Tracking particles in flows near invariant manifolds via balance functions

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 th...

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Bibliographic Details
Published inNonlinear dynamics Vol. 92; no. 3; pp. 983 - 1000
Main Authors Kuehn, Christian, Romanò, Francesco, Kuhlmann, Hendrik C.
Format Journal Article
LanguageEnglish
Published Dordrecht Springer Netherlands 01.05.2018
Springer Nature B.V
Subjects
Automotive Engineering
Classical Mechanics
Computational fluid dynamics
Computer simulation
Control
Dynamical Systems
Engineering
Fluid flow
Incompressible flow
Invariants
Liapunov exponents
Liquid bridges
Manifolds
Mathematical analysis
Mathematical models
Mechanical Engineering
Original Paper
Particle motion
Vibration
Vortices
Perturbation theory
Fluid dynamics
Fast–slow systems
Particle–surface interaction
Entry–exit function
Invariant manifold
Transient dynamics
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Summary: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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ISSN:0924-090X
1573-269X
DOI:10.1007/s11071-018-4104-6