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Tracking particles in flows near invariant manifolds via balance functions

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Abstract

Particle moving inside a fluid near, and interacting with, invariant manifolds is a common phenomenon in a wide variety of applications. One elementary question is whether we can determine once a particle has entered a neighbourhood of an invariant manifold, when it leaves again. Here we approach this problem mathematically by introducing balance functions, which relate the entry and exit points of a particle by an integral variational formula. We define, study, and compare different natural choices for balance functions and conclude that an efficient compromise is to employ normal infinitesimal Lyapunov exponents. We apply our results to two different model flows: a regularized solid-body rotational flow and the asymmetric Kuhlmann–Muldoon model developed in the context of liquid bridges. To test the balance function approach, we also compute the motion of a finite size particle in an incompressible liquid near a shear-stress interface (invariant wall), using fully resolved numerical simulation. In conclusion, our theoretically developed framework seems to be applicable to models as well as data to understand particle motion near invariant manifolds.

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Acknowledgements

We thank three referees for their reviews and helpful comments.

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Authors and Affiliations

  1. Fakultät für Mathematik, Technical University of Munich, Boltzmannstr. 3, 85748, Garching bei München, Germany

    Christian Kuehn

  2. Institute of Fluid Mechanics and Heat Transfer, Vienna University of Technology, Getreidemarkt 9, 1060, Vienna, Austria

    Francesco Romanò & Hendrik C. Kuhlmann

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  1. Christian Kuehn
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Correspondence to Christian Kuehn.

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CK acknowledges support via an APART fellowship of the Austrian Academy of Sciences (ÖAW) and via a Lichtenberg Professorship of the Volkswagen Foundation. Furthermore, CK would like to thank Peter Szmolyan and Daniel Karrasch for general discussions regarding non-autonomous dynamical systems, and Armin Rainer for a clarifying remark regarding differentiability of certain parameterized eigenvalues.

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Kuehn, C., Romanò, F. & Kuhlmann, H.C. Tracking particles in flows near invariant manifolds via balance functions. Nonlinear Dyn 92, 983–1000 (2018). https://doi.org/10.1007/s11071-018-4104-6

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