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Experimental and computational studies of mixing in complex Stokes flows: the vortex mixing flow and multicellular cavity flows

Published online by Cambridge University Press:  26 April 2006

Sadhan C. Jana ,
Guy Metcalfe  and
J. M. Ottino
Sadhan C. Jana
Affiliation:
Department of Chemical Engineering, Robert R. McCormick School of Engineering and Applied Science, Northwestern University, Evanston, IL 60208–3120, USA Present address: The Benjamin Levich Institute, Steinman Hall, 1M, City University of New York, 140th Street & Convent Avenue, New York, NY 10031, USA.
Guy Metcalfe
Affiliation:
Department of Chemical Engineering, Robert R. McCormick School of Engineering and Applied Science, Northwestern University, Evanston, IL 60208–3120, USA
J. M. Ottino
Affiliation:
Department of Chemical Engineering, Robert R. McCormick School of Engineering and Applied Science, Northwestern University, Evanston, IL 60208–3120, USA
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Abstract

A complex Stokes flow has several cells, is subject to bifurcation, and its velocity field is, with rare exceptions, only available from numerical computations. We present experimental and computational studies of two new complex Stokes flows: a vortex mixing flow and multicell flows in slender cavities. We develop topological relations between the geometry of the flow domain and the family of physically realizable flows; we study bifurcations and symmetries, in particular to reveal how the forcing protocol's phase hides or reveals symmetries. Using a variety of dynamical tools, comparisons of boundary integral equation numerical computations to dye advection experiments are made throughout. Several findings challenge commonly accepted wisdom. For example, we show that higher-order periodic points can be more important than period-one points in establishing the advection template and extended regions of large stretching. We demonstrate also that a broad class of forcing functions produces the same qualitative mixing patterns. We experimentally verify the existence of potential mixing zones for adiabatic forcing and investigate the crossover from adiabatic to non-adiabatic behaviour. Finally, we use the entire array of tools to address an optimization problem for a complex flow. We conclude that none of the dynamical tools alone can successfully fulfil the role of a merit function; however, the collection of tools can be applied successively as a dynamical sieve to uncover a global optimum.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 269 , 25 June 1994 , pp. 199 - 246
Copyright
© 1994 Cambridge University Press

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References

Aref, H. 1984 Stirring by chaotic advection. J. Fluid Mech. 143, 1–21.Google Scholar
Aref, H. 1991 Chaotic advection of fluid particles. Phil. Trans. R. Soc. Lond. A 333, 273–281.Google Scholar
Ballal, B. Y. & Rivlin, R. S. 1976 Flow of a Newtonian fluid between eccentric rotating cylinders: inertial effects. Arch. Rat. Mech. Anal. 62, 237–294.Google Scholar
Beige, D., Leonard, A. & Wiggins, S. 1991 Chaotic transport in the homoclinic and heteroclinic tangle regions of quasiperiodically forced two-dimensional dynamical systems. Nonlinearity 4, 775–819.Google Scholar
Burggraf, O. R. 1966 Analytical and numerical studies of the structure of steady separated flows. J. Fluid Mech. 24, 113–151.Google Scholar
Camassa, R. & Wiggins, S. 1991 Chaotic advection in Rayleigh—Bénard flow. Phys. Rev. A 43, 774–797.Google Scholar
Chaiken, J., Chevray, R., Tabor, M. & Tan, Q. M. 1986 Experimental study of Lagrangian turbulence in a Stokes flow. Proc. R. Soc. Lond. A 408, 165–174.Google Scholar
Chien, W.-L., Rising, H. & Ottino, J. M. 1986 Laminar mixing and chaotic mixing in several cavity flows. J. Fluid Mech. 170, 355–377.Google Scholar
Flegg, H. G. 1974 From Geometry to Topology. The English University Press Ltd.
Franjione, J. G., Leong, C.-W. & Ottino, J. M. 1989 Symmetries within chaos: a route to effective mixing. Phys. Fluids A 1, 1772–1783.Google Scholar
Franjione, J. G. & Ottino, J. M. 1992 Symmetry concepts for geometric analysis of mixing flows. Phil. Trans. R. Soc. Lond. A 338, 301–323.Google Scholar
Guckenheimer, J. & Holmes, P. 1983 Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Field. Springer.
Higdon, J. J. L. 1985 Stokes flow in arbitrary two-dimensional domains: shear flow over ridges and cavities. J. Fluid Mech. 159, 194–226.Google Scholar
Hobson, D. 1993 An efficient method for computing invariant manifolds of planar maps. J. Comput. Phys. 104, 14–22.Google Scholar
Jana, S. C. & Ottino, J. M. 1992 Chaos-enhanced transport in cellular flows. Phil. Trans. R. Soc. Lond. A 338, 519–532.Google Scholar
Jana, S. C., Tjahjadi, M. & Ottino, J. M. 1994 Chaotic mixing of viscous fluids by periodic changes in geometry: the baffled cavity flow. AIChE J. (to appear).Google Scholar
Jeffrey, D. J. & Sherwood, J. D. 1980 Streamline patterns and eddies in low-Reynolds-number flow. J. Fluid Mech. 96, 315–334.Google Scholar
Kaper, T. J. & Wiggins, S. 1993 An analytical study of transport in Stokes flows exhibiting largescale chaos in the eccentric journal bearing. J. Fluid Mech. 253, 211–243.Google Scholar
Kim, S. & Karilla, S. J. 1991 Microhydrodynamics: Principles and Selected Applications. Butterworth-Heinemann.
Kusch, H. A. & Ottino, J. M. 1992 Experiments on mixing in continuous flows. J. Fluid Mech. 236, 319–348.Google Scholar
Leong, C. W. 1990 Chaotic mixing of viscous fluids in time-periodic cavity flows. PhD thesis, University of Massachusetts at Amherst.
Leong, C. W. & Ottino, J. M. 1989a Experiments on mixing due to chaotic advection in a cavity. J. Fluid Mech. 209, 463–499.Google Scholar
Leong, C. W. & Ottino, J. M. 1989b Gallery of fluid motion. Phys. Fluids A 1, 1441.Google Scholar
Ling, F. H. & Schmidt, G. 1992 Mixing windows in discontinuous cavity flows. Phys. Lett. A 165, 221–230.Google Scholar
Moffatt, H. K. 1964 Viscous and resistive eddies near a sharp corner. J. Fluid Mech. 18, 1–18.Google Scholar
Muzzio, F. J., Swanson, P. D. & Ottino, J. M. 1991 The statistics of stretching and stirring in chaotic flows. Phys. Fluids A 3, 822–834.Google Scholar
Ng, R. C.-Y. 1989 Semi-dilute polymer solutions in strong flows. PhD dissertation, California Institute of Technology.
Ottino, J. M. 1990 Mixing, chaotic advection, and turbulence. Ann. Rev. Fluid Mech. 22, 207–254.Google Scholar
Pan, F. & Acrivos, A. 1967 Steady flows in rectangular cavities. J. Fluid Mech. 28, 643–655.Google Scholar
Pozrikidis, C. 1992 Boundary Integral and Singularity Methods for Linearized Viscous Flow. 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
Swanson, P. D. 1991 Regular and chaotic mixing of viscous fluids in eccentric rotating cylinders. PhD dissertation, University of Massachusetts at Amherst.
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
Wannier, G. H. 1950 A contribution to the hydrodynamics of lubrication. Q. Appl. Maths. VIII, 1–32.Google Scholar
Wiggins, S. 1988 Global Bifurcations and Chaos. Springer.
Wiggins, S. 1992 Chaotic Transport in Dynamical Systems. Springer.