{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,4]],"date-time":"2026-06-04T21:44:47Z","timestamp":1780609487167,"version":"3.54.1"},"reference-count":50,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2024,8,29]],"date-time":"2024-08-29T00:00:00Z","timestamp":1724889600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Brazilian Council for Scientific and Technological Development (CNPq)","award":["304962\/2022-8"],"award-info":[{"award-number":["304962\/2022-8"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper studies constrained Newtonian fluid flows through porous media, accounting for the drag effect on the fluid, modeled using a Mixture Theory perspective and a constitutive relation for the pressure\u2014namely, a continuous and differentiable function of the saturation that ensures always preserving the problem hyperbolicity. The pressure equation also permits an ultra-small porous matrix supersaturation (that is controlled) and the transition from unsaturated to saturated flow (and vice versa). The mathematical model gives rise to a nonlinear, non-homogeneous hyperbolic system. Its numerical simulation combines Glimm\u2019s method with an operator-splitting strategy to account for the Darcy and Forchheimer terms that cause the system\u2019s non-homogeneity. Despite the Glimm method\u2019s proven convergence, it is not adequate to approximate non-homogeneous hyperbolic systems unless combined with an operator-splitting technique. Although other approaches have already addressed this problem, the novelty is combining Glimm\u2019s method with operator-splitting to account for linear and nonlinear drag effects. Glimm\u2019s scheme marches in time using a formerly selected number of associated Riemann problems. The constitutive relation for the pressure\u2014an increasing function of the saturation, with the first derivative also increasing, convex, and positive, enables us to obtain explicit expressions for the Riemann invariants. The results show the influence of the Darcy and Forchheimer drag terms on the flow.<\/jats:p>","DOI":"10.3390\/axioms13090587","type":"journal-article","created":{"date-parts":[[2024,8,29]],"date-time":"2024-08-29T08:01:47Z","timestamp":1724918507000},"page":"587","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Combining Glimm\u2019s Scheme and Operator Splitting for Simulating Constrained Flows in Porous Media"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6586-3670","authenticated-orcid":false,"given":"Maria Laura","family":"Martins-Costa","sequence":"first","affiliation":[{"name":"Laboratory of Theoretical and Applied Mechanics (LMTA), Mechanical Engineering Graduate Program (TEM-PGMEC), Universidade Federal Fluminense, Rua Passo da P\u00e1tria, 156, Niter\u00f3i 24210-240, RJ, Brazil"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2190-8141","authenticated-orcid":false,"given":"Felipe Bastos de","family":"Freitas Rachid","sequence":"additional","affiliation":[{"name":"Mechanical Engineering Graduate Program (TEM-PGMEC), Universidade Federal Fluminense, Rua Passo da P\u00e1tria, 156, Niter\u00f3i 24210-240, RJ, Brazil"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Rog\u00e9rio Pazetto S. da","family":"Gama","sequence":"additional","affiliation":[{"name":"Mechanical Engineering Graduate Program (FEN), Universidade do Estado do Rio de Janeiro, Rua S\u00e3o Francisco Xavier, 524, Rio de Janeiro 20550-013, RJ, Brazil"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Rog\u00e9rio M.","family":"Saldanha da Gama","sequence":"additional","affiliation":[{"name":"Mechanical Engineering Graduate Program (FEN), Universidade do Estado do Rio de Janeiro, Rua S\u00e3o Francisco Xavier, 524, Rio de Janeiro 20550-013, RJ, Brazil"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2024,8,29]]},"reference":[{"key":"ref_1","unstructured":"Bear, J. 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