Well-posedness of generalized Stokes-Brinkman equations modeling moving solid phases

Nisachon Kumankat; Kanognudge Wuttanachamsri; Nisachon Kumankat; Kanognudge Wuttanachamsri

doi:10.3934/era.2023085

Electronic Research Archive

2023, Volume 31, Issue 3: 1641-1661. doi: 10.3934/era.2023085

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Research article

Well-posedness of generalized Stokes-Brinkman equations modeling moving solid phases

Department of Mathematics, School of Science, King Mongkut's Institute of Technology Ladkrabang, Bangkok 10520, Thailand

Received: 04 November 2022 Revised: 04 January 2023 Accepted: 12 January 2023 Published: 31 January 2023

Fluid flow through a free-fluid region and the adjacent porous medium has been studied in various problems, such as water flow in rice fields. For the problem with self-propelled solid phases, we provide a generalized Stokes equation for the free-fluid domain and the Brinkman equation in a macroscopic scale due to the movement of self-propelled solid phases rather than a single solid in the porous medium. The model is derived with the assumption that the porosity is not a constant. The porosity in the mathematical model varies depending on the propagation of the solid phases. These two models can be matched at the free-fluid/porous-medium interface and are developed for real world problems. We show the proof of the well-posedness of the discretized form of the weak formulation obtained from applying a mixed finite element scheme to the generalized Stokes-Brinkman equations. The proofs of the continuity and coercive property of the linear and bilinear functionals in the discretized equation are illustrated. We present the existence and uniqueness of the generalized Stokes-Brinkman equations for the numerical problem in two dimensions. The system of equations can be applied to fluid flow propelled by moving solid phases, such as mucus flow in the trachea.
- generalized Stokes-Brinkman equations,
- finite element approach,
- well-posedness,
- varied porosity,
- permeability tensor
Citation: Nisachon Kumankat, Kanognudge Wuttanachamsri. Well-posedness of generalized Stokes-Brinkman equations modeling moving solid phases[J]. Electronic Research Archive, 2023, 31(3): 1641-1661. doi: 10.3934/era.2023085

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Abstract

Fluid flow through a free-fluid region and the adjacent porous medium has been studied in various problems, such as water flow in rice fields. For the problem with self-propelled solid phases, we provide a generalized Stokes equation for the free-fluid domain and the Brinkman equation in a macroscopic scale due to the movement of self-propelled solid phases rather than a single solid in the porous medium. The model is derived with the assumption that the porosity is not a constant. The porosity in the mathematical model varies depending on the propagation of the solid phases. These two models can be matched at the free-fluid/porous-medium interface and are developed for real world problems. We show the proof of the well-posedness of the discretized form of the weak formulation obtained from applying a mixed finite element scheme to the generalized Stokes-Brinkman equations. The proofs of the continuity and coercive property of the linear and bilinear functionals in the discretized equation are illustrated. We present the existence and uniqueness of the generalized Stokes-Brinkman equations for the numerical problem in two dimensions. The system of equations can be applied to fluid flow propelled by moving solid phases, such as mucus flow in the trachea.

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