Steady bifurcating solutions of the Couette-Taylor problem for flow in a deformable cylinder. (English) Zbl 1382.76044
Summary: The classical Couette-Taylor problem is to describe the motion of a viscous incompressible fluid in the region between two rigid coaxial cylinders, which rotate at constant angular velocities. This paper treats a generalization of this problem in which the rigid outer cylinder is replaced by a deformable (nonlinearly elastic) cylinder. The inner cylinder is rigid and rotates at a prescribed angular velocity. We study steady rotationally symmetric motions of the fluid coupled with steady axisymmetric motions of the deformable outer cylinder in which it rotates at a prescribed constant angular velocity, typically different from that of the inner cylinder. The motion of the outer cylinder is governed by a geometrically exact theory of shells and the motion of the liquid by the Navier-Stokes equations, with the domain occupied by the liquid depending on the deformation of the outer cylinder. The nonlinear fluid-solid system admits a (trivial) steady solution, termed the Couette solution, which can be found explicitly. This paper treats the global (multiparameter) bifurcation of steady-state solutions from the Couette solution. This problem exhibits technical mathematical difficulties directly due to the fluid-solid interaction: The smoothness of the shell’s configuration restricts the smoothness of the fluid variables, and their boundary values on the shell determine the smoothness of the shell’s configuration. It is essential to ensure that this cycle of implications is consistent.
MSC:
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 35B32 | Bifurcations in context of PDEs |
| 35Q35 | PDEs in connection with fluid mechanics |
| 74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |
| 76U05 | General theory of rotating fluids |
Keywords:
fluid-solid interaction; Navier-Stokes equations; bifurcation; nonlinear elasticity; Couette-Taylor problemReferences:
| [1] | Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975) · Zbl 0314.46030 |
| [2] | Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. Commun. Pure Appl. Math. 12, 623-727 (1959) · Zbl 0093.10401 · doi:10.1002/cpa.3160120405 |
| [3] | Alexander, JC; Fadell, E. (ed.); Fournier, G. (ed.), A primer on connectivity, 455-483 (1981), Berlin · Zbl 0491.47027 · doi:10.1007/BFb0092200 |
| [4] | Alexander, J.C., Antman, S.S.: Global and local behavior of bifurcating multidimensional continua of solutions for multiparameter nonlinear eigenvalue problems. Arch. Rational Mech. Anal. 76, 339-354 (1981) · Zbl 0479.58005 · doi:10.1007/BF00249970 |
| [5] | Alexander, J.C., Yorke, J.A.: The implicit function theorem and global methods of cohomology. J. Funct. Anal. 21, 330-339 (1976) · Zbl 0321.58006 · doi:10.1016/0022-1236(76)90044-6 |
| [6] | Antman, S.S.: Nonlinear Problems of Elasticity, 2nd edn. Springer, New York (2005) · Zbl 1098.74001 |
| [7] | Antman, S.S., Bourne, D.: Rotational symmetry vs. axisymmetry in shell theory. Int. J. Eng. Sci. 48, 991-1005 (2010) · Zbl 1231.74290 · doi:10.1016/j.ijengsci.2010.09.009 |
| [8] | Antman, SS; Lanza de Cristoforis, M.; Ni, W-M (ed.); Peletier, LA (ed.); Vazquez, JL (ed.), Nonlinear, nonlocal problems of fluid-solid interactions, 1-18 (1993), New York · Zbl 0806.76012 · doi:10.1007/978-1-4612-0885-3_1 |
| [9] | Beirão da Veiga, H.: On the existence of strong solutions to a coupled fluid-structure evolution problem. J. Math. Fluid Mech. 6, 21-52 (2004) · Zbl 1068.35087 · doi:10.1007/s00021-003-0082-5 |
| [10] | Bourne, D.: The Taylor-Couette problem for flow in a deformable cylinder, dissertation. Univ, Maryland (2007) |
| [11] | Bourne, D., Antman, S.S.: A non-self-adjoint quadratic eigenvalue problem describing a fluid-solid interaction Part I: formulation, analysis, and computations. Commun. Pure Appl. Anal. 8, 123-142 (2009) · Zbl 1153.74014 · doi:10.3934/cpaa.2009.8.123 |
| [12] | Bourne, D., Elman, H., Osborn, J.E.: A non-self-adjoint quadratic eigenvalue problem describing a fluid-solid interaction Part II: analysis of convergence. Commun. Pure Appl. Anal. 8, 143-160 (2009) · Zbl 1153.74015 |
| [13] | Boyer, F., Fabrie, P.: Mathematical Tools for the Study of the Navier-Stokes Equations. Springer, New York (2013) · Zbl 1286.76005 |
| [14] | Chambolle, A., Desjardins, B., Esteban, M.J., Grandmont, C.: Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate. J. Math. Fluid Mech. 7, 368-404 (2005) · Zbl 1080.74024 · doi:10.1007/s00021-004-0121-y |
| [15] | Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability. Oxford University Press, Oxford (1961) · Zbl 0142.44103 |
| [16] | Chossat, P., Iooss, G.: The Couette-Taylor Problem. Springer, New York (1994) · Zbl 0817.76001 · doi:10.1007/978-1-4612-4300-7 |
| [17] | Cliffe, K.A., Mullin, T., Schaeffer, D.: The onset of steady vortices isn Taylor-Couette flow: the role of approximate symmetry. Phys. Fluids 24, 064102-1-064102-18 (2012) · Zbl 1309.76045 |
| [18] | Constantin, P., Foias, C.: Navier-Stokes Equations. University of Chicago Press, Chicago (1988) · Zbl 0687.35071 |
| [19] | Couette, M.: Études sur le frottement des liquides. Ann. Chim. Phys. 21, 433-510 (1890) · JFM 22.0964.01 |
| [20] | Coleman, B.D., Markovitz, N., Noll, W.: Viscometric Flows of Non-Newtonian Fluids. Springer, New York (1966) · Zbl 0137.21903 · doi:10.1007/978-3-642-88655-3 |
| [21] | Coutand, D., Shkoller, S.: Motion of an elastic solid inside an incompressible viscous fluid. Arch. Rational Mech. Anal. 176, 25-102 (2005) · Zbl 1064.74057 · doi:10.1007/s00205-004-0340-7 |
| [22] | Coutand, D., Shkoller, S.: The interaction between quasilinear elastodynamics and the Navier-Stokes equations. Arch. Rational Mech. Anal. 179, 303-352 (2006) · Zbl 1138.74325 · doi:10.1007/s00205-005-0385-2 |
| [23] | Crandall, M.G., Rabinowitz, P.H.: Bifurcation from simple eigenvalues. J. Funct. Anal. 8, 321-340 (1971) · Zbl 0219.46015 · doi:10.1016/0022-1236(71)90015-2 |
| [24] | Drazin, P.G., Reid, W.H.: Hydrodynamic Stability, 2nd edn. Cambridge University Press, Cambridge (2004) · Zbl 1055.76001 · doi:10.1017/CBO9780511616938 |
| [25] | Fabes, E.B., Kenig, C.E., Verchota, G.C.: The Dirichlet problem for the Stokes flow system on Lipschitz domains. Duke Math. J. 57, 769-793 (1988) · Zbl 0685.35085 · doi:10.1215/S0012-7094-88-05734-1 |
| [26] | Fitzpatrick, P.M., Massabò, I., Pejsachowicz, J.: Global several parameter bifurcation and continuation theorems. Math. Ann. 263, 61-73 (1985) · Zbl 0519.58024 · doi:10.1007/BF01457084 |
| [27] | Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems, 2nd edn. Springer, New York (2011) · Zbl 1245.35002 |
| [28] | Galdi, G.P., Kyed, M.: Steady flow of a Navier-Stokes liquid past an elastic body. Arch. Rational Mech. Anal. 194, 849-875 (2009) · Zbl 1291.76079 · doi:10.1007/s00205-009-0224-y |
| [29] | Galdi, G.P., Rannacher, R.: Fundamental Trends in Fluid-Structure Interaction. World Scientific, Singapore (2010) · Zbl 1410.76010 · doi:10.1142/7675 |
| [30] | Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, New York (1983) · Zbl 0361.35003 · doi:10.1007/978-3-642-61798-0 |
| [31] | Girault, V., Raviart, P.-A.: Finite Element Methods for Navier-Stokes Equations. Springer, New York (1986) · Zbl 0585.65077 |
| [32] | Golubitsky, M., Stewart, I., Schaeffer, D.G.: Singularities and Groups in Bifurcation Theory, vol. II. Springer, New York (1988) · Zbl 0691.58003 · doi:10.1007/978-1-4612-4574-2 |
| [33] | Grandmont, C.: Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate. SIAM J. Math. Anal. 40, 716-737 (2008) · Zbl 1158.74016 · doi:10.1137/070699196 |
| [34] | Guidorzi, M., Padula, M., Plotnikov, P.I.: Hopf solutions to a fluid-elastic interaction model. Math. Models Methods Appl. Sci. 18, 215-269 (2008) · Zbl 1153.76338 · doi:10.1142/S0218202508002668 |
| [35] | Ize, J.; Matzeu, M. (ed.); Vignoli, A. (ed.), Topological bifurcation, 341-463 (1995), Boston · Zbl 0899.58010 · doi:10.1007/978-1-4612-2570-6_5 |
| [36] | Ize, J., Massabò, I., Pejsachowicz, J., Vignoli, A.: Structure and dimension of global branches of solutions to multiparameter nonlinear equations. Trans. Amer. Math. Soc. 291, 383-435 (1985) · Zbl 0578.58005 · doi:10.1090/S0002-9947-1985-0800246-0 |
| [37] | Joseph, D.D.: Domain perturbations: the higher order theory of infinitesimal water waves. Arch. Rational Mech. Anal. 51, 295-303 (1973) · Zbl 0271.76009 · doi:10.1007/BF00250536 |
| [38] | Joseph, D.D.: Stability of Fluid Motions, vol. I. Springer, New York (1976) · Zbl 0345.76023 |
| [39] | Kirchgässner, K.: Bifurcation in nonlinear hydrodynamic stability. SIAM Rev. 17, 652-683 (1975) · Zbl 0328.76035 · doi:10.1137/1017072 |
| [40] | Kirchgässner, K., Kielhöfer, H.: Stability and bifurcation in fluid dynamics. Rocky Mt. J. Math. 3, 275-318 (1973) · Zbl 0265.76063 · doi:10.1216/RMJ-1973-3-2-275 |
| [41] | Ladyzhenskaya, O.A.: The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, New York (1963) · Zbl 0121.42701 |
| [42] | Lichtenstein, L.: Grundlagen der Hydromechanik. Springer, Berlin (1929) · JFM 55.1124.01 |
| [43] | Lin, C.C.: The Theory of Hydrodynamic Stability. Cambridge University Press, Cambridge (1955) · Zbl 0068.39202 |
| [44] | Mitrea, M., Wright, M.: Boundary Value Problems for the Stokes System in Arbitrary Lipschitz Domains, Astérisque, vol. 77. SMF, Paris (2012) · Zbl 1345.35076 |
| [45] | Odqvist, F.K.G.: Über die Randwertaufgaben der Hydrodynamik zäher Flüssigkeiten. Math. Z. 32, 329-375 (1930) · JFM 56.0713.04 · doi:10.1007/BF01194638 |
| [46] | Rabinowitz, P.H.: Some global results for nonlinear eigenvalue problems. J. Funct. Anal. 7, 487-513 (1971) · Zbl 0212.16504 · doi:10.1016/0022-1236(71)90030-9 |
| [47] | Rabinowitz, P.H.: Théorie du Degré Topologique et Applications a des Problémes aux Limites Non Linéaires, Lecture Notes, vol. VI. University of Paris, Paris (1975) |
| [48] | Rouche, N., Mawhin, J.: Ordinary Differential Equations. Pitman, Boston (1980) · Zbl 0433.34001 |
| [49] | Russo, R.: On the existence of solutions to the steady Navier-Stokes equations. Ric. Mat. 52, 285-348 (2003) · Zbl 1121.35104 |
| [50] | Sohr, H.: The Navier-Stokes Equations. Birkhäuser, Boston (2001) · Zbl 1388.35001 · doi:10.1007/978-3-0348-0551-3 |
| [51] | Stakgold, I.: Boundary Value Problems of Mathematical Physics, vol. II. MacMillan, New York (1968) · Zbl 0165.36803 |
| [52] | Stakgold, I.: Green’s Functions and Boundary Value Problems, 2nd edn. Wiley, New York (1998) · Zbl 0897.35001 |
| [53] | Stoker, J.J.: Water Waves. Wiley, New York (1957) · Zbl 0078.40805 |
| [54] | Stupelis, L.: Navier-Stokes Equations in Irregular Domains. Kluwer, Boston (1995) · Zbl 0837.35003 · doi:10.1007/978-94-015-8525-5 |
| [55] | Tagg, R.; Andereck, CD (ed.); Hayot, F. (ed.), A guide to literature related to the Taylor-Couette problem, 303-354 (1992), New York · doi:10.1007/978-1-4615-3438-9_32 |
| [56] | Taylor, G.I.: Stability of a viscous fluid contained between two rotating cylinders. Phil. Trans. Roy. Soc. London Ser. A 223, 289-343 (1923) · JFM 49.0607.01 · doi:10.1098/rsta.1923.0008 |
| [57] | Temam, R.: Navier-Stokes Equations, 3rd edn. AMS Chelsea, Boston (2001) · Zbl 0981.35001 |
| [58] | Velte, W.: Stabilität und Verzweigung stationärer Lösungen der Navier-Stokesschen Gleichungen. Arch. Rational Mech. Anal. 22, 1-14 (1966) · Zbl 0233.76054 · doi:10.1007/BF00281240 |
| [59] | Wehausen, J.V., Laitone, E.V.: Surface Waves, in Handbuch der Physik, vol. IX. Springer, New York (1960) · Zbl 1339.76009 |
| [60] | Zeidler, E.: Nonlinear Functional Analysis and its Applications, vol. IV. Springer, New York (1997) |
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