Topological chaos and mixing in a three-dimensional channel flow

Chen, Jie; Stremler, Mark A.

doi:10.1063/1.3076247

Cited by 13 publications

(11 citation statements)

References 17 publications

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“…In dynamical systems applications, these surfaces are typically two-dimensional physical space and the maps act in time, but under certain conditions maps can also be constructed in three-dimensional space. 32 An isotopy class consists of all the topologically equivalent diffeomorphisms that can be mapped to each other through a "fixed stirrer" diffeomorphism in which the punctures are fixed relative to each other during the mapping. According to the TNCT, each isotopy class contains a representative homeomorphism U, the Thurston-Nielsen (TN) representative, that is one of three possible types: finite order (FO), pseudo-Anosov (PA), or reducible.…”

Section: -21mentioning

confidence: 99%

Topological chaos, braiding and bifurcation of almost-cyclic sets

Grover

Ross

Stremler

et al. 2012

Chaos: An Interdisciplinary Journal of Nonlinear Science

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In certain two-dimensional time-dependent flows, the braiding of periodic orbits provides a way to analyze chaos in the system through application of the Thurston-Nielsen classification theorem (TNCT). We expand upon earlier work that introduced the application of the TNCT to braiding of almost-cyclic sets, which are individual components of almost-invariant sets [Stremler et al., "Topological chaos and periodic braiding of almost-cyclic sets," Phys. Rev. Lett. 106, 114101 (2011)]. In this context, almost-cyclic sets are periodic regions in the flow with high local residence time that act as stirrers or "ghost rods" around which the surrounding fluid appears to be stretched and folded. In the present work, we discuss the bifurcation of the almost-cyclic sets as a system parameter is varied, which results in a sequence of topologically distinct braids. We show that, for Stokes' flow in a lid-driven cavity, these various braids give good lower bounds on the topological entropy over the respective parameter regimes in which they exist. We make the case that a topological analysis based on spatiotemporal braiding of almost-cyclic sets can be used for analyzing chaos in fluid flows. Hence, we further develop a connection between set-oriented statistical methods and topological methods, which promises to be an important analysis tool in the study of complex systems. When a body of fluid moves, whether it be in the atmosphere, an ocean, or a kitchen sink, there are often regions of fluid that move together for an extended period of time. As these coherent sets of fluid trace out trajectories in space and time, they can be thought of as "stirring" the surrounding fluid. The entanglement of these trajectories as they "braid" around each other is connected to the level of chaos present in the fluid system. When the sets return periodically to their initial positions in a twodimensional, time-dependent system, the entanglement of their trajectories can be used to predict a lower bound on the rate of stretching in the surrounding domain. We examine the trajectories of such "almost-cyclic sets" in a lid-driven cavity flow and demonstrate that this combination of topological analysis with set-oriented methods can be an effective means of predicting chaos. The characterization of the entanglement and associated prediction of stretching are achieved through application of the Thurston-Nielsen Classification Theorem, which in general classifies the topological complexity of homeomorphisms of punctured surfaces. While a rigorous lower bound on topological entropy is not available in the absence of exactly periodic braiding structures, our approach finds the "topological skeleton" that can be used to get an approximate rate of stretching.

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Section: -21mentioning

confidence: 99%

Topological chaos, braiding and bifurcation of almost-cyclic sets

Grover

Ross

Stremler

et al. 2012

Chaos: An Interdisciplinary Journal of Nonlinear Science

Self Cite

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“…Moreover, Stremler & Chen [10] suggested that a modified two dimensional cavity flow, featuring vortical motion with three or more well-defined ghost rods, could be used to obtain effective mixing. Chen & Stremler [3] further suggested that this vortical lid driven flow could provide efficient lateral mixing for a pressure driven, continuous flow mixer. Though hypothetical at this stage, this continuous flow mixer shows promising potential and bears a strong qualitative resemblance to the realistic micromixer of Kang et al [7].…”

Section: Ghost Rodsmentioning

confidence: 99%

“…Section 2 begins by proposing an analogue of braiding in three dimensions. Section 3.1, following the methodology of Stremler and Chen [10,3], proposes a hypothetical batch mixer capable of executing these three dimensional 'braiding' protocols. The efficiency of each protocol is assessed using surface tracking (Section 3.2) and compared with theoretical predictions (Section 4).…”

Section: The Next Frontier: From 2d To 3dmentioning

confidence: 99%

A topological approach to three dimensional laminar mixing

Jewell

2012

ANZIAMJ

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In recent years, topological concepts have yielded valuable insights into the long standing problem of laminar fluid mixing. Topologically complex stirring protocols are typically far superior to topologically simple protocols, guaranteeing chaotic advection of fluid particles and the associated exponential dilation of material elements. Furthermore, topological approaches to mixer design are typically intuitive and insensitive to precise geometry or fluid properties. However, results to date have been limited to two dimensional flows (for example, batch stirrers in food or polymer manufacturing) and quasi three dimensional protocols (for example, continuous flow micromixers). Motivated by a simple stretching and folding argument that works well in two dimensions, we propose a topological approach to fully three dimensional fluid mixing. A transition matrix is derived to describe the mapping induced by a three dimensional 'braid' on area elements,

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“…The final feature that we wish to point out is that the ''axial flow'' in this kinematic model is independent of the location in the crosssection. This could be dealt with by modeling the axial flow with an applied Poiseuille flow (as is done in Stroock and McGraw (2004), Chen andStremler (2009), Song et al (2008), for example). However, if the flow was driven by the electro-osmotic force, rather than pressure, this would not be an issue (Pacheco (2008)).…”

Section: A Kinematic Model For Multiple Secondary Flowsmentioning

confidence: 99%

“…This question has been addressed using computational fluid dynamics techniques by Aubin et al (2005) and Yang et al (2005). Models of the SHM based on lid driven cavity flows have been considered by Stroock and McGraw (2004), Chen andStremler (2009), andSong et al (2008), where the walls, with the imposition of no-slip boundary conditions, are moved in a way that simulated the secondary flows generated by the herringbone patterns and the axial motion of the fluid is governed by an imposed pressure driven Poiseuille flow.…”

Section: Introductionmentioning

confidence: 99%

Optimizing mixing in channel flows: kinematic aspects associated with secondary flows in the cross-section

McIlhany

Wiggins

2010

Microfluid Nanofluid

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We consider an Eulerian predictor of optimal mixing appropriate for steady, three-dimensional channel flow of the type that is commonly used in a variety of microfluidic mixing applications. This Eulerian indicator is applied to a kinematic model of the channel flow that allows one to control the flow structure of the secondary flows and the number of distinct secondary flows. The cases of two and three distinct secondary flows are considered and the Eulerian indicator predicts the optimal flow structure in the two secondary flow case and in the three secondary flow case that result in the best mixing. Moreover, it is shown that the case of three secondary flows mixes ''optimally'' (i.e., faster and more completely) than the case of two secondary flows. An explanation of this difference is given in terms of the Eulerian flow structure.

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Topological chaos and mixing in a three-dimensional channel flow

Cited by 13 publications

References 17 publications

Topological chaos, braiding and bifurcation of almost-cyclic sets

Topological chaos, braiding and bifurcation of almost-cyclic sets

A topological approach to three dimensional laminar mixing

Optimizing mixing in channel flows: kinematic aspects associated with secondary flows in the cross-section

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