Nonlinear Dynamics

, Volume 92, Issue 3, pp 983–1000 | Cite as

Tracking particles in flows near invariant manifolds via balance functions

  • Christian Kuehn
  • Francesco Romanò
  • Hendrik C. Kuhlmann
Original Paper


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.


Invariant manifold Fluid dynamics Entry–exit function Perturbation theory Particle–surface interaction Fast–slow systems Transient dynamics 



We thank three referees for their reviews and helpful comments.


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Authors and Affiliations

  1. 1.Fakultät für MathematikTechnical University of MunichGarching bei MünchenGermany
  2. 2.Institute of Fluid Mechanics and Heat TransferVienna University of TechnologyViennaAustria

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