Lagrangian turbulence and spatial complexity in a Stokes flow

Chaiken, J.; Chu, C. K.; Tabor, M.; Tan, Qinxue

doi:10.1063/1.866373

Cited by 86 publications

(42 citation statements)

References 13 publications

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“…For example, Aref & Balachandar (1986) showed that unsteady Stokes flow between eccentric rotating cylinders, in which the rotation rate is modulated periodically in time, can exhibit chaotic particle motions of the Smale horseshoe type, Thus this particular Stokes flow is effectively non-reversible. This same flow has also been studied experimentally as well as theoretically by Chaiken et al (1986Chaiken et al ( , 1987. Ottino and coworkers (see Chien, Rising & Ottino 1986; Khakar, Rising ) studied chaotic fluid particle motions in a variety of flows, both at small and large Reynolds numbers with particular emphasis on using dynamical systems techniques as a theoretical basis for the discussion of mixing processes.…”

Section: Introductionmentioning

confidence: 98%

An analytical study of transport, mixing and chaos in an unsteady vortical flow

1990

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We examine the transport properties of a particular two-dimensional, inviscid incompressible flow using dynamical systems techniques. The velocity field is time periodic and consists of the field induced by a vortex pair plus an oscillating strainrate field. I n the absence of the strain-rate field the vortex pair moves with a constant velocity and carries with it a constant body of fluid. When the strain-rate field is added the picture changes dramatically ; fluid is entrained and detrained from the neighbourhood of the vortices and chaotic particle motion occurs. We investigate the mechanism for this phenomenon and study the transport and mixing of fluid in this flow. Our work consists of both numerical and analytical studies. The analytical studies include the interpretation of the invariant manifolds as the underlying structure which govern the transport. For small values of strain-rate amplitude we use Melnikov's technique to investigate the behaviour of the manifolds as the parameters of the problem change and to prove the existence of a horseshoe map and thus the existence of chaotic particle paths in the flow. Using the Melnikov technique once more we develop an analytical estimate of the flux rate into and out of the vortex neighbourhood. We then develop a technique for determining the residence time distribution for fluid particles near the vortices that is valid for arbitrary strainrate amplitudes. The technique involves an understanding of the geometry of the tangling of the stable and unstable manifolds and results in a dramatic reduction in computational effort required for the determination of the residence time distributions. Additionally, we investigate the total stretch of material elements while they are in the vicinity of the vortex pair, using this quantity as a measure of the effect of the horseshoes on trajectories passing through this region. The numerical work verifies the analytical predictions regarding the structure of the invariant manifolds, the mechanism for entrainment and detrainment and the flux rate.

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Section: Introductionmentioning

confidence: 98%

An analytical study of transport, mixing and chaos in an unsteady vortical flow

1990

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“…There are no internal stagnation points. In this sense, the MHD flow in the gap between two eccentric cylinders is topologically different from the shear-flow induced when the inner cylinder rotates as in Aref & Balachandar (1986) and Chaiken et al (1987).…”

Section: Mhd Flow Between Two Eccentric Cylindersmentioning

confidence: 99%

“…This approximation is commonly used in the context of Stokes flows (i.e. Landau & Lifshitz 1959, p. 91;Aref & Balachander 1986;Chaiken et al 1987).…”

Section: Two Point Electrodesmentioning

confidence: 99%

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A magnetohydrodynamic chaotic stirrer

2002

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A magnetohydrodynamic (MHD) stirrer that exhibits chaotic advection is investigated experimentally and theoretically. The stirrer consists of a circular cavity with an electrode (C) deposited around its periphery. Two additional electrodes (A) and (B) are deposited eccentrically inside the cavity on the bottom. The cavity is positioned in a uniform magnetic field that is parallel to the cylinder's axis, and it is filled with a weak electrolyte solution. Fluid motion is induced in the cavity by applying a potential difference across a pair of electrodes. A closed-form, analytical solution is derived for the MHD creeping flow field in the gap between the two eccentric cylinders. A singular solution is obtained for the special case when the size of the inner electrode shrinks to a point. Subsequently, passive tracers' trajectories are computed when the electric potential differences are applied alternately across electrodes AC and BC with period T. At small periods T, the flow is regular and periodic in most of the cavity. As the period increases, so does the complexity of the motion. At relatively large periods, the passive tracer experiences global chaotic advection. Such a device can serve as an efficient stirrer. Since this device has no moving parts, it is especially suitable for microfluidic applications. This is yet another practical example of a modulated, two-dimensional Stokes flow that exhibits chaotic advection. A magnetohydrodynamic (MHD) stirrer that exhibits chaotic advection is investigated experimentally and theoretically. The stirrer consists of a circular cavity with an electrode (C) deposited around its periphery. Two additional electrodes (A) and (B) are deposited eccentrically inside the cavity on the bottom. The cavity is positioned in a uniform magnetic field that is parallel to the cylinder's axis, and it is filled with a weak electrolyte solution. Fluid motion is induced in the cavity by applying a potential difference across a pair of electrodes. A closed-form, analytical solution is derived for the MHD creeping flow field in the gap between the two eccentric cylinders. A singular solution is obtained for the special case when the size of the inner electrode shrinks to a point. Subsequently, passive tracers' trajectories are computed when the electric potential differences are applied alternately across electrodes AC and BC with period T . At small periods T , the flow is regular and periodic in most of the cavity. As the period increases, so does the complexity of the motion. At relatively large periods, the passive tracer experiences global chaotic advection. Such a device can serve as an efficient stirrer. Since this device has no moving parts, it is especially suitable for microfluidic applications. This is yet another practical example of a modulated, two-dimensional Stokes flow that exhibits chaotic advection. A magnetohydrodynamic chaotic stirrer By M I N G Q I A N G Y I , S H I Z H I Q I A N A N D H A I M H. B A U †

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“…The approximation is commonly used in the context of Stokes flows and chaotic advection. 15,16 When the potential difference is induced across electrode group A n -C, we denote the resulting flow field as n ͓Eq. ͑10͔͒.…”

Section: Time-modulated Flowsmentioning

confidence: 99%

A stirrer for magnetohydrodynamically controlled minute fluidic networks

2002

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Magnetohydrodynamics may potentially provide a convenient means for controlling fluid flow and stirring fluids in minute fluidic networks. The branches of such fluidic networks consist of conduits with rectangular cross sections. Each conduit has two individually controlled electrodes positioned along opposing walls and additional disk-shaped electrodes deposited in the conduit’s interior away from its sidewalls. The network is positioned in a uniform magnetic field. When one applies a potential difference between a disk-shaped electrode and two wall electrodes acting in tandem, circulatory motion is induced in the conduit. When the potential difference alternates periodically across two or more such configurations, complicated (chaotic) motions evolve. As the period of alternation increases, so does the complexity of the flow. We derive a two-dimensional, time-independent expression for the magnetohydrodynamic creeping flow around a centrally positioned disk-shaped electrode in the limit of zero radius. With the aid of this expression, the trajectories of passive tracers are computed as functions of the alternations protocol and the electrodes’ locations. The theoretical results are qualitatively compared with flow visualization experiments.

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Lagrangian turbulence and spatial complexity in a Stokes flow

Cited by 86 publications

References 13 publications

An analytical study of transport, mixing and chaos in an unsteady vortical flow

An analytical study of transport, mixing and chaos in an unsteady vortical flow

A magnetohydrodynamic chaotic stirrer

A stirrer for magnetohydrodynamically controlled minute fluidic networks

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