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Transport and buckling dynamics of an elastic fibre in a viscous cellular flow

Published online by Cambridge University Press:  25 March 2015

N. Quennouz ,
M. Shelley ,
O. du Roure  and
A. Lindner
N. Quennouz
Affiliation:
Physique et Mécanique des Milieux Hétérogènes UMR7636 ESPCI-CNRS-Université Pierre et Marie Curie-Université Paris Diderot-10 rue Vauquelin F-75005 Paris, France
M. Shelley*
Affiliation:
Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA
O. du Roure*
Affiliation:
Physique et Mécanique des Milieux Hétérogènes UMR7636 ESPCI-CNRS-Université Pierre et Marie Curie-Université Paris Diderot-10 rue Vauquelin F-75005 Paris, France
A. Lindner
Affiliation:
Physique et Mécanique des Milieux Hétérogènes UMR7636 ESPCI-CNRS-Université Pierre et Marie Curie-Université Paris Diderot-10 rue Vauquelin F-75005 Paris, France
*
†Email addresses for correspondence: [email protected], [email protected]
†Email addresses for correspondence: [email protected], [email protected]
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Abstract

We study, using both experiment and theory, the coupling of transport and shape dynamics for elastomeric fibres moving through an inhomogeneous flow. The cellular flow, created electromagnetically in our experiment, comprises many identical cells of counter-rotating vortices, with a global flow geometry characterized by a backbone of stable and unstable manifolds connecting hyperbolic stagnation points. Our mathematical model is based upon slender-body theory for the Stokes equations, with the fibres modelled as inextensible elastica. Above a certain threshold of the control parameter, the elasto-viscous number, transport of fibres is mediated by their episodic buckling by compressive stagnation point flows, lending an effectively chaotic component to their dynamics. We use simulations of the model to construct phase diagrams of the fibre state (buckled or not) near stagnation points in terms of two variables that arise in characterizing the transport dynamics. We show that this reduced statistical description quantitatively captures our experimental observations. By carefully reproducing the experimental protocols and time scales of observation within our numerical simulations, we also quantitatively explain features of the measured buckling probability curve as a function of the effective flow forcing. Finally, we show within both experiment and simulation the existence of short and long time scales in the evolution of fibre conformation.

JFM classification

Low-Reynolds-number flows: Low-Reynolds-number flows Low-Reynolds-number flows: Slender-body theory
Type
Papers
Information
Journal of Fluid Mechanics , Volume 769 , 25 April 2015 , pp. 387 - 402
Copyright
© 2015 Cambridge University Press 

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