Journal articles: 'Chaotic stirring'

1

Rypina, Irina I., Lawrence J. Pratt, Julie Pullen, Julia Levin, and Arnold L. Gordon. "Chaotic Advection in an Archipelago*." Journal of Physical Oceanography 40, no. 9 (September 1, 2010): 1988–2006. http://dx.doi.org/10.1175/2010jpo4336.1.

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Abstract Techniques from dynamical systems theory have been applied to study horizontal stirring of fluid in the Philippine Archipelago. The authors’ analysis is based on velocity fields produced by two high-resolution (3 and 6 km) numerical models. Particular attention is paid to identifying robust surface flow patterns and associating them with dominant Lagrangian coherent structures (LCSs). A recurrent wind-driven dipole in the lee of the coastline is considered in detail. The associated LCSs form a template for stirring, exchange, and biological transport in and around the dipole. Chaotic advection is argued to provide a relevant framework for interpreting mesoscale horizontal stirring processes in an archipelago as a whole. Implications for the formation of filaments, the production of tracer variance, and the scale at which stirring leads to mixing are discussed in connection with an observed temperature record.

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Lekien, Francois, and Chad Coulliette. "Chaotic stirring in quasi-turbulent flows." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 365, no. 1861 (September 14, 2007): 3061–84. http://dx.doi.org/10.1098/rsta.2007.0020.

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Transport in laminar flows is governed by chaotic stirring and striation in long thin filaments. In turbulent flows, isotropic mixing dominates and tracers behave like stochastic variables. In this paper, we investigate the quasi-turbulent, intermediate regime where both chaotic stirring and turbulent mixing coexist. In these flows, the most common in nature, aperiodic Lagrangian coherent structures (LCSs) delineate particle transport and chaotic stirring. We review the recent developments in LCS theory and apply these techniques to measured surface currents in Monterey Bay, California. In the bay, LCSs can be used to optimize the release of drifting buoys or to minimize the impact of a coastal pollution source.

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Ridderinkhof, H., and J. T. F. Zimmerman. "Chaotic Stirring in a Tidal System." Science 258, no. 5085 (November 13, 1992): 1107–11. http://dx.doi.org/10.1126/science.258.5085.1107.

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Brown, Michael G., and Kevin B. Smith. "Ocean stirring and chaotic low‐order dynamics." Physics of Fluids A: Fluid Dynamics 3, no. 5 (May 1991): 1186–92. http://dx.doi.org/10.1063/1.858047.

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Pratt, L. J., I. I. Rypina, T. M. Özgökmen, P. Wang, H. Childs, and Y. Bebieva. "Chaotic advection in a steady, three-dimensional, Ekman-driven eddy." Journal of Fluid Mechanics 738 (December 5, 2013): 143–83. http://dx.doi.org/10.1017/jfm.2013.583.

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AbstractWe investigate and quantify stirring due to chaotic advection within a steady, three-dimensional, Ekman-driven, rotating cylinder flow. The flow field has vertical overturning and horizontal swirling motion, and is an idealization of motion observed in some ocean eddies. The flow is characterized by strong background rotation, and we explore variations in Ekman and Rossby numbers, $E$ and ${R}_{o} $, over ranges appropriate for the ocean mesoscale and submesoscale. A high-resolution spectral element model is used in conjunction with linear analytical theory, weakly nonlinear resonance analysis and a kinematic model in order to map out the barriers, manifolds, resonance layers and other objects that provide a template for chaotic stirring. As expected, chaos arises when a radially symmetric background state is perturbed by a symmetry-breaking disturbance. In the background state, each trajectory lives on a torus and some of the latter survive the perturbation and act as barriers to chaotic transport, a result consistent with an extension of the KAM theorem for three-dimensional, volume-preserving flow. For shallow eddies, where $E$ is $O(1)$, the flow is dominated by thin resonant layers sandwiched between KAM-type barriers, and the stirring rate is weak. On the other hand, eddies with moderately small $E$ experience thicker resonant layers, wider-spread chaos and much more rapid stirring. This trend reverses for sufficiently small $E$, corresponding to deep eddies, where the vertical rigidity imposed by strong rotation limits the stirring. The bulk stirring rate, estimated from a passive tracer release, confirms the non-monotonic variation in stirring rate with $E$. This result is shown to be consistent with linear Ekman layer theory in conjunction with a resonant width calculation and the Taylor–Proudman theorem. The theory is able to roughly predict the value of $E$ at which stirring is maximum. For large disturbances, the stirring rate becomes monotonic over the range of Ekman numbers explored. We also explore variation in the eddy aspect ratio.

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BOYLAND, PHILIP L., HASSAN AREF, and MARK A. STREMLER. "Topological fluid mechanics of stirring." Journal of Fluid Mechanics 403 (January 25, 2000): 277–304. http://dx.doi.org/10.1017/s0022112099007107.

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A new approach to regular and chaotic fluid advection is presented that utilizes the Thurston–Nielsen classification theorem. The prototypical two-dimensional problem of stirring by a finite number of stirrers confined to a disk of fluid is considered. The theory shows that for particular ‘stirring protocols’ a significant increase in complexity of the stirred motion – known as topological chaos – occurs when three or more stirrers are present and are moved about in certain ways. In this sense prior studies of chaotic advection with at most two stirrers, that were, furthermore, usually fixed in place and simply rotated about their axes, have been ‘too simple’. We set out the basic theory without proofs and demonstrate the applicability of several topological concepts to fluid stirring. A key role is played by the representation of a given stirring protocol as a braid in a (2+1)-dimensional space–time made up of the flow plane and a time axis perpendicular to it. A simple experiment in which a viscous liquid is stirred by three stirrers has been conducted and is used to illustrate the theory.

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7

Gilpin, William. "Cryptographic hashing using chaotic hydrodynamics." Proceedings of the National Academy of Sciences 115, no. 19 (April 23, 2018): 4869–74. http://dx.doi.org/10.1073/pnas.1721852115.

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Fluids may store and manipulate information, enabling complex applications ranging from digital logic gates to algorithmic self-assembly. While controllable hydrodynamic chaos has previously been observed in viscous fluids and harnessed for efficient mixing, its application to the manipulation of digital information has been sparsely investigated. We show that chaotic stirring of a viscous fluid naturally produces a characteristic signature of the stirring process in the arrangement of particles in the fluid, and that this signature directly satisfies the requirements for a cryptographic hash function. This includes strong divergence between similar stirring protocols’ hashes and avoidance of collisions (identical hashes from distinct stirs), which are facilitated by noninvertibility and a broad chaotic attractor that samples many points in the fluid domain. The hashing ability of the chaotic fluidic map implicates several unexpected mechanisms, including incomplete mixing at short time scales that produces a hyperuniform hash distribution. We investigate the dynamics of hashing using interparticle winding statistics, and find that hashing starts with large-scale winding of kinetically disjoint regions of the chaotic attractor, which gradually gives way to smaller scale braiding of single-particle trajectories. In addition to providing a physically motivated approach to implementing and analyzing deterministic chaotic maps for cryptographic applications, we anticipate that our approach has applications in microfluidic proof-of-work systems and characterizing large-scale turbulent flows from sparse tracer data.

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Abraham, Edward R., and Melissa M. Bowen. "Chaotic stirring by a mesoscale surface-ocean flow." Chaos: An Interdisciplinary Journal of Nonlinear Science 12, no. 2 (June 2002): 373–81. http://dx.doi.org/10.1063/1.1481615.

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9

Gleeson, James P. "Transient micromixing: Examples of laminar and chaotic stirring." Physics of Fluids 17, no. 10 (2005): 100614. http://dx.doi.org/10.1063/1.1928627.

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10

Kweon Suh, Yong. "A Chaotic Stirring by an Oscillating Point Vortex." Journal of the Physical Society of Japan 60, no. 3 (March 15, 1991): 896–906. http://dx.doi.org/10.1143/jpsj.60.896.

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11

Fan, Yuewei, Shibo Wang, Hua Wang, Jianxin Xu, Qingtai Xiao, and Yonggang Wei. "Formation mechanism and chaotic reinforcement elimination of the mechanical stirring isolated mixed region." International Journal of Chemical Reactor Engineering 19, no. 3 (February 18, 2021): 239–50. http://dx.doi.org/10.1515/ijcre-2020-0213.

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Abstract The isolated mixed region (IMR) is gradually formed during stirring and reduces the mixing efficiency. The unsteady-state formation process of the IMR was modeled and its formation mechanism was analyzed. The rotating frequency of the impeller was optimized using the chaos mathematical theory to improve the stirring efficiency without increasing the power consumption. The calculated results demonstrate that the IMR is a coherent structure, and its formation process is based on the free shear effect of the mixed layer. The chaotic stirring method can accelerate the momentum dissipation process by 37% by eliminating the IMR, and increase the speed by up to 31%. Therefore, chaotic mixing can eliminate the IMR in a shorter time and lower the power consumption.

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12

Muzzio, F. J., P. D. Swanson, and J. M. Ottino. "The statistics of stretching and stirring in chaotic flows." Physics of Fluids A: Fluid Dynamics 3, no. 5 (May 1991): 822–34. http://dx.doi.org/10.1063/1.858013.

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13

Beerens, S. P., H. Ridderinkhof, and J. T. F. Zimmerman. "An analytical study of chaotic stirring in tidal areas." Chaos, Solitons & Fractals 4, no. 6 (June 1994): 1011–29. http://dx.doi.org/10.1016/0960-0779(94)90136-8.

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14

Brett, Genevieve Jay, Larry Pratt, Irina Rypina, and Peng Wang. "Competition between chaotic advection and diffusion: stirring and mixing in a 3-D eddy model." Nonlinear Processes in Geophysics 26, no. 2 (April 5, 2019): 37–60. http://dx.doi.org/10.5194/npg-26-37-2019.

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Abstract. The importance of chaotic advection relative to turbulent diffusion is investigated in an idealized model of a 3-D swirling and overturning ocean eddy. Various measures of stirring and mixing are examined in order to determine when and where chaotic advection is relevant. Turbulent diffusion is alternatively represented by (1) an explicit, observation-based, scale-dependent diffusivity, (2) stochastic noise, added to a deterministic velocity field, or (3) explicit and implicit diffusion in a spectral numerical model of the Navier–Stokes equations. Lagrangian chaos in our model occurs only within distinct regions of the eddy, including a large chaotic “sea” that fills much of the volume near the perimeter and central axis of the eddy and much smaller “resonant” bands. The size and distribution of these regions depend on factors such as the degree of axial asymmetry of the eddy and the Ekman number. The relative importance of chaotic advection and turbulent diffusion within the chaotic regions is quantified using three measures: the Lagrangian Batchelor scale, the rate of dispersal of closely spaced fluid parcels, and the Nakamura effective diffusivity. The role of chaotic advection in the stirring of a passive tracer is generally found to be most important within the larger chaotic seas, at intermediate times, with small diffusivities, and for eddies with strong asymmetry. In contrast, in thin chaotic regions, turbulent diffusion at oceanographically relevant rates is at least as important as chaotic advection. Future work should address anisotropic and spatially varying representations of turbulent diffusion for more realistic models.

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15

Jones, Scott W., Oran M. Thomas, and Hassan Aref. "Chaotic advection by laminar flow in a twisted pipe." Journal of Fluid Mechanics 209 (December 1989): 335–57. http://dx.doi.org/10.1017/s0022112089003137.

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The appearance of chaotic particle trajectories in steady, laminar, incompressible flow through a twisted pipe of circular cross-section is demonstrated using standard dynamical systems diagnostics and a model flow based on Dean's perturbation solutions. A study is performed to determine the parameters that control fluid stirring in this mixing device that has no moving parts. Insight into the chaotic dynamics are provided by a simple one-dimensional map of the pipe boundary onto itself. The results of numerical experiments illustrating the stretching of material lines, stirring of blobs of material, and the three-dimensional trajectories of fluid particles are presented. Finally, enhanced longitudinal particle dispersal due to the coupling between chaos in the transverse direction and the non-uniform longitudinal transport of particles is shown.

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16

GAO, Dianrong. "PIV EXPERIMENTAL INVESTIGATION OF CHAOTIC MIXING USING NON-CONSTANT SPEED STIRRING." Chinese Journal of Mechanical Engineering 42, no. 08 (2006): 44. http://dx.doi.org/10.3901/jme.2006.08.044.

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17

Leibensperger, Eric M., and R. Alan Plumb. "Effective Diffusivity in Baroclinic Flow." Journal of the Atmospheric Sciences 71, no. 3 (February 27, 2014): 972–84. http://dx.doi.org/10.1175/jas-d-13-0217.1.

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Abstract Large-scale chaotic stirring stretches tracer contours into filaments containing fine spatial scales until small-scale diffusive processes dissipate tracer variance. Quantification of tracer transport in such circumstances is possible through the use of Nakamura’s “effective diffusivity” diagnostics, which make clear the controlling role of stirring, rather than small-scale dissipation, in large-scale transport. Existing theory of effective diffusivity is based on a layerwise approach, in which tracer variance is presumed to cascade via horizontal (or isentropic) stirring to small-scale horizontal (or isentropic) diffusion. In most geophysical flows of interest, however, baroclinic shear will tilt stirred filamentary structures into almost-horizontal sheets, in which case the thinnest dimension is vertical; accordingly, it will be vertical (or diabatic) diffusion that provides the ultimate dissipation of variance. Here new theoretical developments define effective diffusivity in such flows. In the frequently relevant case of isentropic stirring, it is shown that the theory is, in most respects, unchanged from the case of isentropic diffusion: effective isentropic diffusivity is controlled by the isentropic stirring and, it is argued, largely independent of the nature of the ultimate dissipation. Diabatic diffusion is not amplified by the stirring, although it can be modestly enhanced through eddy modulation of static stability. These characteristics are illustrated in numerical simulations of a stratospheric flow; in regions of strong stirring, the theoretical predictions are well supported, but agreement is less good where stirring is weaker.

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Mukiibi, Daniel, Gualtiero Badin, and Nuno Serra. "Three-Dimensional Chaotic Advection by Mixed Layer Baroclinic Instabilities." Journal of Physical Oceanography 46, no. 5 (May 2016): 1509–29. http://dx.doi.org/10.1175/jpo-d-15-0121.1.

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AbstractThree-dimensional (3D) finite-time Lyapunov exponents (FTLEs) are computed from numerical simulations of a freely evolving mixed layer (ML) front in a zonal channel undergoing baroclinic instability. The 3D FTLEs show a complex structure, with features that are less defined than the two-dimensional (2D) FTLEs, suggesting that stirring is not confined to the edges of vortices and along filaments and posing significant consequences on mixing. The magnitude of the FTLEs is observed to be strongly determined by the vertical shear. A scaling law relating the local FTLEs and the nonlocal density contrast used to initialize the ML front is derived assuming thermal wind balance. The scaling law only converges to the values found from the simulations within the pycnocline, while it displays differences within the ML, where the instabilities show a large ageostrophic component. The probability distribution functions of 2D and 3D FTLEs are found to be non-Gaussian at all depths. In the ML, the FTLEs wavenumber spectra display −1 slopes, while in the pycnocline, the FTLEs wavenumber spectra display −2 slopes, corresponding to frontal dynamics. Close to the surface, the geodesic Lagrangian coherent structures (LCSs) reveal a complex stirring structure, with elliptic structures detaching from the frontal region. In the pycnocline, LCSs are able to detect filamentary structures that are not captured by the Eulerian fields.

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Qian, Shizhi, and Haim H. Bau. "Theoretical investigation of electro-osmotic flows and chaotic stirring in rectangular cavities." Applied Mathematical Modelling 29, no. 8 (August 2005): 726–53. http://dx.doi.org/10.1016/j.apm.2004.10.006.

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20

Hwu, Tzong-Yih. "Chaotic stirring in a new type of mixer with rotating rigid blades." European Journal of Mechanics - B/Fluids 27, no. 3 (May 2008): 239–50. http://dx.doi.org/10.1016/j.euromechflu.2007.05.002.

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Smrčinová, Miroslava, Preben Graae Sørensen, Július Krempasky, and Peter Ballo. "Chaotic Oscillations in a Chloroplast System Under Constant Illumination." International Journal of Bifurcation and Chaos 08, no. 12 (December 1998): 2467–70. http://dx.doi.org/10.1142/s0218127498001984.

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We show that nonlinear, aperiodic oscillatory behavior can be observed in a chloroplast system by measuring the absorption of light. This behavior depends on illumination of the system with light which is absorbed by chlorophyll. The oscillations can be stopped by addition of acetone indicating that chemical reactions are involved. These observations confirm the role of photosynthesis. The effect of stirring and the statistics of the oscillating part of the spectrum suggest that the buoyancy of oxygen bubbles formed during photosynthesis is an important art of the mechanism.

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Metcalfe, Guy, Daniel Lester, Alison Ord, Pandurang Kulkarni, Murray Rudman, Mike Trefry, Bruce Hobbs, Klaus Regenaur-Lieb, and Jeffery Morris. "An experimental and theoretical study of the mixing characteristics of a periodically reoriented irrotational flow." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 368, no. 1918 (May 13, 2010): 2147–62. http://dx.doi.org/10.1098/rsta.2010.0037.

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The minimum-energy method to generate chaotic advection should be to use an irrotational flow. However, irrotational flows have no saddle connections to perturb in order to generate chaotic orbits. To the early work of Jones & Aref (Jones & Aref 1988 Phys. Fluids 31 , 469–485 ( doi:10.1063/1.866828 )) on potential flow chaos, we add periodic reorientation to generate chaotic advection with irrotational experimental flows. Our experimental irrotational flow is a dipole potential flow in a disc-shaped Hele-Shaw cell called the rotated potential mixing flow; it leads to chaotic advection and transport in the disc. We derive an analytical map for the flow. This is a partially open flow, in which parts of the flow remain in the cell forever, and parts of it pass through with residence-time and exit-time distributions that have self-similar features in the control parameter space of the stirring. The theory compares well with the experiment.

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Metcalfe, Guy, Daniel Lester, Alison Ord, Pandurang Kulkarni, Mike Trefry, Bruce E. Hobbs, Klaus Regenaur-Lieb, and Jeffery Morris. "A partially open porous media flow with chaotic advection: towards a model of coupled fields." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 368, no. 1910 (January 13, 2010): 217–30. http://dx.doi.org/10.1098/rsta.2009.0198.

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In nature, dissipative fluxes of fluid, heat and/or reacting species couple to each other and may also couple to deformation of a surrounding porous matrix. We use the well-known analogy of Hele–Shaw flow to Darcy flow to make a model porous medium with porosity proportional to local cell height. Time- and space-varying fluid injection from multiple source/sink wells lets us create many different kinds of chaotic flows and chemical concentration patterns. Results of an initial time-dependent potential flow model illustrate that this is a partially open flow, in which parts of the material transported by the flow remain in the cell forever and parts pass through with residence time and exit time distributions that have self-similar features in the control parameter space of the stirring. We derive analytically the existence boundary in stirring control parameter space between where isolated fluid regions can and cannot remain forever in the open flow. Experiments confirm the predictions.

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Fan, Yuewei, Chunlin Li, Shibo Wang, Hua Wang, Yonggang Wei, Jianxin Xu, and Qingtai Xiao. "Enhancement of mixing efficiency in mechanical stirring reactors via chaotic stirring techniques: Application to the treatment of zinc-containing solid waste." Chemical Engineering Science 249 (February 2022): 117367. http://dx.doi.org/10.1016/j.ces.2021.117367.

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Lukovich, Jennifer Verlaine, and Theodore G. Shepherd. "Stirring and Mixing in Two-Dimensional Divergent Flow." Journal of the Atmospheric Sciences 62, no. 11 (November 1, 2005): 3933–54. http://dx.doi.org/10.1175/jas3580.1.

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Abstract While stirring and mixing properties in the stratosphere are reasonably well understood in the context of balanced (slow) dynamics, as is evidenced in numerous studies of chaotic advection, the strongly enhanced presence of high-frequency gravity waves in the mesosphere gives rise to a significant unbalanced (fast) component to the flow. The present investigation analyses result from two idealized shallow-water numerical simulations representative of stratospheric and mesospheric dynamics on a quasi-horizontal isentropic surface. A generalization of the Hua–Klein Eulerian diagnostic to divergent flow reveals that velocity gradients are strongly influenced by the unbalanced component of the flow. The Lagrangian diagnostic of patchiness nevertheless demonstrates the persistence of coherent features in the zonal component of the flow, in contrast to the destruction of coherent features in the meridional component. Single-particle statistics demonstrate t2 scaling for both the stratospheric and mesospheric regimes in the case of zonal dispersion, and distinctive scaling laws for the two regimes in the case of meridional dispersion. This is in contrast to two-particle statistics, which in the mesospheric (unbalanced) regime demonstrate a more rapid approach to Richardson’s t3 law in the case of zonal dispersion and is evidence of enhanced meridional dispersion.

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Aboelkassem, Yasser. "Chaotic mixing by oscillating a Stokeslet in a circular Hele-Shaw microfluidic device." Mathematics of Quantum Technologies 5, no. 1 (January 1, 2016): 1–8. http://dx.doi.org/10.1515/nsmmt-2016-0001.

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AbstractChaotic mixing by oscillating a Stokeslet in a circular Hele-Shaw microffluidic device is presented in this article. Mathematical modeling for the induced flow motions by moving a Stokeslet along the x-axis is derived using Fourier expansion method. The solution is formulated in terms of the velocity stream function. The model is then used to explore different stirring dynamics as function of the Stokeslet parameters. For instance, the effects of using various oscillation amplitudes and force strengths are investigated. Mixing patterns using Poincaré maps are obtained numerically and have been used to characterize the mixing efficiency. Results have shown that, for a given Stokeslet’s strength, efficient mixing can be obtained when small oscillation amplitudes are used. The present mixing platform is expected to be useful for many of biomicrofluidic applications.

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Beron-Vera, F. J., and M. J. Olascoaga. "An Assessment of the Importance of Chaotic Stirring and Turbulent Mixing on the West Florida Shelf." Journal of Physical Oceanography 39, no. 7 (July 1, 2009): 1743–55. http://dx.doi.org/10.1175/2009jpo4046.1.

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Abstract Application of dynamical systems tools has recently revealed in surface ocean currents produced by a Hybrid-Coordinate Ocean Model (HYCOM) simulation the presence of a persistent large-scale Lagrangian coherent structure (LCS) on the southern portion of the west Florida shelf (WFS). Consistent with satellite-tracked drifter trajectories, this LCS constitutes a cross-shelf barrier for the lateral transport of passive tracers. Because of the constraints that the above LCS, as well as smaller-scale LCSs lying shoreside, can impose on pollutant dispersal and its potentially very important biological consequences, a study was carried out on the nature of the surface ocean Lagrangian motion on the WFS. The analysis is based on the same simulated surface ocean velocity field that has been able to sustain the aforementioned persistent cross-shelf transport barrier. Examination of several diagnostics suggests that chaotic stirring dominates over turbulent mixing on time scales of up to two months or so. More specifically, it is found on those time scales that tracer evolution at a given length scale is governed to a nonnegligible extent by coarser-scale velocity field features, fluid particle dispersion is spatially inhomogeneous, and the Lagrangian evolution is more irregular than the driving Eulerian flow.

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Strizhak, Peter E. "Stirring-induced bifurcation driven by the chaotic regime in the Belousov—Zhabotinsky reaction in a CSTR." Chemical Physics Letters 243, no. 5-6 (September 1995): 540–44. http://dx.doi.org/10.1016/0009-2614(95)00883-6.

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RIZZI, F., and L. CORTELEZZI. "Stirring, stretching and transport generated by a pair of like-signed vortices." Journal of Fluid Mechanics 674 (March 30, 2011): 244–80. http://dx.doi.org/10.1017/s002211201000652x.

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We consider a pair of like-signed, initially elliptical vortices with uniform vorticity distribution embedded in an incompressible, inviscid fluid occupying a two-dimensional, infinite domain. We characterize this finite-time, aperiodic, dynamical system in terms of its fixed points and separatrices, which divide the flow into inner core, inner recirculation, outer recirculation regions and outer flow. We numerically simulate the time evolution of the vortex pair using a contour dynamics algorithm. The rotational and co-rotational motion of the vortices perturb the separatrices, which undergo to deformations, yielding a tangle whose complexity increases as the amplitude of the perturbation increases. We analyse the dynamics of the tangle and explain the transport of fluid between different regions. We use two diagnostics to quantify stirring: stretching of the interface and the mix-norm. These two diagnostics characterize stirring in contradicting ways and present different sensitivity to the parameters considered. We find that stretching is dominated by the chaotic advection induced within the inner core and inner recirculation regions, whereas the mix-norm is dominated by the laminar transport induced within the outer recirculation regions. For pairs of vortices of small aspect ratio, stretching is piecewise linear and the mix-norm does not decrease monotonically. We show that these two effects are strongly coupled and synchronized with the rotational motion of the vortices. Since the nominal domain is unbounded, we quantify stirring on three concentric, circular domains. One domain nearly encloses the outer separatrices of the vortex pair, one is smaller and one larger than the first one. We show that the mix-norm is very sensitive to the size of the domain, while stretching is not. To quantify the sensitivity of stirring to the geometry of the initial concentration field, we consider, as an initial scalar field, two concentrations delimited by a straight-line interface of adjustable orientation. We show that the interface passing through the centroids of the vortices is the one most efficiently stretched, while the initial concentration field with an orthogonal interface is the most efficiently stirred. Finally, we investigate the effects of the angular impulse on the stirring performance of the vortex pair. Stretching is very sensitive to the angular impulse, while the mix-norm is not. We show that there is a value of the angular impulse which maximizes stretching and argue that this is due to two competing mechanisms.

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Mostefa, Telha, Aissaoui Djamel Eddine, Naas Toufik Tayeb, Shakhawat Hossain, Arifur Rahman, Bachiri Mohamed, and Kwang-Yong Kim. "Kinematic Properties of a Twisted Double Planetary Chaotic Mixer: A Three-Dimensional Numerical Investigation." Micromachines 13, no. 9 (September 17, 2022): 1545. http://dx.doi.org/10.3390/mi13091545.

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In this study, a numerical investigation based on the CFD method is carried out to study the unsteady laminar flow of Newtonian fluid with a high viscosity in a three-dimensional simulation of a twisted double planetary mixer, which is composed of two agitating rods inside a moving tank. The considered stirring protocol is a “Continuous sine squared motion” by using the dynamic mesh model and user-defined functions (UDFs)to define the velocity profiles. The chaotic advection is obtained in our active mixers by the temporal modulation of rotational velocities of the moving walls in order to enhance the mixing of the fluid for a low Reynolds number and a high Peclet number. For this goal, we applied the Poincaré section and Lyapunov exponent as reliable mathematic tools for checking mixing quality by tracking a number of massless particles inside the fluid domain. Additionally, we investigated the development of fluid kinematics proprieties, such as vorticity, helicity, strain rate and elongation rate, at various time periods in order to view the impact of temporal modulation on the flow properties. The results of the mentioned simulation showed that it is possible to obtain a chaotic advection after a relatively short time, which can deeply enhance mixing fluid efficiency.

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Du, Kai, Jingni Song, Weiyu Liu, Ye Tao, and Yukun Ren. "Multifrequency Induced-Charge Electroosmosis." Micromachines 10, no. 7 (July 3, 2019): 447. http://dx.doi.org/10.3390/mi10070447.

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We present herein a unique concept of multifrequency induced-charge electroosmosis (MICEO) actuated directly on driving electrode arrays, for highly-efficient simultaneous transport and convective mixing of fluidic samples in microscale ducts. MICEO delicately combines transversal AC electroosmotic vortex flow, and axial traveling-wave electroosmotic pump motion under external dual-Fourier-mode AC electric fields. The synthetic flow field associated with MICEO is mathematically analyzed under thin layer limit, and the particle tracing experiment with a special powering technique validates the effectiveness of this physical phenomenon. Meanwhile, the simulation results with a full-scale 3D computation model demonstrate its robust dual-functionality in inducing fully-automated analyte transport and chaotic stirring in a straight fluidic channel embedding double-sided quarter-phase discrete electrode arrays. Our physical demonstration with multifrequency signal control on nonlinear electroosmosis provides invaluable references for innovative designs of multifunctional on-chip analytical platforms in modern microfluidic systems.

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Koshel, Konstantin, Eugene Ryzhov, and Xavier Carton. "Vortex Interactions Subjected to Deformation Flows: A Review." Fluids 4, no. 1 (January 18, 2019): 14. http://dx.doi.org/10.3390/fluids4010014.

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Deformation flows are the flows incorporating shear, strain and rotational components. These flows are ubiquitous in the geophysical flows, such as the ocean and atmosphere. They appear near almost any salience, such as isolated coherent structures (vortices and jets) and various fixed obstacles (submerged obstacles and continental boundaries). Fluid structures subject to such deformation flows may exhibit drastic changes in motion. In this review paper, we focus on the motion of a small number of coherent vortices embedded in deformation flows. Problems involving isolated one and two vortices are addressed. When considering a single-vortex problem, the main focus is on the evolution of the vortex boundary and its influence on the passive scalar motion. Two vortex problems are addressed with the use of point vortex models, and the resulting stirring patterns of neighbouring scalars are studied by a combination of numerical and analytical methods from the dynamical system theory. Many dynamical effects are reviewed with emphasis on the emergence of chaotic motion of the vortex phase trajectories and the scalars in their immediate vicinity.

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Kim, Ho Jun, and Ali Beskok. "Numerical Modeling of Chaotic Mixing in Electroosmotically Stirred Continuous Flow Mixers." Journal of Heat Transfer 131, no. 9 (June 24, 2009). http://dx.doi.org/10.1115/1.3139109.

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We present numerical studies of particle dispersion and species mixing in a ζ potential patterned straight microchannel. A continuous flow is generated by superposition of a steady pressure-driven flow and time-periodic electroosmotic flow induced by a streamwise ac electric field. ζ potential patterns are placed critically in the channel to achieve spatially asymmetric time-dependent flow fields that lead to chaotic stirring. Parametric studies are performed as a function of the Strouhal number (normalized ac frequency), while the mixer geometry, ratio of the Poiseuille flow and electroosmotic velocities, and the flow kinematics (Reynolds number) are kept constant. Lagrangian particle tracking is employed for observations of particle dispersion. Poincaré sections are constructed to identify the chaotic and regular zones in the mixer. Filament stretching and the probability density function of the stretching field are utilized to quantify the “locally optimum” stirring conditions and to demonstrate the statistical behavior of fully and partially chaotic flows. Numerical solutions of the species transport equation are performed as a function of the Peclet number (Pe) at fixed kinematic conditions. Mixing efficiency is quantified using the mixing index, based on standard deviation of the scalar species distribution. The mixing length (lm) is characterized as a function of the Peclet number and lm∝ln(Pe) scaling is observed for the fully chaotic flow case. Objectives of this study include the presentation and characterization of the new continuous flow mixer concept and the demonstration of the Lagrangian-based particle tracking tools for quantification of chaotic strength and stirring efficiency in continuous flow systems.

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Gepner, S. W., and J. M. Floryan. "Use of Surface Corrugations for Energy-Efficient Chaotic Stirring in Low Reynolds Number Flows." Scientific Reports 10, no. 1 (June 17, 2020). http://dx.doi.org/10.1038/s41598-020-66800-5.

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Rao, Pradeep, Andrew Duggleby, and Mark A. Stremler. "Mixing Analysis in a Lid-Driven Cavity Flow at Finite Reynolds Numbers." Journal of Fluids Engineering 134, no. 4 (April 1, 2012). http://dx.doi.org/10.1115/1.4006361.

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The influence of inertial effects on chaotic advection and mixing is investigated for a two-dimensional, time-dependent lid-driven cavity flow. Previous work shows that this flow exhibits exponential stretching and folding of material lines due to the presence of figure-eight stirring patterns in the creeping flow regime. The high sensitivity to initial conditions and the exponential growth of errors in chaotic flows necessitate an accurate solution of the flow in order to calculate metrics based on Lagrangian particle tracking. The streamfunction-vorticity formulation of the Navier-Stokes equations is solved using a Fourier-Chebyshev spectral method, providing the necessary exponential convergence and machine-precision accuracy. Poincaré sections and mixing measures are used to analyze chaotic advection and quantify the mixing efficiency. The calculated mixing characteristics are almost identical for Re ≤ 1. For the time range investigated, the best mixing in this system is observed for Re = 10. Interestingly, increasing the Reynolds number to the range 10 < Re ≤ 100 results in an observed decrease in mixing efficacy.

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Kailasham, R., and Aditya S. Khair. "Dynamics of forced and unforced autophoretic particles." Journal of Fluid Mechanics 948 (September 14, 2022). http://dx.doi.org/10.1017/jfm.2022.728.

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Chemically active, or autophoretic, particles that isotropically emit or absorb solute molecules undergo spontaneous self-propulsion when their activity is increased beyond a critical Péclet number ( $Pe$ ). Here, we conduct numerical computations, using a spectral element based method, of a rigid, spherical autophoretic particle in unsteady rectilinear translation. The particle can be freely suspended (or ‘unforced’) or subject to an external force field (or ‘forced’). The motion of an unforced particle progresses through four regimes as $Pe$ is increased: quiescent, steady, stirring and chaos. The particle is stationary in the quiescent regime, and the solute profile is isotropic about the particle. At $Pe=4$ the fore–aft symmetry in the solute profile is broken, resulting in its steady self-propulsion. Our computations indicate that the self-propulsion speed scales linearly with $Pe-4$ near the onset of self-propulsion, as has been predicted in previous studies. A further increase in $Pe$ gives rise to the stirring regime at $Pe\approx 27$ , where the fluid undergoes recirculation, while the particle remains essentially stationary. As $Pe$ is increased even further, the particle dynamics is marked by chaotic oscillations at $Pe\approx 55$ and higher, which we characterize in terms of the mean square displacement and velocity autocorrelation of the particle. Our results for an autophoretic particle under a weak external force are in good agreement with recent asymptotic predictions (Saha, Yariv & Schnitzer, J. Fluid Mech., vol. 916, 2021, p. A47). Additionally, we demonstrate that the strength and temporal scheduling of the external force may be tuned to modulate the chaotic dynamics at large $Pe$ .

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Zhang, Lian, Kai Yang, Meng Li, Qingtai Xiao, and Hua Wang. "Chaotic characterization of macromixing effect in a gas–liquid stirring system using modified 0–1 test." Canadian Journal of Chemical Engineering, April 22, 2021. http://dx.doi.org/10.1002/cjce.24100.

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Journal articles on the topic 'Chaotic stirring'

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