{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,14]],"date-time":"2025-10-14T00:41:25Z","timestamp":1760402485359,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2022,1,7]],"date-time":"2022-01-07T00:00:00Z","timestamp":1641513600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Key projects of universities in Guangdong Province (NATURAL SCIENCE)","award":["2019KZDXM042"],"award-info":[{"award-number":["2019KZDXM042"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we consider the Brinkman equations pipe flow, which includes the salinity and the temperature. Assuming that the fluid satisfies nonlinear boundary conditions at the finite end of the cylinder, using the symmetry of differential inequalities and the energy analysis methods, we establish the exponential decay estimates for homogeneous Brinkman equations. That is to prove that the solutions of the equation decay exponentially with the distance from the finite end of the cylinder. To make the estimate of decay explicit, the bound for the total energy is also derived.<\/jats:p>","DOI":"10.3390\/sym14010098","type":"journal-article","created":{"date-parts":[[2022,1,9]],"date-time":"2022-01-09T23:35:09Z","timestamp":1641771309000},"page":"98","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Spatial Decay Bounds for the Brinkman Fluid Equations in Double-Diffusive Convection"],"prefix":"10.3390","volume":"14","author":[{"given":"Xuejiao","family":"Chen","sequence":"first","affiliation":[{"name":"School of Data Science, Guangzhou Huashang College, Guangzhou 511300, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9314-4104","authenticated-orcid":false,"given":"Yuanfei","family":"Li","sequence":"additional","affiliation":[{"name":"School of Data Science, Guangzhou Huashang College, Guangzhou 511300, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dandan","family":"Li","sequence":"additional","affiliation":[{"name":"School of Data Science, Guangzhou Huashang College, Guangzhou 511300, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,1,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Nield, D.A., and Bejan, A. 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