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Chaotic streamlines inside drops immersed in steady Stokes flows

Published online by Cambridge University Press:  26 April 2006

H. A. Stone ,
Ali Nadim  and
Steven H. Strogatz
H. A. Stone
Affiliation:
Division of Applied Sciences, Harvard University, Cambridge, MA 02138, USA
Ali Nadim
Affiliation:
Department of Mathematics, MIT, Cambridge, MA 02139, USA
Steven H. Strogatz
Affiliation:
Department of Mathematics, MIT, Cambridge, MA 02139, USA
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Abstract

Motivated by the recent work of Bajer & Moffatt (1990), we investigate the kinematics of bounded steady Stokes flows. Specifically, we consider the streamlines inside a neutrally buoyant spherical drop immersed in a general linear flow. The Eulerian velocity field internal to the drop, known analytically, is a cubic function of position. For a wide range of parameters the internal streamlines, hence the fluid particle paths, may wander chaotically. Typical Poincaré sections show both ordered and chaotic regions. The extent and existence of chaotic wandering is related to (i) the orientation of the vorticity vector relative to the principal axes of strain of the undisturbed flow and (ii) the magnitude of the vorticity relative to the magnitude of the rate-of-strain tensor. In the limit of small vorticity, we use the method of averaging to predict the size of the dominant island region. This yields the critical orientation of the vorticity vector at which this dominant island disappears so that particle paths fill almost the entire Poincaré section. The problem studied here appears to be one of the simplest, physically realizable, bounded steady Stokes flows which produces chaotic streamlines.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 232 , November 1991 , pp. 629 - 646
Copyright
© 1991 Cambridge University Press

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