radiating thixotropic nanofluids nearby a horizontal stretchable surface: Passive control strategy","volume":"42","author":"Rasool","year":"2023","journal-title":"Case Stud. Therm. Eng."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"174","DOI":"10.19026\/rjaset.15.5849","article-title":"Magnetogravitodynamic Stability of Three Dimensional Streaming Velocities of Fluid Cylinder under the Effect of Capillary Force","volume":"15","author":"Hasan","year":"2018","journal-title":"Res. J. Appl. Sci. Eng. Technol."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"101887","DOI":"10.1016\/j.asej.2022.101887","article-title":"Analytical and numerical investigation of Darcy-Forchheimer flow of a nonlinear-radiative non-Newtonian fluid over a Riga plate with entropy optimization","volume":"14","author":"Eswaramoorthi","year":"2023","journal-title":"Ain. Shams Eng. J."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"6763","DOI":"10.1002\/mma.5190","article-title":"Uniqueness and decay in local thermal non-equilibrium double porosity thermos elasticity","volume":"41","author":"Franchi","year":"2018","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"768","DOI":"10.1088\/0253-6102\/68\/6\/768","article-title":"General solutions for hydromagnetic free convection flow over an infinite plate with Newtonian heating, mass diffusion and chemical reaction","volume":"68","author":"Fetecau","year":"2017","journal-title":"Commun. Theor. Phys."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"779","DOI":"10.1108\/MMMS-11-2018-0189","article-title":"Cattaneo\u2013LTNE porous ferroconvection","volume":"15","author":"Ravisha","year":"2019","journal-title":"Multidiscip. Model. Mater. Struct."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"3613","DOI":"10.1002\/htj.21791","article-title":"Double diffusive LTNE porous convection with Cattaneo effects in the solid","volume":"49","author":"Hema","year":"2020","journal-title":"Heat Transf."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"e202100360","DOI":"10.1002\/zamm.202100360","article-title":"Analysis of heat transfer using Local thermal non-equilibrium conditions for a non-Newtonian fluid flow containing Ti6Al4V and AA7075 nanoparticles in a porous media","volume":"103","author":"Alsulami","year":"2023","journal-title":"ZAMM-J. Appl. Math. Mech. Z. F\u00fcr Angew. Math. Mech."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"443","DOI":"10.1016\/j.powtec.2020.04.005","article-title":"Numerical study on natural convection of Ag\u2013MgO hybrid\/water nanofluid inside a porous enclosure: A local thermal non-equilibrium model","volume":"367","author":"Mehryan","year":"2020","journal-title":"Powder Technol."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Sarada, K., Gowda, R.J.P., Sarris, I.E., Kumar, R.N., and Prasannakumara, B.C. (2021). Effect of Magnetohydrodynamics on Heat Transfer Behaviour of a Non-Newtonian Fluid Flow over a Stretching Sheet under Local Thermal Non-Equilibrium Condition. Fluids, 6.","DOI":"10.3390\/fluids6080264"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1140\/epjp\/s13360-023-03717-8","article-title":"Onset of LTNE anisotropic porous convection: Effect of asymmetric temperature boundary conditions","volume":"138","author":"Shyabal","year":"2023","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"103889","DOI":"10.1016\/j.ijnonlinmec.2021.103889","article-title":"Onset of convection in LTNE Darcy-Brinkman anisotropic porous layer: Cattaneo effect in the solid","volume":"139","author":"Capone","year":"2022","journal-title":"Int. J. Nonlinear Mech."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"441","DOI":"10.1017\/S0022112075002662","article-title":"The onset of transient convective instability","volume":"71","author":"Nield","year":"1975","journal-title":"J. Fluid Mech."},{"key":"ref_24","first-page":"154","article-title":"Effects of nonuniform thermal gradient and adiabatic boundaries on convection in porous media","volume":"102","author":"Rudraiah","year":"1980","journal-title":"J. Hcut. Trmfcr."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"4199","DOI":"10.1016\/0017-9310(93)90082-H","article-title":"The Brinkman model for natural convection in a porous layer: Effects of nonuniform thermal gradient","volume":"36","author":"Vasseur","year":"1993","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"781","DOI":"10.1016\/S0017-9310(02)00352-6","article-title":"Influence of anisotropy on convection in porous media with nonuniform thermal gradient","volume":"46","author":"Degan","year":"2003","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1016\/j.icheatmasstransfer.2008.11.009","article-title":"On effect of non-uniform basic temperature gradient on B\u00e9nard\u2013Marangoni convection in micropolar fluid","volume":"36","author":"Idris","year":"2009","journal-title":"Int. Commun. Heat Mass Transf."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"906","DOI":"10.1016\/j.icheatmasstransfer.2011.04.023","article-title":"Effects of thermal nonequilibrium and non-uniform temperature gradients on the onset of convection in a heterogeneous porous medium","volume":"38","author":"Shivakumara","year":"2011","journal-title":"Int. Commun. Heat Mass Transf."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"659","DOI":"10.1615\/JPorMedia.v14.i8.10","article-title":"Effect of non-uniform temperature gradients on thermo-gravitational convection in a porous layer using a thermal non-equilibrium model","volume":"14","author":"Lee","year":"2011","journal-title":"J. Porous Med."},{"key":"ref_30","first-page":"43","article-title":"Effect of Nonlinear Temperature Profile on Thermal Convection in a Binary Fluid Saturated an Anisotropic Porous Medium","volume":"56","author":"Hamid","year":"2019","journal-title":"J. Adv. Res. Fluid Mech. Ther. Sci."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/9\/1695\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:46:15Z","timestamp":1760129175000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/9\/1695"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,9,4]]},"references-count":30,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2023,9]]}},"alternative-id":["sym15091695"],"URL":"https:\/\/doi.org\/10.3390\/sym15091695","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,9,4]]}}}