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Prandtl–Batchelor theorem for flows with quasiperiodic time dependence

  • Hassan Arbabi (a1) and Igor Mezić (a2)


The classical Prandtl–Batchelor theorem (Prandtl, Proc. Intl Mathematical Congress, Heidelberg, 1904, pp. 484–491; Batchelor, J. Fluid Mech., vol. 1 (02), 1956, pp. 177–190) states that in the regions of steady 2D flow where viscous forces are small and streamlines are closed, the vorticity is constant. In this paper, we extend this theorem to recirculating flows with quasiperiodic time dependence using ergodic and geometric analysis of Lagrangian dynamics. In particular, we show that 2D quasiperiodic viscous flows, in the limit of zero viscosity, cannot converge to recirculating inviscid flows with non-uniform vorticity distribution. A corollary of this result is that if the vorticity contours form a family of closed curves in a quasiperiodic viscous flow, then at the limit of zero viscosity, vorticity is constant in the area enclosed by those curves at all times.


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Arbabi, H. & Mezić, I. 2017 Study of dynamics in post-transient flows using Koopman mode decomposition. Phys. Rev. Fluids 2, 124402.
Arnold, V. I. 1989 Mathematical Methods of Classical Mechanics, vol. 60. Springer.
Batchelor, G. K. 1956 On steady laminar flow with closed streamlines at large Reynolds number. J. Fluid Mech. 1 (02), 177190.
Blennerhassett, P. J. 1979 A three-dimensional analogue of the Prandtl–Batchelor closed streamline theory. J. Fluid Mech. 93 (2), 319324.
Burggraf, O. R. 1966 Analytical and numerical studies of the structure of steady separated flows. J. Fluid Mech. 24 (1), 113151.
Childress, S., Landman, M. & Strauss, H. 1990 Steady motion with helical symmetry at large Reynolds number. In Proc. IUTAM Symp. on Topological Fluid dynamics (ed. Moffatt, H. K. & Tsinober, A.), pp. 216224. Cambridge University Press.
Govindarajan, N., Mohr, R., Chandrasekaran, S. & Mezić, I.2018 On the approximation of Koopman spectra for measure preserving transformations. arXiv:1803.03920.
Grimshaw, R. 1969 On steady recirculating flows. J. Fluid Mech. 39 (4), 695703.
Lee, J. M. 2003 Introduction to Smooth Manifolds. Springer.
Mane, R. 1987 Ergodic Theory and Differentiable Dynamics. Springer.
Masmoudi, N. 2007 Remarks about the inviscid limit of the Navier–Stokes system. Commun. Math. Phys. 270 (3), 777788.
Mezić, I.1994 On geometrical and statistical properties of dynamical systems: theory and applications. PhD thesis, California Institute of Technology.
Mezić, I. 2002 An extension of Prandtl–Batchelor theory and consequences for chaotic advection. Phys. Fluids 14 (9), L61L64.
Pan, F. & Acrivos, A. 1967 Steady flows in rectangular cavities. J. Fluid Mech. 28 (4), 643655.
Petersen, K. E. 1989 Ergodic Theory, vol. 2. Cambridge University Press.
Prandtl, L. 1904 Uber Flüssigkeitsbewegung bei sehr kleiner Reibung. In Proceedings of the International Mathematical Congress, Heidelberg, pp. 484–491, see Gesammelte Abhandlungen II, 575 (1961).
Sandoval, M. & Chernyshenko, S. 2010 Extension of the Prandtl–Batchelor theorem to three-dimensional flows slowly varying in one direction. J. Fluid Mech. 654, 351361.
Susuki, Y. & Mezić, I.2018 Uniformly bounded sets in qausiperiodically forced dynamical systems. arXiv:1808.08340.
Trefethen, L. N. 2000 Spectral Methods in MATLAB, vol. 10. SIAM.
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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
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