Existence and uniqueness of solutions of unsteady Darcy-Brinkman problem for modelling miscible reactive flows in porous media

[2506.22225] Existence and uniqueness of solutions of unsteady Darcy-Brinkman problem for modelling miscible reactive flows in porous media Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Analysis of PDEs arXiv:2506.22225 (math) [Submitted on 27 Jun 2025 ( v1 ), last revised 31 Mar 2026 (this version, v2)] Title: Existence and uniqueness of solutions of unsteady Darcy-Brinkman problem for modelling miscible reactive flows in porous media Authors: Pankaj Roy , Satyajit Pramanik View a PDF of the paper titled Existence and uniqueness of solutions of unsteady Darcy-Brinkman problem for modelling miscible reactive flows in porous media, by Pankaj Roy and Satyajit Pramanik View PDF HTML (experimental) Abstract: In this work, we investigate a model describing flow through porous media with permeability heterogeneity, combining an advection-reaction-diffusion equation for solute concentration with an unsteady Darcy-Brinkman equation with Korteweg stresses in the presence of external body forces for the flow field. Such models are appropriate in describing flows in fractured karst reservoirs, mineral wool, industrial foam, coastal mud, etc. These equations are coupled with Neumann boundary conditions for the solute concentration and no-flow conditions for the fluid velocity. For a broad class of initial data, we proved the existence of weak solutions. In the presence of a second-order nonlinear reaction, we show that the long-time behaviour of the solution depends on the initial concentration \(C_0\). More precisely, the solution exists for all time if \(0\leq C_0\leq 1\), and blows up at finite time if $C_0>1$. Furthermore, the uniqueness of the solution is proved for a two-dimensional domain. Finally, numerical simulations based on the finite element method have been presented that illustrate non-negativity of the concentration, long-time decay, and finite-time blow-up in agreement with theoretical estimates. Comments: 2 figures Subjects: Analysis of PDEs (math.AP) MSC classes: 35A01, 35A02, 35A15, 76D03 Cite as: arXiv:2506.22225 [math.AP] (or arXiv:2506.22225v2 [math.AP] for this version) https://doi.org/10.48550/arXiv.2506.22225 Focus to learn more arXiv-issued DOI via DataCite Related DOI : https://doi.org/10.1016/j.cnsns.2026.109997 Focus to learn more DOI(s) linking to related resources Submission history From: Satyajit Pramanik Dr. [ view email ] [v1] Fri, 27 Jun 2025 13:41:12 UTC (34 KB) [v2] Tue, 31 Mar 2026 08:45:07 UTC (1,085 KB) Full-text links: Access Paper: View a PDF of the paper titled Existence and uniqueness of solutions of unsteady Darcy-Brinkman problem for modelling miscible reactive flows in porous media, by Pankaj Roy and Satyajit Pramanik View PDF HTML (experimental) TeX Source view license Current browse context: math.AP < prev | next > new | recent | 2025-06 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

Related documents

• Newtonian fluid invasion into a Yield-Stress Fluid-Saturated Medium#1070286
• Robust stability and preconditioning of Darcy-Forchheimer equations#987492
• Reactive transport codes for subsurface environmental simulation#436244
• Physics-Informed DeepONet Coupled with FEM for Convective Transport in Porous Media with Sharp Gaussian Sources#1070699
• Lattice Boltzmann Simulation of Non-Darcy Flow#640996