Topological chaos in inviscid and viscous mixers

Topological chaos may be used to generate highly effective laminar mixing in a simple batch stirring device. Boyland, Aref & Stremler (2000) have computed a material stretch rate that holds in a chaotic flow, provided it has appropriate topological properties, irrespective of the details of the flow. Their theoretical approach, while widely applicable, cannot predict the size of the region in which this stretch rate is achieved. Here, we present numerical simulations to support the observation of Boyland et al. that the region of high stretch is comparable with that through which the stirring elements move during operation of the device. We describe a fast technique for computing the velocity field for either inviscid, irrotational or highly viscous flow, which enables accurate numerical simulation of dye advection. We calculate material stretch rates, and find close agreement with those of Boyland et al., irrespective of whether the fluid is modelled as inviscid or viscous, even though there are significant differences between the flow fields generated in the two cases.

[1]  T. Mullin,et al.  Numerical and experimental characterization of a family of two–roll–mill flows , 2003, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[2]  Mark A. Stremler,et al.  Topological fluid mechanics of stirring , 2000, Journal of Fluid Mechanics.

[3]  M. Pollicott,et al.  Dynamical Systems and Ergodic Theory , 1998 .

[4]  Mitchell A. Berger,et al.  Topological Ideas and Fluid Mechanics , 1996 .

[5]  Metcalfe,et al.  Autocatalytic processes in mixing flows. , 1994, Physical review letters.

[6]  F. Muzzio,et al.  The intermaterial area density generated by time- and spatially periodic 2D chaotic flows , 2000 .

[7]  S. Cox,et al.  Lamellar modelling of reaction, diffusion and mixing in a two-dimensional flow , 1998 .

[8]  Ottino,et al.  Diffusion and reaction in a lamellar system: Self-similarity with finite rates of reaction. , 1990, Physical review. A, Atomic, molecular, and optical physics.

[9]  A. Vikhansky,et al.  Simulation of topological chaos in laminar flows. , 2004, Chaos.

[10]  Hassan Aref,et al.  Chaotic advection by laminar flow in a twisted pipe , 1989, Journal of Fluid Mechanics.

[11]  Julio M. Ottino,et al.  Experiments on mixing in continuous chaotic flows , 1992, Journal of Fluid Mechanics.

[12]  S. Cox,et al.  The Influence of a Lamellar Structure upon the Yield of a Chemical Reaction , 2000 .

[13]  M. Giona,et al.  Global geometry and coarse-grained formulation of the evolution of pointwise intermaterial interface measure in chaotic flows , 2001 .

[14]  Julio M. Ottino,et al.  Experimental and computational studies of mixing in complex Stokes flows: the vortex mixing flow and multicellular cavity flows , 1994, Journal of Fluid Mechanics.

[15]  Hsueh-Chia Chang,et al.  Effect of chaotic interfacial stretching on bimolecular chemical reaction in helical-coil reactors , 1996 .

[16]  Ottino,et al.  Dynamics of a lamellar system with diffusion and reaction: Scaling analysis and global kinetics. , 1989, Physical review. A, General physics.

[17]  V. Meleshkoa,et al.  Three-dimensional mixing in Stokes flow : the partitioned pipe mixer problem revisited , 1999 .

[18]  Julio M. Ottino,et al.  The statistics of stretching and stirring in chaotic flows , 1991 .

[19]  Nicholas B. Tufillaro,et al.  Experimental approach to nonlinear dynamics and chaos , 1992, Studies in nonlinearity.

[20]  Lloyd N. Trefethen,et al.  Schwarz-Christoffel Mapping , 2002 .

[21]  J. Andre,et al.  On the solution of Stokes’ equations between confocal ellipses , 1994 .

[22]  P. Boyland.  Fluid Mechanics and Mathematical Structures , 2001 .

[23]  Chaotic mixing and heat transfer between confocal ellipses: Experimental and numerical results , 1996 .

[24]  J. Butler,et al.  Effect of convective stretching and folding on aerosol mixing deep in the lung, assessed by approximate entropy. , 1997, Journal of applied physiology.

[25]  Hassan Aref,et al.  Chaotic advection in a Stokes flow , 1986 .

[26]  Hassan Aref,et al.  Topological fluid mechanics of point vortex motions , 1999 .

[27]  R. A. Frazer.  On the Motion of Circular Cylinders in a Viscous Fluid , 1926 .

[28]  D. Acheson.  Elementary Fluid Dynamics , 1990 .

[29]  G. G. Peters,et al.  Mixing in Stokes flow in an annular wedge cavity , 1999 .

[30]  Jozef L. Kokini,et al.  Estimation and prediction of shear rate distribution as a model mixer , 2000 .

[31]  S. Cox,et al.  Stokes flow in a mixer with changing geometry , 2001 .

[32]  Circle theorems for steady Stokes flow , 1989 .

[33]  D. A. Zumbrunnen,et al.  Novel sub-micron highly multi-layered polymer films formed by continuous flow chaotic mixing , 2001 .

[34]  Scott W. Jones,et al.  Enhanced separation of diffusing particles by chaotic advection , 1989 .

[35]  J Christensen,et al.  Fluid Mechanics of the Duodenum , 1980 .

[36]  Renzo L. Ricca,et al.  An introduction to the geometry and topology of fluid flows , 2001 .