Abstract
Stents are medical devices designed to modify blood flow in aneurysm sacs, in order to prevent their rupture. Some of them can be considered as a locally periodic rough boundary. In order to approximate blood flow in arteries and vessels of the cardio-vascular system containing stents, we use multi-scale techniques to construct boundary layers and wall laws. Simplifying the flow we turn to consider a 2-dimensional Poisson problem that conserves essential features related to the rough boundary. Then, we investigate convergence of boundary layer approximations and the corresponding wall laws in the case of Neumann type boundary conditions at the inlet and outlet parts of the domain. The difficulty comes from the fact that correctors, for the boundary layers near the rough surface, may introduce error terms on the other portions of the boundary. In order to correct these spurious oscillations, we introduce a vertical boundary layer. Trough a careful study of its behavior, we prove rigorously decay estimates.We then construct complete boundary layers that respect the macroscopic boundary conditions. We also derive error estimates in terms of the roughness size ε either for the full boundary layer approximation and for the corresponding averaged wall law.
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References
Amrouche, C., Girault, V., Giroire, J.: Weighted sobolev spaces and laplace’s equation in ℝn. J. Math. Pur. Appl. 73, 579–606 (1994)
Babuška, I.: Solution of interface problems by homogenization. Parts I and II. SIAM J. Math. Anal. 7(5), 603–645 (1976)
Bonnetier, E., Vogelius, M.: An elliptic regularity result for a composite medium with “touching” fibers of circular cross-section. SIAM J. Math. Anal. 31, 651–677 (2000)
Bresch, D., Milišić, V.: High order multi-scale wall laws: Part i, the periodic case. Accepted for publication in Quart. Appl. Math. (2008)
Bresch, D., Milišić, V.: Towards implicit multi-scale wall laws C.R. Math. 346 (15–16), 833–838 (2008)
Galdi, P.: An Introduction to the Mathematical Theory of the NS Equations, vol. I & II. Springer, New York (1994)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer-Verlag, Berlin (2001). Reprint of the 1998 edition
Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Monographs and Studies in Mathematics, vol. 24. Pitman (Advanced Publishing Program), Boston, MA (1985)
Hanouzet, B.: Espaces de Sobolev avec poids application au problème de Dirichlet dans un demi espace. Rend. Sem. Mat. Univ. Padova 46, 227–272 (1971)
Jäger, W., Mikelić, A.: On the interface boundary condition of Beavers, Joseph, and Saffman. SIAM J. Appl. Math. 60(4), 1111–1127 (2000)
Jäger, W., Mikelić, A.: On the roughness-induced effective boundary condition for an incompressible viscous flow. J. Diff. Eqns. 170, 96–122 (2001)
Kress, R.: Linear Integral Equations. Appl. Math. Sci., vol. 82, second edn. Springer-Verlag, New York (1999)
Kudrjavcev, L.D.: An imbedding theorem for a class of functions defined in the whole space or in the half-space. I. Mat. Sb. (N.S.) 69(111), 616–639 (1966)
Kudrjavcev, L.D.: Imbedding theorems for classes of functions defined in the whole space or in the half-space. II. Mat. Sb. (N.S.) 70(112), 3–35 (1966)
Kufner, A.: Weighted Sobolev Spaces. Teubner-Texte zur Mathematik edn. BSB B. G. Teubner Verlagsgesellschaft (1980)
Milišić, V.: Very weak estimates for a rough Poisson-Dirichlet problem with natural vertical boundary conditions. Meth. Appl. Anal. 16(2), 157–186 (2009)
Moskow, S., Vogelius, M.: First-order corrections to the homogenised eigenvalues of a periodic composite medium. A convergence proof. Proc. Roy. Soc. Edinburgh Sect. 127(6), 1263–1299 (1997)
Nečas, J.: Les Méthodes Directes en Théorie des Équations Elliptiques. Masson et Cie, Éditeurs, Paris (1967)
Sánchez-Palencia, E.: Nonhomogeneous Media and Vibration Theory. Lecture Notes in Physics, vol. 127. Springer-Verlag, Berlin (1980)
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Bonnetier, E., Bresch, D., Milišić, V. (2010). A Priori Convergence Estimates for a Rough Poisson-Dirichlet Problem with Natural Vertical Boundary Conditions. In: Rannacher, R., Sequeira, A. (eds) Advances in Mathematical Fluid Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04068-9_7
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DOI: https://doi.org/10.1007/978-3-642-04068-9_7
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