Derivation of Stokes-plate-equations modeling fluid flow interaction with thin porous elastic layers

References

• Clopeau T, Ferrín JL, Gilbert RP, et al. Homogenizing the acoustic properties of the seabed: part II. Math Comput Model. 2001;33:821–841.
   Web of Science ®Google Scholar
• Ferrin J, Mikelić A. Homogenizing the acoustic properties of a porous matrix containing an incompressible inviscid fluid. Math Methods Appl Sci. 2003;26(10):831–859.
   Web of Science ®Google Scholar
• Gilbert RP, Mikelić A. Homogenizing the acoustic properties of the seabed: part I. Nonlinear Anal. 2000;40:185–212.
   Web of Science ®Google Scholar
• Jäger W, Mikelić A, Neuss-Radu M. Analysis of differential equations modelling the reactive flow through a deformable system of cells. Arch Rational Mech Anal. 2009;192:331–374.
   Web of Science ®Google Scholar
• Jäger W, Mikelić A, Neuss-Radu M. Homogenization limit of a model system for interaction of flow, chemical reaction, and mechanics in cell tissues. SIAM J Math Anal. 2011;43(3):1390–1435.
   Web of Science ®Google Scholar
• Jäger W, Mikelić A. On the effective equations of a viscous incompressible fluid flow through a filter of finite thickness. Commun Pure Appl Math. 1998;51:1073–1121.
   Web of Science ®Google Scholar
• Mikelić A, Wheeler MF. On the interface law between a deformable porous medium containing a viscous fluid and an elastic body. Math Models Meth Appl Sci. 2012;22(11):32.
   Web of Science ®Google Scholar
• Beavers GS, Joseph DD. Boundary conditions at a naturally permeable wall. J Fluid Mech. 1967;30:197–207.
   Web of Science ®Google Scholar
• Jäger W, Mikelić A. On the boundary conditions at the contact interface between a porous medium and a free fluid. Ann Scuola Norm Sup Pisa Cl Sci. 1996;23:403–465.
   Google Scholar
• Jäger W, Mikelic A, Neuss N. Asymptotic analysis of the laminar viscous flow over a porous bed. SIAM J Sci Comput. 2001;22(6):2006–2028.
   Web of Science ®Google Scholar
• Marciniak-Czochra A, Mikelić A. A rigorous derivation of the equations for the clamped biot-kirchhoff-love poroelastic plate. Arch Ration Mech Anal. 2015;215(3):1035–1062.
   Web of Science ®Google Scholar
• Neuss-Radu M, Jäger W. Effective transmission conditions for reaction-diffusion processes in domains separated by an interface. SIAM J Math Anal. 2007;39:687–720.
   Web of Science ®Google Scholar
• Marušić S, Marušić-Paloka E. Two-scale convergence for thin domains and its applications to some lower-dimensional model in fluid mechanics. Asymptotic Analysis. 2000;23:23–58.
   Web of Science ®Google Scholar
• Gahn M, Jäger W, Neuss-Radu M. Two-scale tools for homogenization and dimension reduction of perforated thin layers: Extensions, Korn-inequalities, and two-scale compactness of scale-dependent sets in Sobolev spaces. Submitted Preprint: Available from: arXiv:2112.00559.
   Google Scholar
• Boulakia M. Existence of weak solutions for the three-dimensional motion of an elastic structure in an incompressible fluid. J Math Fluid Mech. 2007;9:262–294.
   Web of Science ®Google Scholar
• Coutand D, Shkoller S. Motion of an elastic solid inside an incompressible viscous fluid. Arch Rational Mech Anal. 2005;176:25–102.
   Web of Science ®Google Scholar
• Coutand D, Shkoller S. The interaction between quasilinear elastodynamics and the Navier–Stokes equations. Arch Rational Mech Anal. 2006;179:303–352.
   Web of Science ®Google Scholar
• Raymond J-P, Vanninathan M. A fluid–structure coupling the Navier–Stokes equations and the Lamé system. Journal De Math Pures Et Appl. 2014;102:546–596.
   Web of Science ®Google Scholar
• Muha B, Canić S. 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. Archive Rational Mech Anal. 2013;207(3):919–968.
   Web of Science ®Google Scholar
• Sánchez-Palencia E. Boundary value problems in domains containing perforated walls. Nonlinear partial differential equations and their applications. Vol. 3, College de France Seminar; 1982. p. 309–325.
   Google Scholar
• Conca C. Étude d'un fluid traversant une paroi perforeé I. comportement limite près de la paroi. J Math Pures Et Appl. 1987;66:1–43.
   Google Scholar
• Conca C. Étude d'un fluid traversant une paroi perforeé II. comportement limite loin de la paroi. J Math Pures Et Appl. 1987;66:45–69.
   Google Scholar
• Bourgeat A, Gipouloux O, Marušić-Paloka E. Mathematical modelling and numerical simulation of a non-Newtonian viscous flow through a thin filter. SIAM J Appl Math. 2001;62(2):597–626.
   Web of Science ®Google Scholar
• Allaire G. Homogenization of the Navier–Stokes equations in open sets perforated with tiny holes II: non-critical sizes of the holes for a volume distribution and a surface distribution of holes. Arch Rational Mech Anal. 1991;113:261–298.
   Web of Science ®Google Scholar
• Marušić S. Low concentration limit for a fluid flow through a filter. Math Models Methods Appl Sci. 1998;8(4):623–643.
   Web of Science ®Google Scholar
• Ciarlet PG. Mathematical elasticity: volume II: theory of plates. New York: Elsevier; 1997.
   Google Scholar
• Caillerie D, Nedelec J. Thin elastic and periodic plates. Math Methods Appl Sci. 1984;6(1):159–191.
   Google Scholar
• Griso G, Khilkova L, Orlik J, et al. Homogenization of perforated elastic structures. J Elast. 2020;141:181–225.
   Web of Science ®Google Scholar
• Orlik J, Panasenko G, Stavre R. Asymptotic analysis of a viscous fluid layer separated by a thin stiff stratified elastic plate. Appl Anal. 2021;100(3):589–629.
   Web of Science ®Google Scholar
• Fabricius J. Stokes flow with kinematic and dynamic boundary conditions. preprint 2017. Available from: arXiv:1702.03155.
   Google Scholar
• Cattabriga L. Su un problema al contorno relativo al sistema di equazioni di stokes. Rendiconti Del Seminario Matematico Della Università Di Padova. 1961;31:308–340.
   Google Scholar
• Sohr H. The Navier–Stokes equations: an elementary functional analytic approach. Basel: Birkhäuser; 2001.
   Google Scholar
• Oleinik OA, Shamaev AS, Yosifian GA. Mathematical problems in elasticity and homogenization. Amsterdam: North Holland; 1992.
   Google Scholar
• Wloka J. Partial differential equations. Cambridge: Cambridge University Press; 1982.
   Google Scholar
• Lions JL. Quelques méthodes de résolution des problèmes aux limites non linéaires. Paris: Dunod; 1969.
   Google Scholar
• Allaire G. Homogenization and two-scale convergence. SIAM J Math Anal. 1992;23:1482–1518.
   Web of Science ®Google Scholar
• Grisvard P. Elliptic problems in nonsmooth domains. Marshfield (MA): Pitman Advanced Publishing Program; 1985.
   Google Scholar
• Bhattacharya A, Gahn M, Neuss-Radu M. Effective transmission conditions for reaction–diffusion processes in domains separated by thin channels. Appl Anal. 2020;101:1896–1910.
   Web of Science ®Google Scholar
• Gahn M, Neuss-Radu M, Knabner P. Derivation of effective transmission conditions for domains separated by a membrane for different scaling of membrane diffusivity. Discrete Contin Dyn Syst. 2017;10(4):773.
   Web of Science ®Google Scholar