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

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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

Grigory Panasenko ^1, and Ruxandra Stavre ^2,

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

Abstract
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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.

Keywords: asymptotic expansion, boundary layer method, elastic membrane, variable viscosity, viscous fluid, Fluid-structure interaction, non periodic case, error estimate..

Mathematics Subject Classification: Primary: 76M45; Secondary: 74F1.

Citation: Grigory Panasenko, Ruxandra Stavre. Asymptotic analysis of the Stokes flow with variable viscosity in a thin elastic channel. Networks and 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, Série IIb, 330 (2002), 661-666.
[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), 867-883.
[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), 1019-1042.
[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), 523-538.
[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), 59-71. doi: 10.1007/s002050050136.
[6]	G. P. Galdi, "An Introduction to the Mathematical Theory of Navier-Stokes Equations," Vol. I, Springer-Verlag, New York, 1994.
[7]	V. Girault and P. A. Raviart, "Finite Element Methods for Navier-Stokes Equations," Springer-Verlag, Berlin, 1986.
[8]	C. Grandmont and Y. Maday, Existence for an unsteady fluid-structure interaction problem, M²AN Math. Model. Numer. Anal., 34 (2000), 609-636. doi: 10.1051/m2an:2000159.
[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), 1-9. doi: 10.1088/1478-3975/5/4/046003.
[10]	J-L. Lions, "Quelques Mèthodes de Résolution des Problèmes aux Limites Non Linéaires," Dunod, Gauthier-Villars, Paris, 1969.
[11]	S. A. Nazarov and B. A. Plamenevskii, "Elliptic Problems in Domains with Piecewise Smooth Boundaries," Walter de Gruyter, Berlin, 1994.
[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), 558-579. doi: 10.1016/j.matpur.2005.10.011.
[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), 651-673.
[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), 473-482. doi: 10.1615/IntJMultCompEng.v5.i6.40.
[15]	D. Serre, Chute libre d'un solide dans un fluide visqueux incompressible. Existence, Japan J. Appl. Math., 4 (1987), 99-110.
[16]	R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis," 3rd edition, North-Holland, Amsterdam, 1984.

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, Série IIb, 330 (2002), 661-666.
[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), 867-883.
[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), 1019-1042.
[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), 523-538.
[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), 59-71. doi: 10.1007/s002050050136.
[6]	G. P. Galdi, "An Introduction to the Mathematical Theory of Navier-Stokes Equations," Vol. I, Springer-Verlag, New York, 1994.
[7]	V. Girault and P. A. Raviart, "Finite Element Methods for Navier-Stokes Equations," Springer-Verlag, Berlin, 1986.
[8]	C. Grandmont and Y. Maday, Existence for an unsteady fluid-structure interaction problem, M²AN Math. Model. Numer. Anal., 34 (2000), 609-636. doi: 10.1051/m2an:2000159.
[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), 1-9. doi: 10.1088/1478-3975/5/4/046003.
[10]	J-L. Lions, "Quelques Mèthodes de Résolution des Problèmes aux Limites Non Linéaires," Dunod, Gauthier-Villars, Paris, 1969.
[11]	S. A. Nazarov and B. A. Plamenevskii, "Elliptic Problems in Domains with Piecewise Smooth Boundaries," Walter de Gruyter, Berlin, 1994.
[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), 558-579. doi: 10.1016/j.matpur.2005.10.011.
[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), 651-673.
[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), 473-482. doi: 10.1615/IntJMultCompEng.v5.i6.40.
[15]	D. Serre, Chute libre d'un solide dans un fluide visqueux incompressible. Existence, Japan J. Appl. Math., 4 (1987), 99-110.
[16]	R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis," 3rd edition, North-Holland, Amsterdam, 1984.

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