# American Institute of Mathematical Sciences

December  2010, 5(4): 783-812. doi: 10.3934/nhm.2010.5.783

## Asymptotic analysis of the Stokes flow with variable viscosity in a thin elastic channel

 1 Laboratory of Mathematics of the University of Saint-Etienne (LaMUSE), University Jean Monnet, 23, rue Dr Paul Michelon 42023 Saint-Etienne 2 Institute of Mathematics “Simion Stoilow”, Romanian Academy, P.O. Box 1-764, 014 700 Bucharest

Received  February 2010 Revised  October 2010 Published  November 2010

The non-steady viscous flow in a thin channel with elastic wall is considered. The viscosity is constant everywhere except for some small neighborhood of the origin of the coordinate system, where it is a variable function. The problem contains two small parameters: $\varepsilon$, that is the ratio of the thickness of the channel and its length, and $\delta = \varepsilon^\gamma,$ $\gamma \geq 3 ,$ that is the "softness of the wall", i.e. its inverse (rigidity) is great. An asymptotic expansion of the solution is constructed and, in particular, the leading term is described. An important new element of this paper is the procedure of construction of the boundary layer in the neighborhood of the origin of the coordinate system, generated by the variable viscosity. The error estimates for the difference of a truncated asymptotic ansatz and the exact solution are obtained. To this end, the existence and uniqueness of the solution are studied and some a priori estimates are proved.
Citation: Grigory Panasenko, Ruxandra Stavre. Asymptotic analysis of the Stokes flow with variable viscosity in a thin elastic channel. Networks & Heterogeneous Media, 2010, 5 (4) : 783-812. doi: 10.3934/nhm.2010.5.783
##### References:
 [1] S. Čanić and A. Mikelić, Effective equations describing the flow of a viscous incompressible fluid through a long elastic tube,, C. R. Acad. Sci. Paris, 330 (2002), 661. Google Scholar [2] S. Čanić and A. Mikelić, A two-dimensional effective model describing fluid-structure interaction in blood flow: Analysis, simulation and experimental validation,, C. R. Acad. Sci. Mécanique, 333 (2005), 867. Google Scholar [3] C. Conca, J. San Martin and M. Tucsnak, Existence of solutions for the equations modeling the motion of a rigid body in a viscous fluid,, Comm. Partial Diff. Eqns., 25 (2000), 1019. Google Scholar [4] B. Desjardins, M. J. Esteban, C. Grandmont and P. le Talec, Weak solutions for a fluid-structure interaction model,, Rev. Mat. Comput., 14 (2001), 523. Google Scholar [5] B. Desjardins and M. J. Esteban, Existence of weak solutions for the motion of rigid bodies in a viscous fluid,, Arch. Rational Mech. Anal., 146 (1999), 59. doi: 10.1007/s002050050136. Google Scholar [6] G. P. Galdi, "An Introduction to the Mathematical Theory of Navier-Stokes Equations," Vol. I,, Springer-Verlag, (1994). Google Scholar [7] V. Girault and P. A. Raviart, "Finite Element Methods for Navier-Stokes Equations,", Springer-Verlag, (1986). Google Scholar [8] C. Grandmont and Y. Maday, Existence for an unsteady fluid-structure interaction problem,, M2AN Math. Model. Numer. Anal., 34 (2000), 609. doi: 10.1051/m2an:2000159. Google Scholar [9] B. M. Haines, I. S. Aranson, L. Berlyand and D. A. Karpeev, Effective viscosity of dilute bacterial suspensions: A two dimensional model,, Phys. Biol., 5 (2008), 1. doi: 10.1088/1478-3975/5/4/046003. Google Scholar [10] J-L. Lions, "Quelques Mèthodes de Résolution des Problèmes aux Limites Non Linéaires,", Dunod, (1969). Google Scholar [11] S. A. Nazarov and B. A. Plamenevskii, "Elliptic Problems in Domains with Piecewise Smooth Boundaries,", Walter de Gruyter, (1994). Google Scholar [12] G. P. Panasenko and R. Stavre, Asymptotic analysis of a periodic flow in a thin channel with visco-elastic wall,, J. Math. Pures Appl., 85 (2006), 558. doi: 10.1016/j.matpur.2005.10.011. Google Scholar [13] G. P. Panasenko and R. Stavre, Asymptotic analysis of a non-periodic flow in a thin channel with visco-elastic wall,, Networks and Heterogeneous Media, 3 (2008), 651. Google Scholar [14] G. P. Panasenko, Y. Sirakov and R. Stavre, Asymptotic and numerical modelling of a flow in a thin channel with visco-elastic wall,, Int. J. Multiscale Comput. Engng., 5 (2007), 473. doi: 10.1615/IntJMultCompEng.v5.i6.40. Google Scholar [15] D. Serre, Chute libre d'un solide dans un fluide visqueux incompressible. Existence,, Japan J. Appl. Math., 4 (1987), 99. Google Scholar [16] R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis,", 3rd edition, (1984). Google Scholar

show all references

##### References:
 [1] S. Čanić and A. Mikelić, Effective equations describing the flow of a viscous incompressible fluid through a long elastic tube,, C. R. Acad. Sci. Paris, 330 (2002), 661. Google Scholar [2] S. Čanić and A. Mikelić, A two-dimensional effective model describing fluid-structure interaction in blood flow: Analysis, simulation and experimental validation,, C. R. Acad. Sci. Mécanique, 333 (2005), 867. Google Scholar [3] C. Conca, J. San Martin and M. Tucsnak, Existence of solutions for the equations modeling the motion of a rigid body in a viscous fluid,, Comm. Partial Diff. Eqns., 25 (2000), 1019. Google Scholar [4] B. Desjardins, M. J. Esteban, C. Grandmont and P. le Talec, Weak solutions for a fluid-structure interaction model,, Rev. Mat. Comput., 14 (2001), 523. Google Scholar [5] B. Desjardins and M. J. Esteban, Existence of weak solutions for the motion of rigid bodies in a viscous fluid,, Arch. Rational Mech. Anal., 146 (1999), 59. doi: 10.1007/s002050050136. Google Scholar [6] G. P. Galdi, "An Introduction to the Mathematical Theory of Navier-Stokes Equations," Vol. I,, Springer-Verlag, (1994). Google Scholar [7] V. Girault and P. A. Raviart, "Finite Element Methods for Navier-Stokes Equations,", Springer-Verlag, (1986). Google Scholar [8] C. Grandmont and Y. Maday, Existence for an unsteady fluid-structure interaction problem,, M2AN Math. Model. Numer. Anal., 34 (2000), 609. doi: 10.1051/m2an:2000159. Google Scholar [9] B. M. Haines, I. S. Aranson, L. Berlyand and D. A. Karpeev, Effective viscosity of dilute bacterial suspensions: A two dimensional model,, Phys. Biol., 5 (2008), 1. doi: 10.1088/1478-3975/5/4/046003. Google Scholar [10] J-L. Lions, "Quelques Mèthodes de Résolution des Problèmes aux Limites Non Linéaires,", Dunod, (1969). Google Scholar [11] S. A. Nazarov and B. A. Plamenevskii, "Elliptic Problems in Domains with Piecewise Smooth Boundaries,", Walter de Gruyter, (1994). Google Scholar [12] G. P. Panasenko and R. Stavre, Asymptotic analysis of a periodic flow in a thin channel with visco-elastic wall,, J. Math. Pures Appl., 85 (2006), 558. doi: 10.1016/j.matpur.2005.10.011. Google Scholar [13] G. P. Panasenko and R. Stavre, Asymptotic analysis of a non-periodic flow in a thin channel with visco-elastic wall,, Networks and Heterogeneous Media, 3 (2008), 651. Google Scholar [14] G. P. Panasenko, Y. Sirakov and R. Stavre, Asymptotic and numerical modelling of a flow in a thin channel with visco-elastic wall,, Int. J. Multiscale Comput. Engng., 5 (2007), 473. doi: 10.1615/IntJMultCompEng.v5.i6.40. Google Scholar [15] D. Serre, Chute libre d'un solide dans un fluide visqueux incompressible. Existence,, Japan J. Appl. Math., 4 (1987), 99. Google Scholar [16] R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis,", 3rd edition, (1984). Google Scholar
 [1] Grégoire Allaire, Alessandro Ferriero. Homogenization and long time asymptotic of a fluid-structure interaction problem. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 199-220. doi: 10.3934/dcdsb.2008.9.199 [2] Serge Nicaise, Cristina Pignotti. Asymptotic analysis of a simple model of fluid-structure interaction. Networks & Heterogeneous Media, 2008, 3 (4) : 787-813. doi: 10.3934/nhm.2008.3.787 [3] Igor Kukavica, Amjad Tuffaha. Solutions to a fluid-structure interaction free boundary problem. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1355-1389. doi: 10.3934/dcds.2012.32.1355 [4] Emine Kaya, Eugenio Aulisa, Akif Ibragimov, Padmanabhan Seshaiyer. A stability estimate for fluid structure interaction problem with non-linear beam. Conference Publications, 2009, 2009 (Special) : 424-432. doi: 10.3934/proc.2009.2009.424 [5] Qiang Du, M. D. Gunzburger, L. S. Hou, J. Lee. Analysis of a linear fluid-structure interaction problem. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 633-650. doi: 10.3934/dcds.2003.9.633 [6] Andro Mikelić, Giovanna Guidoboni, Sunčica Čanić. Fluid-structure interaction in a pre-stressed tube with thick elastic walls I: the stationary Stokes problem. Networks & Heterogeneous Media, 2007, 2 (3) : 397-423. doi: 10.3934/nhm.2007.2.397 [7] George Avalos, Roberto Triggiani. Uniform stabilization of a coupled PDE system arising in fluid-structure interaction with boundary dissipation at the interface. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 817-833. doi: 10.3934/dcds.2008.22.817 [8] Mehdi Badra, Takéo Takahashi. Feedback boundary stabilization of 2d fluid-structure interaction systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2315-2373. doi: 10.3934/dcds.2017102 [9] Daniele Boffi, Lucia Gastaldi. Discrete models for fluid-structure interactions: The finite element Immersed Boundary Method. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 89-107. doi: 10.3934/dcdss.2016.9.89 [10] George Avalos, Roberto Triggiani. Fluid-structure interaction with and without internal dissipation of the structure: A contrast study in stability. Evolution Equations & Control Theory, 2013, 2 (4) : 563-598. doi: 10.3934/eect.2013.2.563 [11] Oualid Kafi, Nader El Khatib, Jorge Tiago, Adélia Sequeira. Numerical simulations of a 3D fluid-structure interaction model for blood flow in an atherosclerotic artery. Mathematical Biosciences & Engineering, 2017, 14 (1) : 179-193. doi: 10.3934/mbe.2017012 [12] Salim Meddahi, David Mora. Nonconforming mixed finite element approximation of a fluid-structure interaction spectral problem. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 269-287. doi: 10.3934/dcdss.2016.9.269 [13] Martina Bukač, Sunčica Čanić. Longitudinal displacement in viscoelastic arteries: A novel fluid-structure interaction computational model, and experimental validation. Mathematical Biosciences & Engineering, 2013, 10 (2) : 295-318. doi: 10.3934/mbe.2013.10.295 [14] George Avalos, Thomas J. Clark. A mixed variational formulation for the wellposedness and numerical approximation of a PDE model arising in a 3-D fluid-structure interaction. Evolution Equations & Control Theory, 2014, 3 (4) : 557-578. doi: 10.3934/eect.2014.3.557 [15] Henry Jacobs, Joris Vankerschaver. Fluid-structure interaction in the Lagrange-Poincaré formalism: The Navier-Stokes and inviscid regimes. Journal of Geometric Mechanics, 2014, 6 (1) : 39-66. doi: 10.3934/jgm.2014.6.39 [16] Emine Kaya, Eugenio Aulisa, Akif Ibragimov, Padmanabhan Seshaiyer. FLUID STRUCTURE INTERACTION PROBLEM WITH CHANGING THICKNESS NON-LINEAR BEAM Fluid structure interaction problem with changing thickness non-linear beam. Conference Publications, 2011, 2011 (Special) : 813-823. doi: 10.3934/proc.2011.2011.813 [17] George Avalos, Roberto Triggiani. Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 417-447. doi: 10.3934/dcdss.2009.2.417 [18] I. D. Chueshov. Interaction of an elastic plate with a linearized inviscid incompressible fluid. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1759-1778. doi: 10.3934/cpaa.2014.13.1759 [19] Eugenio Aulisa, Akif Ibragimov, Emine Yasemen Kaya-Cekin. Fluid structure interaction problem with changing thickness beam and slightly compressible fluid. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1133-1148. doi: 10.3934/dcdss.2014.7.1133 [20] Chiu-Ya Lan, Chi-Kun Lin. Asymptotic behavior of the compressible viscous potential fluid: Renormalization group approach. Discrete & Continuous Dynamical Systems - A, 2004, 11 (1) : 161-188. doi: 10.3934/dcds.2004.11.161

2018 Impact Factor: 0.871