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.
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@article{CRMECA_2023__351_S1_A17_0,
author = {Jeffrey Kuan and Sun\v{c}ica \v{C}ani\'c and Boris Muha},
title = {Existence of a weak solution to a regularized moving boundary fluid-structure interaction problem with poroelastic media},
journal = {Comptes Rendus. M\'ecanique},
publisher = {Acad\'emie des sciences, Paris},
year = {2023},
doi = {10.5802/crmeca.190},
language = {en},
note = {Online first},
}
TY - JOUR AU - Jeffrey Kuan AU - Sunčica Čanić AU - Boris Muha TI - Existence of a weak solution to a regularized moving boundary fluid-structure interaction problem with poroelastic media JO - Comptes Rendus. Mécanique PY - 2023 PB - Académie des sciences, Paris N1 - Online first DO - 10.5802/crmeca.190 LA - en ID - CRMECA_2023__351_S1_A17_0 ER -
%0 Journal Article %A Jeffrey Kuan %A Sunčica Čanić %A Boris Muha %T Existence of a weak solution to a regularized moving boundary fluid-structure interaction problem with poroelastic media %J Comptes Rendus. Mécanique %D 2023 %I Académie des sciences, Paris %Z Online first %R 10.5802/crmeca.190 %G en %F CRMECA_2023__351_S1_A17_0
Jeffrey Kuan; Sunčica Čanić; Boris Muha. Existence of a weak solution to a regularized moving boundary fluid-structure interaction problem with poroelastic media. Comptes Rendus. Mécanique, Online first (2023), pp. 1-30. doi : 10.5802/crmeca.190.
[1] Fluid-poroelastic structure interaction with moving boundary (2023) (In draft form. To be submitted)
[2] Homogenization of Reticulated Structures, Applied Mathematical Sciences, 136, Springer, 1998 | Zbl
[3] Existence of a weak solution to a nonlinear fluid-structure interaction problem modeling the flow of an incompressible, viscous fluid in a cylinder with deformable walls, Arch. Ration. Mech. Anal., Volume 207 (2013) no. 3, pp. 919-968 | DOI | MR | Zbl
[4] Fluid-structure interaction between an incompressible, viscous 3D fluid and an elastic shell with nonlinear Koiter membrane energy, Interfaces Free Bound., Volume 17 (2015) no. 4, pp. 465-495 | DOI | MR | Zbl
[5] Existence of a weak solution to a fluid-elastic structure interaction problem with the Navier slip boundary condition, J. Differ. Equations, Volume 260 (2016) no. 12, pp. 8550-8589 | DOI | MR | Zbl
[6] Existence of a solution to a fluid-multi-layered-structure interaction problem, J. Differ. Equations, Volume 256 (2014) no. 2, pp. 658-706 | DOI | MR | Zbl
[7] Analysis of a 3D nonlinear, moving boundary problem describing fluid-mesh-shell interaction, Trans. Am. Math. Soc., Volume 373 (2020) no. 9, pp. 6621-6681 | DOI | MR | Zbl
[8] A note on optimal regularity and regularizing effects of point mass coupling for a heat-wave system, J. Math. Anal. Appl., Volume 425 (2015) no. 2, pp. 1134-1147 | DOI | MR | Zbl
[9] A variational approach to hyperbolic evolutions and fluid-structure interactions (2020) (https://arxiv.org/abs/2008.04796)
[10] Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model, Fluids and waves. Recent trends in applied analysis (Contemporary Mathematics), Volume 440, American Mathematical Society, 2007, pp. 55-82 | DOI | MR | Zbl
[11] Smoothness of weak solutions to a nonlinear fluid-structure interaction model, Indiana Univ. Math. J., Volume 57 (2008) no. 3, pp. 1173-1207 | DOI | MR | Zbl
[12] Strong solutions for a fluid structure interaction system, Adv. Differ. Equ., Volume 15 (2010) no. 3-4, pp. 231-254 | MR | Zbl
[13] On the existence of strong solutions to a coupled fluid-structure evolution problem, J. Math. Fluid Mech., Volume 6 (2004) no. 1, pp. 21-52 | DOI | MR | Zbl
[14] Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate, J. Math. Fluid Mech., Volume 7 (2005) no. 3, pp. 368-404 | DOI | MR | Zbl
[15] Navier-Stokes equations interacting with a nonlinear elastic biofluid shell, SIAM J. Math. Anal., Volume 39 (2007) no. 3, pp. 742-800 | DOI | MR | Zbl
[16] The interaction of the 3D Navier-Stokes equations with a moving nonlinear Koiter elastic shell, SIAM J. Math. Anal., Volume 42 (2010) no. 3, pp. 1094-1155 | DOI | MR | Zbl
[17] Motion of an elastic solid inside an incompressible viscous fluid, Arch. Ration. Mech. Anal., Volume 176 (2005) no. 1, pp. 25-102 | DOI | MR | Zbl
[18] The interaction between quasilinear elastodynamics and the Navier-Stokes equations, Arch. Ration. Mech. Anal., Volume 179 (2006) no. 3, pp. 303-352 | DOI | MR | Zbl
[19] Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate, SIAM J. Math. Anal., Volume 40 (2008) no. 2, pp. 716-737 | DOI | MR | Zbl
[20] Existence of global strong solutions to a beam-fluid interaction system, Arch. Ration. Mech. Anal., Volume 220 (2016) no. 3, pp. 1283-1333 | DOI | MR | Zbl
[21] Mathematical and numerical analysis of some FSI problems, Fluid-structure interaction and biomedical applications (T. Bodnár; G. P. Galdi; Š. Nečasová, eds.) (Advances in Mathematical Fluid Mechanics), Birkhäuser, 2014, pp. 1-77 | DOI | Zbl
[22] On well-posedness for a free boundary fluid-structure model, J. Math. Phys., Volume 53 (2012) no. 11, 115624, 13 pages | DOI | MR | Zbl
[23] On well-posedness and small data global existence for an interface damped free boundary fluid-structure model, Nonlinearity, Volume 27 (2014) no. 3, pp. 467-499 | DOI | MR | Zbl
[24] Solutions to a fluid-structure interaction free boundary problem, Discrete Contin. Dyn. Syst., Volume 32 (2012) no. 4, pp. 1355-1389 | DOI | MR | Zbl
[25] Weak solutions for an incompressible Newtonian fluid interacting with a Koiter type shell, Arch. Ration. Mech. Anal., Volume 211 (2014) no. 1, pp. 205-255 | DOI | MR | Zbl
[26] Existence of strong solutions to a fluid-structure system, SIAM J. Math. Anal., Volume 43 (2011) no. 1, pp. 389-410 | DOI | MR | Zbl
[27] A nonlinear, 3D fluid-structure interaction problem driven by the time-dependent dynamic pressure data: a constructive existence proof, Commun. Inf. Syst., Volume 13 (2013) no. 3, pp. 357-397 | DOI | MR | Zbl
[28] Existence of a solution to a fluid-multi-layered-structure interaction problem, J. Differ. Equations, Volume 256 (2014) no. 2, pp. 658-706 | DOI | MR | Zbl
[29] A fluid-structure model coupling the Navier-Stokes equations and the Lamé system, J. Math. Pures Appl., Volume 102 (2014) no. 3, pp. 546-596 | DOI | Zbl
[30] Effective equations modeling the flow of a viscous incompressible fluid through a long elastic tube arising in the study of blood flow through small arteries, SIAM J. Appl. Dyn. Syst., Volume 2 (2003) no. 3, pp. 431-463 | DOI | MR | Zbl
[31] Effective equations describing the flow of a viscous incompressible fluid through a long elastic tube, C. R. Méc. Acad. Sci. Paris, Volume 300 (2002) no. 10, pp. 661-666 | Zbl
[32] Self-Consistent Effective Equations Modeling Blood Flow in Medium-to-Large Compliant Arteries, Multiscale Model. Simul., Volume 3 (2005) no. 3, pp. 559-596 | DOI | MR | Zbl
[33] A two-dimensional effective model describing fluid-structure interaction in blood flow: analysis, simulation and experimental validation, C. R. Méc. Acad. Sci. Paris, Volume 333 (2005) no. 12, pp. 867-883 | Zbl
[34] Effective Model of the Fluid Flow through Elastic Tube with Variable Radius, Grazer Math. Ber., Volume 348 (2005), pp. 91-112 | MR | Zbl
[35] Blood Flow in Compliant Arteries: An Effective Viscoelastic Reduced Model, Numerics and Experimental Validation, Ann. Biomed. Eng., Volume 34 (2006), pp. 575-592 | DOI
[36] Homogenization Closure for a Two-Dimensional Effective Model Describing Fluid-Structure Interaction in Blood Flow, Math everywhere. Deterministic and stochastic modelling in biomedicine, economics and industry, Springer, 2007, pp. 193-205 | Zbl
[37] Modeling viscoelastic behavior of arterial walls and their interaction with pulsatile blood flow, SIAM J. Appl. Math., Volume 67 (2006) no. 1, pp. 164-193 | DOI | MR | Zbl
[38] Fluid-Structure Interaction in a Pre-Stressed Tube with Thick Elastic Walls I: The Stationary Stokes Problem, Netw. Heterog. Media, Volume 2 (2007) no. 3, pp. 397-423 | DOI | MR | Zbl
[39] General theory of three-dimensional consolidation, J. Appl. Phys., Volume 12 (1941) no. 2, pp. 155-164 | DOI | Zbl
[40] Theory of elasticity and consolidation for a porous anisotropic solid, J. Appl. Phys., Volume 26 (1955) no. 2, pp. 182-185 | DOI | MR | Zbl
[41] Poroelastic media, Homogenization and porous media (Ulrich Hornung, ed.) (Interdisciplinary Applied Mathematics), Volume 6, Springer, 1997, pp. 63-182 | DOI | MR
[42] Mechanics and Physics of Porous Solids, John Wiley & Sons, 2011
[43] Theory of the dynamic Biot-Allard equations and their link to the quasi-static Biot system, J. Math. Phys., Volume 53 (2012) no. 12, 123702, 15 pages | DOI | MR | Zbl
[44] A lubrication fracture model in a poro-elastic medium, Math. Models Methods Appl. Sci., Volume 25 (2015) no. 4, pp. 587-645 | DOI | MR | Zbl
[45] A multiscale Darcy-Brinkman model for fluid flow in fractured porous media, Numer. Math., Volume 117 (2011) no. 4, pp. 717-752 | DOI | MR | Zbl
[46] A mathematical model of intestinal oedema formation, Math. Med. Biol., Volume 31 (2014) no. 1, pp. 1-15 | DOI | MR | Zbl
[47] Theoretical aspects of wave propagation for Biot’s consolidation problem, 8th Zaragoza-Pau conference on applied mathematics and statistics, Jaca, Spain, September 15–17, 2003 (Monografías del Seminario Matemático “García de Galdeano”), Volume 31, Universidad de Zaragoza, Zaragoza, 2004, pp. 449-458
[48] On nonlinear Biot’s consolidation models, Nonlinear Anal., Theory Methods Appl., Volume 63 (2005) no. 5-7, p. e985-e995 | DOI | Zbl
[49] A Galerkin method for Biot consolidation model, Math. Mech. Solids, Volume 15 (2010) no. 1, pp. 42-56 | DOI | MR | Zbl
[50] Diffusion in poro-elastic media, J. Math. Anal. Appl., Volume 251 (2000) no. 1, pp. 310-340 | DOI | MR | Zbl
[51] Partially saturated flow in a poroelastic medium, Discrete Contin. Dyn. Syst., Ser. B, Volume 1 (2001) no. 4, pp. 403-420 | DOI | MR | Zbl
[52] The existence and uniqueness theorem in Biot’s consolidation theory, Apl. Mat., Volume 29 (1984) no. 3, pp. 194-211 | DOI | MR | Zbl
[53] Analysis of nonlinear poro-elastic and poro-visco-elastic models, Arch. Ration. Mech. Anal., Volume 222 (2016) no. 3, pp. 1445-1519 | DOI | MR | Zbl
[54] Nonlinear quasi-static poroelasticity, J. Differ. Equations, Volume 296 (2021), pp. 242-278 | DOI | MR | Zbl
[55] Weak solutions in nonlinear poroelasticity with incompressible constituents, Nonlinear Anal., Real World Appl., Volume 67 (2022) no. 22, 103563 | DOI | MR | Zbl
[56] Filtration in Porous Media and Industrial Applications, Lecture Notes in Mathematics, 1734, Springer, 2000 | DOI
[57] Homogenization and Porous Media, Interdisciplinary Applied Mathematics, 6, Springer, 1997 | DOI | Zbl
[58] Thermoporoelasticity via homogenization I. Modeling and formal two-scale expansions, Int. J. Eng. Sci., Volume 138 (2019), pp. 1-25 | DOI | MR | Zbl
[59] A monolithic phase-field model of a fluid-driven fracture in a nonlinear poroelastic medium, Math. Mech. Solids, Volume 24 (2019) no. 5, pp. 1530-1555 | DOI | MR | Zbl
[60] Derivation of a poroelastic elliptic membrane shell model, Appl. Anal., Volume 98 (2019) no. 1-2, pp. 136-161 | DOI | MR | Zbl
[61] Phase-field modeling of two phase fluid filled fractures in a poroelastic medium, Multiscale Model. Simul., Volume 16 (2018) no. 4, pp. 1542-1580 | DOI | MR | Zbl
[62] Effective pressure boundary condition for the filtration through porous medium via homogenization, Nonlinear Anal., Real World Appl., Volume 44 (2018), pp. 149-172 | DOI | MR | Zbl
[63] Derivation of a poroelastic flexural shell model, Multiscale Model. Simul., Volume 14 (2016) no. 1, pp. 364-397 | DOI | MR | Zbl
[64] Effective interface conditions for the forced infiltration of a viscous fluid into a porous medium using homogenization, Comput. Methods Appl. Mech. Eng., Volume 292 (2015), pp. 195-220 | DOI | MR | Zbl
[65] A Rigorous Derivation of the Equations for the Clamped Biot-Kirchhoff-Love Poroelastic plate, Arch. Ration. Mech. Anal., Volume 215 (2015) no. 3, pp. 1035-1062 | DOI | MR | Zbl
[66] Pressure jump interface law for the Stokes-Darcy coupling: Confirmation by direct numerical simulations, J. Fluid Mech., Volume 732 (2013), pp. 510-536 | DOI | MR | Zbl
[67] Effective pressure interface law for transport phenomena between an unconfined fluid and a porous medium using homogenization, Multiscale Model. Simul., Volume 10 (2012) no. 2, pp. 285-305 | DOI | MR | Zbl
[68] A global existence result for the equations describing unsaturated flow in porous media with dynamic capillary pressure, J. Differ. Equations, Volume 248 (2010) no. 6, pp. 1561-1577 | DOI | MR | Zbl
[69] Polynomial filtration laws for low Reynolds number flows through porous media, Transp. Porous Med., Volume 81 (2010) no. 1, pp. 35-60 | DOI | MR
[70] Modeling effective interface laws for transport phenomena between an unconfined fluid and a porous medium using homogenization, Transp. Porous Med., Volume 78 (2009) no. 3, pp. 489-508 | DOI | MR
[71] On the justification of the Reynolds equation, describing isentropic compressible flows through a tiny pore, Ann. Univ. Ferrara, Sez. VII, Sci. Mat., Volume 53 (2007) no. 1, pp. 95-106 | DOI | MR | Zbl
[72] DG approximation of coupled Navier–Stokes and Darcy equations by Beaver–Joseph–Saffman interface condition, SIAM J. Numer. Anal., Volume 47 (2009) no. 3, pp. 2052-2089 | DOI | MR | Zbl
[73] Navier–Stokes/Darcy coupling: modeling, analysis, and numerical approximation, Rev. Mat. Complut., Volume 22 (2009) no. 2, pp. 315-426 | DOI | MR | Zbl
[74] Numerical analysis of the Navier–Stokes/Darcy coupling, Numer. Math., Volume 115 (2010) no. 2, pp. 195-227 | DOI | MR | Zbl
[75] Analysis of time-dependent Navier–Stokes flow coupled with Darcy flow, J. Numer. Math., Volume 16 (2008) no. 4, pp. 249-280 | DOI | MR | Zbl
[76] Coupled Stokes–Darcy model with Beavers–Joseph interface boundary condition, Commun. Math. Sci., Volume 8 (2010) no. 1, pp. 1-25 | DOI | MR | Zbl
[77] Time-dependent coupling of Navier–Stokes and Darcy flows, ESAIM, Math. Model. Numer. Anal., Volume 47 (2013) no. 2, pp. 539-554 | DOI | Numdam | MR | Zbl
[78] A nonlinear Stokes–Biot model for the interaction of a non-Newtonian fluid with poroelastic media, ESAIM, Math. Model. Numer. Anal., Volume 53 (2019) no. 6, pp. 1915-1955 | DOI | MR | Zbl
[79] Analysis of the coupled Navier–Stokes/Biot problem, J. Math. Anal. Appl., Volume 456 (2017) no. 2, pp. 970-991 | DOI | MR | Zbl
[80] Poroelastic filtration coupled to Stokes flow, Control theory of partial differential equations (Lecture Notes in Pure and Applied Mathematics), Volume 242, Chapman & Hall/CRC, 2005, pp. 229-241 | DOI | MR | Zbl
[81] Multilayered poroelasticity interacting with Stokes flow, SIAM J. Math. Anal., Volume 53 (2021) no. 6, pp. 6243-6279 | DOI | MR | Zbl
[82] Finite element methods for incompressible viscous flow, Numerical methods for fluids (Part 3) (P. G. Ciarlet; J.-L. Lions, eds.) (Handbook of Numerical Analysis), Volume 9, North-Holland, 2003 | DOI | Zbl
[83] Numerical modeling of the fluid-porohyperelastic structure interaction, SIAM J. Sci. Comput., Volume 43 (2021) no. 4, p. A2923-A2948 | DOI | MR | Zbl
[84] A computational framework for fluid-porous structure interaction with large structural deformation, Meccanica, Volume 54 (2019), pp. 101-121 | DOI | MR
[85] Mathematical Elasticity Volume I: Three-Dimensional Elasticity, Studies in Mathematics and its Applications, 20, Elsevier, 1988 | Zbl
[86] On the boundary conditions at the contact interface between a porous medium and a free fluid, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 23 (1996) no. 3, pp. 403-465 | Numdam | MR | Zbl
[87] On the interface boundary condition of Beavers, Joseph, and Saffman, SIAM J. Appl. Math., Volume 60 (2000) no. 4, pp. 1111-1127 | DOI | MR | Zbl
[88] Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate, J. Math. Fluid Mech., Volume 7 (2005) no. 3, pp. 368-404 | DOI | MR | Zbl
[89] Sobolev spaces, Pure and Applied Mathematics, 65, Academic Press Inc., 1975 | MR | Zbl
[90] A generalization of the Aubin-Lions-Simon compactness lemma for problems on moving domains, J. Differ. Equations, Volume 266 (2019) no. 12, pp. 8370-8418 | DOI | MR | Zbl
[91] Fluid-structure interaction with incompressible fluids, Progress in Mathematical Fluid Dynamics (Luigi C. Berselli et al., eds.) (Lecture Notes in Mathematics), Volume 2272, Springer, 2020, pp. 15-87 | DOI | MR | Zbl
[92] Compact families of piecewise constant functions in , Nonlinear Anal., Theory Methods Appl., Volume 75 (2012) no. 6, pp. 3072-3077 | DOI | MR | Zbl
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