{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,19]],"date-time":"2026-01-19T17:48:33Z","timestamp":1768844913624,"version":"3.49.0"},"reference-count":39,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2022,3,25]],"date-time":"2022-03-25T00:00:00Z","timestamp":1648166400000},"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 this recent work, the continuous dependence of double diffusive convection was studied theoretically in a porous medium of the Forchheimer model along with a variable viscosity. The analysis depicts that the density of saturating fluid under consideration shows a linear relationship with its concentration and a cubic dependence on the temperature. In this model, the equations for convection fluid motion were examined when viscosity changed with temperature linearly. This problem allowed the possibility of resonance between internal layers in thermal convection. Furthermore, we investigated the continuous dependence of this solution based on the changes in viscosity. Throughout the paper, we found an \u201ca priori estimate\u201d with coefficients that relied only on initial values, boundary data, and the geometry of the problem that demonstrated the continuous dependence of the solution on changes in the viscosity, which also helped us to state the relationship between the continuous dependence of the solution and the changes in viscosity. Moreover, we deduced a convergence result based on the Forchheimer model at the stage when the variable viscosity trends toward a constant value by assuming a couple of solutions to the boundary-initial-value problems and defining a difference solution of variables that satisfy a given boundary-initial-value problem.<\/jats:p>","DOI":"10.3390\/sym14040682","type":"journal-article","created":{"date-parts":[[2022,3,27]],"date-time":"2022-03-27T21:31:25Z","timestamp":1648416685000},"page":"682","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":20,"title":["A Study of Continuous Dependence and Symmetric Properties of Double Diffusive Convection: Forchheimer Model"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2959-4212","authenticated-orcid":false,"given":"Ali","family":"Ali","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Education for Pure Sciences, University of Basrah, Basrah 61001, Iraq"},{"name":"Doctoral School of Mathematical and Computational Sciences, University of Debrecen, H-4002 Debrecen, Hungary"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0900-9303","authenticated-orcid":false,"given":"Ghazi","family":"Meften","sequence":"additional","affiliation":[{"name":"Basrah Education Directorate, Ministry of Education, Basrah 61001, Iraq"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7251-9608","authenticated-orcid":false,"given":"Omar","family":"Bazighifan","sequence":"additional","affiliation":[{"name":"Section of Mathematics, International Telematic University Uninettuno, Corso Vittorio Emanuele II, 39, 00186 Roma, Italy"},{"name":"Department of Mathematics, Faculty of Education, Seiyun University, Hadhramout 50512, Yemen"},{"name":"Department of Mathematics, Faculty of Science, Hadhramout University, Hadhramout 50512, Yemen"}]},{"given":"Mehak","family":"Iqbal","sequence":"additional","affiliation":[{"name":"Doctoral School of Mathematical and Computational Sciences, University of Debrecen, H-4002 Debrecen, Hungary"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7250-0392","authenticated-orcid":false,"given":"Sergio","family":"Elaskar","sequence":"additional","affiliation":[{"name":"Departamento de Aeron\u00e1utica, FCEFyN, Instituto de Estudios Avanzados en Ingenier\u00eda y Tecnolog\u00eda (IDIT), Universidad Nacional de C\u00f3rdoba and CONICET, C\u00f3rdoba 5000, Argentina"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0387-921X","authenticated-orcid":false,"given":"Jan","family":"Awrejcewicz","sequence":"additional","affiliation":[{"name":"Department of Automation, Biomechanics and Mechatronics, Lodz University of Technology, 1\/15 Stefanowskiego Str., 90-924 Lodz, Poland"}]}],"member":"1968","published-online":{"date-parts":[[2022,3,25]]},"reference":[{"key":"ref_1","unstructured":"Straughan, B. 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