Glasnik Matematicki, Vol. 47, No. 2 (2012), 381-400.

ON A FREE PISTON PROBLEM FOR STOKES AND NAVIER-STOKES EQUATIONS

Boris Muha and Zvonimir Tutek

Abstract.   Our goal is to model and analyze a stationary fluid flow through the junction of two pipes in the gravity field. Inside 'vertical' pipe there is a heavy piston which can freely move along the pipe. We are interested in the equilibrium position of the piston in dependence on geometry of junction. Fluid is modeled with the Navier-Stokes equations and the piston is modeled as a rigid body. We formulate corresponding boundary value problem and prove an existence result. The problem is nonlinear even in case of the Stokes equations for fluid flow; we prove non-uniqueness of solutions and illustrate it with some numerical examples. Furthermore, derivation and analysis of the linearized problem are presented.

2010 Mathematics Subject Classification.   35Q30, 76D05, 74F10.

Key words and phrases.   Navier-Stokes equations, free piston problem, fluid-rigid body interaction.

DOI: 10.3336/gm.47.2.12

References:

1. J. M. Bernard, Non-standard Stokes and Navier-Stokes problems: existence and regularity in stationary case, Math. Methods Appl. Sci. 25 (2002), 627-661.
   MathSciNet     CrossRef

2. S. Blazy, S. Nazarov and M. Specovius-Neugebauer, Artificial boundary conditions of pressure type for viscous flows in a system of pipes, J. Math. Fluid Mech. 9 (2007), 1-33.
   MathSciNet     CrossRef

3. C. Conca, F. Murat and O. Pironneau, The Stokes and Navier-Stokes equations with boundary conditions involving the pressure, Japan. J. Math. (N.S.) 20 (1994), 279-318.
   MathSciNet    

4. C. Conca, J. San Martín H. and M. Tucsnak. Motion of a rigid body in a viscous fluid, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), 473-478.
   MathSciNet     CrossRef

5. B. D'Acunto and S. Rionero, A note on the existence and uniqueness of solutions to a free piston problem, Rend. Accad. Sci. Fis. Mat. Napoli (4) 66 (1999), 75-84.
   MathSciNet    

6. R. Dautray and J.-L. Lions. Mathematical analysis and numerical methods for science and technology. Vol. 1, Physical origins and classical methods, With the collaboration of Philippe Bénilan, Michel Cessenat, André Gervat, Alain Kavenoky and Hélène Lanchon, Translated from the French by Ian N. Sneddon, With a preface by Jean Teillac, Springer-Verlag, Berlin, 1990.
   MathSciNet    

7. B. Desjardins and M. J. Esteban, On weak solutions for fluid-rigid structure interaction: compressible and incompressible models, Comm. Partial Differential Equations 25 (2000), 1399-1413.
   MathSciNet     CrossRef

8. G. P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations. Vol. I. Linearized steady problems, Springer-Verlag, New York, 1994.
   MathSciNet    

9. G. P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations. Vol. II. Nonlinear steady problems, Springer-Verlag, New York, 1994.
   MathSciNet    

10. G. P. Galdi, Mathematical problems in classical and non-Newtonian fluid mechanics, in: Hemodynamical flows, Birkhäuser, Basel, 2008, 121-273.
    MathSciNet     CrossRef

11. V. Girault and P.-A. Raviart, Finite element methods for Navier-Stokes equations. Theory and algorithms, Springer-Verlag, Berlin, 1986.
    MathSciNet     CrossRef

12. M. Hillairet and D. Serre, Chute stationnaire d'un solide dans un fluide visqueux incompressible le long d'un plan incliné, Ann. Inst. H. Poincaré Anal. Non Linéaire 20 (2003), 779-803.
    MathSciNet     CrossRef

13. H. Kielhöfer, Bifurcation theory. An introduction with applications to PDEs, Springer-Verlag, New York, 2004.
    MathSciNet    

14. E. Marušić-Paloka, Rigorous justification of the Kirchhoff law for junction of thin pipes filled with viscous fluid, Asymptot. Anal. 33 (2003), 51-66.
    MathSciNet    

15. V. Maz'ya and J. Rossmann, L_p estimates of solutions to mixed boundary value problems for the Stokes system in polyhedral domains, Math. Nachr. 280 (2007), 751-793.
    MathSciNet     CrossRef

16. V. Maz'ya and J. Rossmann, Mixed boundary value problems for the stationary Navier-Stokes system in polyhedral domains, Arch. Ration. Mech. Anal. 194 (2009), 669-712.
    MathSciNet     CrossRef

17. J. Nečas, Les méthodes directes en théorie des équations elliptiques, Masson et Cie, Éditeurs, Paris, 1967.
    MathSciNet    

18. V. G. Osmolovskiĭ, Linear and nonlinear perturbations of the operator div. Translated from the 1995 Russian original by Tamara Rozhkovskaya, American Mathematical Society, Providence, 1997.
    MathSciNet    

19. S. Takeno, Free piston problem for isentropic gas dynamics, Japan J. Indust. Appl. Math. 12 (1995), 163-194.
    MathSciNet     CrossRef

20. R. Temam, Navier-Stokes equations. Theory and numerical analysis, North-Holland Publishing Co., Amsterdam, 1977.
    MathSciNet