×

Mixing with piecewise isometries on a hemispherical shell. (English) Zbl 1375.37128

Summary: We introduce mixing with piecewise isometries (PWIs) on a hemispherical shell, which mimics features of mixing by cutting and shuffling in spherical shells half-filled with granular media. For each PWI, there is an inherent structure on the hemispherical shell known as the exceptional set \(E\), and a particular subset of \(E\), \(E_{+}\), provides insight into how the structure affects mixing. Computer simulations of PWIs are used to visualize mixing and approximations of \(E_{+}\) to demonstrate their connection. While initial conditions of unmixed materials add a layer of complexity, the inherent structure of \(E_{+}\) defines fundamental aspects of mixing by cutting and shuffling.{
©2016 American Institute of Physics}

MSC:

37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
37M05 Simulation of dynamical systems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Sturman, R., The role of discontinuities in mixing, Adv. Appl. Mech., 45, 51 (2012) · doi:10.1016/B978-0-12-380876-9.00002-1
[2] Juarez, G.; Christov, I. C.; Ottino, J. M.; Lueptow, R. M., Mixing by cutting and shuffling 3D granular flow in spherical tumblers, Chem. Eng. Sci., 73, 195 (2012) · doi:10.1016/j.ces.2012.01.044
[3] Boyer, S. E.; Elliott, D., Thrust systems, AAPG Bull., 66, 1196 (1982)
[4] Sturman, R.; Meier, S. W.; Ottino, J. M.; Wiggins, S., Linked twist map formalism in two and three dimensions applied to mixing in tumbled granular flows, J. Fluid Mech., 602, 129 (2008) · Zbl 1144.76057 · doi:10.1017/S002211200800075X
[5] Krotter, M. K.; Christov, I. C.; Ottino, J. M.; Lueptow, R. M., Cutting and shuffling a line segment: Mixing by interval exchange transformations, Int. J. Bifurcat. Chaos, 22, 1230041 (2012) · doi:10.1142/S0218127412300418
[6] Golomb, S. W., Permutations by cutting and shuffling, SIAM Rev., 3, 293 (1961) · Zbl 0104.00803 · doi:10.1137/1003059
[7] Keane, M., Interval exchange transformations, Math. Z., 141, 25 (1975) · Zbl 0278.28010 · doi:10.1007/BF01236981
[8] Keane, M., Non-ergodic interval exchange transformations, Isr. J. Math., 26, 188 (1977) · Zbl 0351.28012 · doi:10.1007/BF03007668
[9] Veech, W. A., Interval exchange transformations, J. Anal. Math., 33, 222 (1978) · Zbl 0455.28006 · doi:10.1007/BF02790174
[10] Katok, A., Interval exchange transformations and some special flows are not mixing, Isr. J. Math, 35, 301 (1980) · Zbl 0437.28009 · doi:10.1007/BF02760655
[11] Masur, H., Interval exchange transformations and measured foliations, Ann. Math., 115, 169 (1982) · Zbl 0497.28012 · doi:10.2307/1971341
[12] Aldous, D.; Diaconis, P., Shuffling cards and stopping times, Am. Math. Mon., 93, 333 (1986) · Zbl 0603.60006 · doi:10.2307/2323590
[13] Trefethen, L. N.; Trefethen, L. M., How many shuffles to randomize a deck of cards?, Proc. R. Soc. Lond. A, 456, 2561 (2000) · Zbl 0968.60080 · doi:10.1098/rspa.2000.0625
[14] Viana, M., Ergodic theory of interval exchange maps, Rev. Mat. Complut., 19, 7 (2006) · Zbl 1112.37003 · doi:10.5209/rev_REMA.2006.v19.n1.16621
[15] Avila, A.; Forni, G., Weak mixing for interval exchange transformations and translation flows, Ann. Math., 165, 637 (2007) · Zbl 1136.37003 · doi:10.4007/annals.2007.165.637
[16] Novak, C. F., Discontinuity-growth of interval-exchange maps, J. Mod. Dyn., 3, 379 (2009) · Zbl 1183.37077 · doi:10.3934/jmd.2009.3.379
[17] Hmili, H., Non topologically weakly mixing interval exchanges, Discret. Contin. Dyn. S., 27, 1079 (2010) · Zbl 1195.37025 · doi:10.3934/dcds.2010.27.1079
[18] Goetz, A., Piecewise isometries—An emerging area of dynamical systems, Trends in Mathematics: Fractals in Graz 2001, 135 (2003) · Zbl 1030.37009
[19] Goetz, A.; Poggiaspalla, G., Rotations by π/7, Nonlinearity, 17, 1787 (2004) · Zbl 1066.37028 · doi:10.1088/0951-7715/17/5/013
[20] Kahng, B., Singularities of two-dimensional invertible piecewise isometric dynamics, Chaos, 19, 023115 (2009) · Zbl 1309.37039 · doi:10.1063/1.3119464
[21] Lowenstein, J. H.; Vivaldi, F., Anomalous transport in a model of hamiltonian round-off, Nonlinearity, 11, 1321 (1998) · Zbl 0917.58035 · doi:10.1088/0951-7715/11/5/009
[22] Lowenstein, J. H.; Vivaldi, F., Embedding dynamics for round-off errors near a periodic orbit, Chaos, 10, 747 (2000) · Zbl 0969.37038 · doi:10.1063/1.1322027
[23] Kahng, B., Dynamics of symplectic piecewise affine elliptic rotation maps on tori, Ergod. Theor. Dyn. Syst., 22, 483 (2002) · Zbl 1079.37050 · doi:10.1017/S0143385702000238
[24] Kahng, B., The unique ergodic measure of the symmetric piecewise toral isometry of rotation angle θ = kπ/5 is the Hausdorff measure of its singular set, Dynam. Syst., 19, 245 (2004) · Zbl 1064.37020 · doi:10.1080/14689360410001729595
[25] Sturman, R.; Ottino, J. M.; Wiggins, S., The Mathematical Foundations of Mixing: The Linked Twist Map as a Paradigm in Applications: Micro to Macro, Fluids to Solids (2006) · Zbl 1111.37300
[26] Ottino, J. M.; Ranz, W. E.; Macosko, C. W., A framework for description of mechanical mixing of fluids, AIChE J., 27, 565 (1981) · doi:10.1002/aic.690270406
[27] Danckwerts, P. V., The definition and measurement of some characteristics of mixtures, Appl. Sci. Res., 3, 279 (1952) · doi:10.1007/BF03184936
[28] Ottino, J. M., Mixing, chaotic advection, and turbulence, Annu. Rev. Fluid Mech., 22, 207 (1990) · doi:10.1146/annurev.fl.22.010190.001231
[29] Schlick, C. P.; Christov, I. C.; Umbanhowar, P. B.; Ottino, J. M.; Lueptow, R. M., A mapping method for distributive mixing with diffusion: Interplay between chaos and diffusion in time-periodic sine flow, Phys. Fluids, 25, 052102 (2013) · Zbl 1315.76025 · doi:10.1063/1.4803897
[30] Fu, X.-C.; Duan, J., On global attractors for a class of nonhyperbolic piecewise affine maps, Physica D, 237, 3369 (2008) · Zbl 1153.37336 · doi:10.1016/j.physd.2008.07.012
[31] Juarez, G.; Lueptow, R. M.; Ottino, J. M.; Sturman, R.; Wiggins, S., Mixing by cutting and shuffling, Europhys. Lett., 91, 20003 (2010) · doi:10.1209/0295-5075/91/20003
[32] Meier, S. W.; Lueptow, R. M.; Ottino, J. M., A dynamical systems approach to mixing and segregation of granular materials in tumblers, Adv. Phys., 56, 757 (2007) · doi:10.1080/00018730701611677
[33] Christov, I. C.; Lueptow, R. M.; Ottino, J. M.; Sturman, R., A study in three-dimensional chaotic dynamics: Granular flow and transport in a bi-axial spherical tumbler, SIAM J. Appl. Dyn. Syst., 13, 901 (2014) · Zbl 1348.37056 · doi:10.1137/130934076
[34] Christov, I. C.; Ottino, J. M.; Lueptow, R. M., Streamline jumping: A mixing mechanism, Phys. Rev. E, 81, 046307 (2010) · doi:10.1103/PhysRevE.81.046307
[35] Christov, I. C.; Ottino, J. M.; Lueptow, R. M., Chaotic mixing via streamline jumping in quasi-two-dimensional tumbled granular flows, Chaos, 20, 023102 (2010) · doi:10.1063/1.3368695
[36] Christov, I. C.; Lueptow, R. M.; Ottino, J. M., Stretching and folding versus cutting and shuffling: An illustrated perspective on mixing and deformations of continua, Am. J. Phys., 79, 359 (2011) · doi:10.1119/1.3533213
[37] Goetz, A., Stability of piecewise rotations and affine maps, Nonlinearity, 14, 205 (2001) · Zbl 1031.37039 · doi:10.1088/0951-7715/14/2/302
[38] Ashwin, P.; Goetz, A., Invariant curves and explosion of periodic islands in systems of piecewise rotations, SIAM J. Appl. Dyn. Syst., 4, 437 (2005) · Zbl 1090.37029 · doi:10.1137/040605394
[39] Yu, M.; Umbanhowar, P. B.; Ottino, J. M.; Lueptow, R. M., Cutting and shuffling of a line segment: Effect of variation in cut location, Chaos · Zbl 1357.37004
[40] Rasband, S. N., Chaotic Dynamics of Nonlinear Systems (1990) · Zbl 0691.58004
[41] Ott, E., Chaos in Dynamical Systems (2002) · Zbl 1006.37001
[42] Strogatz, S. H., Nonlinear Dynamics and Chaos (2015) · Zbl 1343.37001
[43] Jana, S. C.; Metcalfe, G.; Ottino, J. M., Experimental and computational studies of mixing in complex Stokes flows: The vortex mixing flow and multicelluar cavity flows, J. Fluid Mech., 269, 199 (1994) · doi:10.1017/S0022112094001539
[44] Scott, A.; Holmes, C. A.; Milburn, G. J., Hamiltonian mappings and circle packing phase spaces, Physica D, 155, 34 (2001) · Zbl 0981.37017 · doi:10.1016/S0167-2789(01)00263-9
[45] Scott, A. J., Hamiltonian mappings and circle packing phase spaces: Numerical investigations, Physica D, 181, 45 (2003) · Zbl 1098.70519 · doi:10.1016/S0167-2789(03)00095-2
[46] Goetz, A., Dynamics of piecewise isometries, Illinois J. Math., 44, 465 (2000) · Zbl 0964.37009
[47] Adler, R.; Kitchens, B.; Tresser, C., Dynamics of non-ergodic piecewise affine maps of the torus, Ergod. Theor. Dyn. Syst., 21, 959 (2001) · Zbl 1055.37048 · doi:10.1017/S0143385701001468
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.