×

Two-dimensional Euler flows in slowly deforming domains. (English) Zbl 1210.76049

Summary: We consider the evolution of an incompressible two-dimensional perfect fluid as the boundary of its domain is deformed in a prescribed fashion. The flow is taken to be initially steady, and the boundary deformation is assumed to be slow compared to the fluid motion. The Eulerian flow is found to remain approximately steady throughout the evolution. At leading order, the velocity field depends instantaneously on the shape of the domain boundary, and it is determined by the steadiness and vorticity-preservation conditions. This is made explicit by reformulating the problem in terms of an area-preserving diffeomorphism \(g_\varLambda \) which transports the vorticity. The first-order correction to the velocity field is linear in the boundary velocity, and we show how it can be computed from the time derivative of \(g_\varLambda \).
The evolution of the Lagrangian position of fluid particles is also examined. Thanks to vorticity conservation, this position can be specified by an angle-like coordinate along vorticity contours. An evolution equation for this angle is derived, and the net change in angle resulting from a cyclic deformation of the domain boundary is calculated. This includes a geometric contribution which can be expressed as the integral of a certain curvature over the interior of the circuit that is traced by the parameters defining the deforming boundary.
A perturbation approach using Lie series is developed for the computation of both the Eulerian flow and geometric angle for small deformations of the boundary. Explicit results are presented for the evolution of nearly axisymmetric flows in slightly deformed discs.

MSC:

76D05 Navier-Stokes equations for incompressible viscous fluids
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
35Q31 Euler equations
76E99 Hydrodynamic stability
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Arnold, V. I., Conditions for nonlinear stability of stationary plane curvilinear flows of an ideal fluid, Dokl. Akad. Nauk SSSR, 162, 975-978 (1965), (in Russian); English translation in: Soviet Math. 6, 773-777
[2] Arnold, V. I., On an a priori estimate in the theory of hydrodynamical stability, Izv. Vyssh. Uchebn. Zaved. Matematika, 53, 3-5 (1966), (in Russian); English translation in: Amer. Math. Soc. Transl. Ser. 2, 79, 267-269
[3] Arnold, V. I., Mathematical Methods of Classical Mechanics (1989), Springer-Verlag
[4] Arnol’d, V. I.; Khesin, B. A., Topological Methods in Hydrodynamics (1998), Springer-Verlag · Zbl 0902.76001
[5] Berry, M. V., Classical adiabatic angles and quantal adiabatic phase, J. Phys. A: Math. Gen., 18, 1, 15-27 (1985) · Zbl 0569.70020
[6] Drazin, P. G.; Reid, W. H., Hydrodynamic Stability (1981), Cambridge Univ. Press · Zbl 0449.76027
[7] Frankel, T., The Geometry of Physics (2004), Springer-Verlag
[8] Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order (1977), Springer-Verlag · Zbl 0691.35001
[9] Hannay, J. H., Angle variable holonomy in adiabatic excursion of an integrable Hamiltonian, J. Phys. A: Math. Gen., 18, 2, 221-230 (1985)
[10] Holm, D. D.; Marsden, J. E.; Ratiu, T. S.; Weinstein, A., Nonlinear stability of fluid and plasma equilibria, Phys. Rep., 123, 1-2, 1-116 (1985) · Zbl 0717.76051
[11] Jeffreys, H.; Jeffreys, B., Methods of Mathematical Physics (1974), Cambridge Univ. Press · Zbl 0037.31704
[12] Landau, L. D.; Lifshitz, E. M., Mechanics (1960), Pergamon · Zbl 0081.22207
[13] Lichtenberg, A. J.; Lieberman, M. A., Regular and Chaotic Dynamics (1992), Springer-Verlag · Zbl 0748.70001
[14] Marsden, J. E.; Montgomery, R.; Ratiu, T. S., Reduction, symmetry and phases in mechanics, Mem. AMS, 436, 110 (1990) · Zbl 0713.58052
[15] Maslowe, S. A., Critical layers in shear flows, Ann. Rev. Fluid Mech., 18, 405-432 (1986) · Zbl 0634.76046
[16] Montgomery, R., The connection whose holonomy is the classical adiabatic angles of Hannay and Berry and its generalization to the non-integrable case, Commun. Math. Phys., 120, 269-294 (1988) · Zbl 0689.58043
[17] Morrison, P. J., Hamiltonian description of the ideal fluid, Rev. Modern Phys., 70, 2, 467-521 (1998) · Zbl 1205.37093
[18] Saffman, P. G., Vortex Dynamics (1992), Cambridge Univ. Press · Zbl 0777.76004
[19] Salmon, R., Hamiltonian fluid mechanics, Ann. Rev. Fluid Mech., 20, 225-256 (1988)
[20] Shapere, A.; Wilczek, F., Gauge kinematics of deformable bodies, (Shapere, A.; Wilczek, F., Geometric Phases in Physics (1989), World Scientific), 449-459
[21] Shapere, A.; Wilczek, F., Geometry of self-propulsion at low Reynolds number, J. Fluid Mech., 198, 557-585 (1989) · Zbl 0674.76114
[22] Shashikanth, B. N.; Newton, P. K., Vortex motion and the geometric phase. Part I. Basic configurations and asymptotics, J. Nonlinear Sci., 8, 183-214 (1997) · Zbl 0913.76016
[23] Shashikanth, B. N.; Newton, P. K., Vortex motion and the geometric phase. Part II. Slowly varying spiral structures, J. Nonlinear Sci., 9, 233-254 (1999)
[24] Stewartson, K., The evolution of the critical layer of a Rossby wave, Geophys. Astrophys. Fluid Dynamics, 9, 185-200 (1978) · Zbl 0374.76024
[25] Wirosoetisno, D.; Vanneste, J., Persistence of steady flows of a two-dimensional perfect fluid in deformed domains, Nonlinearity, 18, 2657-2680 (2005) · Zbl 1084.76011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.