Existence of a weak solution to a regularized moving boundary fluid-structure interaction problem with poroelastic media
We study the existence of a weak solution to a regularized, moving boundary, fluid-structure interaction problem with multi-layered poroelastic media consisting of a reticular plate located at the interface between the free flow of an incompressible, viscous fluid modeled by the 2D Navier–Stokes equ...
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| Format: | Article |
| Language: | English |
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Académie des sciences
2023-05-01
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| Series: | Comptes Rendus. Mécanique |
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| Online Access: | https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.190/ |
| _version_ | 1797651424515981312 |
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| author | Kuan, Jeffrey Čanić, Sunčica Muha, Boris |
| author_facet | Kuan, Jeffrey Čanić, Sunčica Muha, Boris |
| author_sort | Kuan, Jeffrey |
| collection | DOAJ |
| description | We study the existence of a weak solution to a regularized, moving boundary, fluid-structure interaction problem with multi-layered poroelastic media consisting of a reticular plate located at the interface between the free flow of an incompressible, viscous fluid modeled by the 2D Navier–Stokes equations, and a poroelastic medium modeled by the 2D Biot equations. The existence result holds for both the elastic and viscoelastic Biot model. The free fluid flow and the poroelastic medium are coupled via the moving interface (the reticular plate) through the kinematic and dynamic coupling conditions. The reticular plate is “transparent” to fluid flow. The nonlinear coupling over the moving interface presents a major difficulty since both the fluid domain and the poroelastic medium domain are functions of time, and the finite energy spaces do not provide sufficient regularity for the corresponding weak formulation to be well-defined. This is why in this manuscript we consider a regularized problem by employing convolution with a smooth kernel only where needed. The resulting problem is still very challenging due to the nonlinear coupling and the motion of the fluid and Biot domains. We provide a constructive existence proof for this regularized fluid-poroelastic structure interaction problem. This regularized problem is consistent with the original, nonregularized problem in the sense that the weak solutions constructed here, converge, as the regularization parameter tends to zero, to a classical solution of the original, nonregularized problem when such a classical solution exists, assuming viscoelasticity in the Biot poroelastic matrix [1]. Furthermore, the existence result presented in this manuscript, is a crucial stepping stone for the singular limit problem in which the thickness of the reticular plate tends to zero. Namely, in [1] we will show that, in the case of a Biot poroviscoelastic matrix, the moving boundary problem obtained in the singular limit as the thickness of the plate tends to zero, has a weak solution. |
| first_indexed | 2024-03-11T16:15:41Z |
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| id | doaj.art-e56bff7b79db4da580c815ffd9ebd8f8 |
| institution | Directory Open Access Journal |
| issn | 1873-7234 |
| language | English |
| last_indexed | 2024-03-11T16:15:41Z |
| publishDate | 2023-05-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mécanique |
| spelling | doaj.art-e56bff7b79db4da580c815ffd9ebd8f82023-10-24T14:21:34ZengAcadémie des sciencesComptes Rendus. Mécanique1873-72342023-05-0113010.5802/crmeca.19010.5802/crmeca.190Existence of a weak solution to a regularized moving boundary fluid-structure interaction problem with poroelastic mediaKuan, Jeffrey0Čanić, Sunčica1Muha, Boris2Department of Mathematics, University of California, Berkeley, USADepartment of Mathematics, University of California, Berkeley, USADepartment of Mathematics, University of Zagreb, CroatiaWe study the existence of a weak solution to a regularized, moving boundary, fluid-structure interaction problem with multi-layered poroelastic media consisting of a reticular plate located at the interface between the free flow of an incompressible, viscous fluid modeled by the 2D Navier–Stokes equations, and a poroelastic medium modeled by the 2D Biot equations. The existence result holds for both the elastic and viscoelastic Biot model. The free fluid flow and the poroelastic medium are coupled via the moving interface (the reticular plate) through the kinematic and dynamic coupling conditions. The reticular plate is “transparent” to fluid flow. The nonlinear coupling over the moving interface presents a major difficulty since both the fluid domain and the poroelastic medium domain are functions of time, and the finite energy spaces do not provide sufficient regularity for the corresponding weak formulation to be well-defined. This is why in this manuscript we consider a regularized problem by employing convolution with a smooth kernel only where needed. The resulting problem is still very challenging due to the nonlinear coupling and the motion of the fluid and Biot domains. We provide a constructive existence proof for this regularized fluid-poroelastic structure interaction problem. This regularized problem is consistent with the original, nonregularized problem in the sense that the weak solutions constructed here, converge, as the regularization parameter tends to zero, to a classical solution of the original, nonregularized problem when such a classical solution exists, assuming viscoelasticity in the Biot poroelastic matrix [1]. Furthermore, the existence result presented in this manuscript, is a crucial stepping stone for the singular limit problem in which the thickness of the reticular plate tends to zero. Namely, in [1] we will show that, in the case of a Biot poroviscoelastic matrix, the moving boundary problem obtained in the singular limit as the thickness of the plate tends to zero, has a weak solution.https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.190/moving boundary problemfluid-poroelastic structure interactionwell-posednessnonlinear couplingBiot equationsNavier–Stokes equations |
| spellingShingle | Kuan, Jeffrey Čanić, Sunčica Muha, Boris Existence of a weak solution to a regularized moving boundary fluid-structure interaction problem with poroelastic media Comptes Rendus. Mécanique moving boundary problem fluid-poroelastic structure interaction well-posedness nonlinear coupling Biot equations Navier–Stokes equations |
| title | Existence of a weak solution to a regularized moving boundary fluid-structure interaction problem with poroelastic media |
| title_full | Existence of a weak solution to a regularized moving boundary fluid-structure interaction problem with poroelastic media |
| title_fullStr | Existence of a weak solution to a regularized moving boundary fluid-structure interaction problem with poroelastic media |
| title_full_unstemmed | Existence of a weak solution to a regularized moving boundary fluid-structure interaction problem with poroelastic media |
| title_short | Existence of a weak solution to a regularized moving boundary fluid-structure interaction problem with poroelastic media |
| title_sort | existence of a weak solution to a regularized moving boundary fluid structure interaction problem with poroelastic media |
| topic | moving boundary problem fluid-poroelastic structure interaction well-posedness nonlinear coupling Biot equations Navier–Stokes equations |
| url | https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.190/ |
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