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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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References

Abef, H. 1984 Stirring by chaotic advection. J. Fluid Mech. 143, 1–21.Google Scholar
Aref, H. & Balachandar, S. 1986 Chaotic advection in a Stokes flow. Phys. Fluids 29, 3515–3521.Google Scholar
Bajer, K. & Moffatt, H. K. 1990 On a class of steady confined Stokes flows with chaotic streamlines. J. Fluid Mech. 212, 337–363.Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Batchelor, G. K. 1979 Mass transfer from a particle suspended in fluid with a steady linear ambient velocity distribution. J. Fluid Mech. 95, 369–400.Google Scholar
Dombre, T., Frisch, U., Greene, J. M., Hénon, M., Mehr, A. & Sowahds, A. M. 1986 ChaoticÉ streamlines in the ABC flow. J. Fluid Mech. 167, 353–391.Google Scholar
Hinch, E. J. 1988 Hydrodynamics at low Reynolds numbers. In Disorder and Mixing (ed. E. Guyon, J. P. Nadal & Y. Pomeau). NATO ASI Series, Series E, vol. 152, pp. 43–55.
Leong, C. W. & Ottino, J. M. 1989 Experiments on mixing due to chaotic advection in a cavity. J. Fluid Mech. 209, 463–499.Google Scholar
Lichtenberg, A. J. & Lieberman, M. A. 1983 Regular and Stochastic Motion. Springer.
Moffatt, H. K. 1991 Electromagnetic stirring. Phys. Fluids A 3, 1336–1343.Google Scholar
Nadim, A. & Stone, H. A. 1991 The motion of small particles and droplets in quadratic flows. Stud. Appl. Maths 85, 53–73.Google Scholar
Ottino, J. M. 1989 The Kinematics of Mixing: Stretching, Chaos, and Transport. Cambridge University Press.
Press, W. H., Flannery, B. P., Teukolsky, S. A. & Vetterling, W. T. 1986 Numerical Recipes. Cambridge University Press.
Rom-Kedar, V., Leonard, A. & Wiggins, S. 1990 An analytical study of transport, mixing and chaos in an unsteady vortical flow. J. Fluid Mech. 214, 347–394.Google Scholar
Sanders, J. A. & Verhulst, F. 1985 Averaging Methods in Nonlinear Dynamical Systems. Springer.
Swanson, P. D. & Ottino, J. M. 1990 A comparative computational and experimental study of chaotic mixing of viscous fluids. J. Fluid Mech. 213, 227–249.Google Scholar
Tsang, K. Y., Mirollo, R. E., Strogatz, S. H. & Wiesenfeld, K. 1991 Dynamics of a globally coupled oscillator array. Physica D 48, 102–112.Google Scholar
Wiggins, S. 1988 Global Bifurcations and Chaos. Springer.