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Spatial Decay of the Vorticity Field of Time-Periodic Viscous Flow Past a Body

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Abstract

We study the asymptotic spatial behavior of the vorticity field, \(\omega (x,t)\), associated to a time-periodic Navier–Stokes flow past a body, \({\mathscr {B}}\), in the class of weak solutions satisfying a Serrin-like condition. We show that, outside the wake region, \({\mathcal {R}}\), \(\omega \) decays pointwise at an exponential rate, uniformly in time. Moreover, denoting by \({\bar{\omega }}\) its time-average over a period and by \(\omega _P:=\omega -{\bar{\omega }}\) its purely periodic component, we prove that inside \({\mathcal {R}}\), \({\bar{\omega }}\) has the same algebraic decay as that known for the associated steady-state problem, whereas \(\omega _P\) decays even faster, uniformly in time. This implies, in particular, that “sufficiently far” from \({\mathscr {B}}\), \(\omega (x,t)\) behaves like the vorticity field of the corresponding steady-state problem.

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Authors and Affiliations

  1. Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstraße 39, 10117, Berlin, Germany

    Thomas Eiter

  2. Department of Mechanical Engineering and Materials Science, University of Pittsburgh, Pittsburgh, PA, 15261, USA

    Giovanni P. Galdi

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  1. Thomas Eiter
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  2. Giovanni P. Galdi
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Correspondence to Giovanni P. Galdi.

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Communicated by P. Constantin

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T. Eiter: Partially supported by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), Project Number: 427538878

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Eiter, T., Galdi, G.P. Spatial Decay of the Vorticity Field of Time-Periodic Viscous Flow Past a Body. Arch Rational Mech Anal 242, 149–178 (2021). https://doi.org/10.1007/s00205-021-01690-z

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