Chaotic advection in a Stokes flow

Aref, Hassan; Balachandar, S.

doi:10.1063/1.865828

Cited by 253 publications

(138 citation statements)

References 15 publications

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“…By adopting such usage consistently, we hope that no confusion can arise. We furthermore note that the definition of how one should measure the quality of mixing achieved by a fluid flow is by no means unique -we shall compute the stretch rate of a finite material line, but many other measures are routinely used, including measurement of the degree of chaotic motion in the Poincaré section (Aref & Balachandar 1986;Jones, Thomas & Aref 1989), the calculation of Liapunov exponents (Muzzio & Swanson 1991), the return percentage of diffusive tracer particles , the growth rate of intermaterial density (Muzzio et al 1999;Giona & Adrover 2001), and the distribution of lamellae generated by mixing a blob of fluid (Clifford, Cox & Roberts 1998Muzzio & Ottino 1989, 1990Sawyers, Sen & Chang 1996).…”

Section: Introductionmentioning

confidence: 99%

Topological chaos in inviscid and viscous mixers

2003

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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. IntroductionStatic and dynamic mixing devices are important in many industries, e. ) have demonstrated, in an unusual blend of ad hoc experimentation and abstract mathematics, that flows with the topology of certain braids achieve a material stretch rate which can be determined quantitatively, given only the topology of the flow. However, a key feature not predicted by their theoretical considerations is the size of the domain in which this stretch rate is attained. Indeed, according to the theory, this domain may have measure zero, and if this were the case then the theory would have little practical impact. Here we provide numerical results that support the observations of Boyland et al., that the chaotic region is in fact commensurate with the region of fluid through which the stirring elements move during operation of the device. We should make clear at the outset that we use the terminology 'topological chaos' in the same sense as Boyland et al. (2000), to

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Section: Introductionmentioning

confidence: 99%

Topological chaos in inviscid and viscous mixers

2003

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“…Our paper is also related to the studies of chaotic advection in two-dimensional, time-dependent Stokes flows induced by two rotating eccentric cylinders (i.e. Aref & Balachandar 1986;Chaiken et al 1986Chaiken et al , 1987. We provide here yet another practical example of chaotic advection in a simple flow system.…”

Section: Introductionmentioning

confidence: 99%

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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“…Chaotic behavior is expected in systems with turbulent flow. However, low Reynolds-number flows, as often found in most polymer-processing equipment, can also exhibit chaotic features [4,5]. This chaotic behavior arises from the geometric and operational complexity present in these systems.…”

Section: Introductionmentioning

confidence: 99%

Entropy Time Evolution In A Twin-flight Single-screw Extruder And Its Relationship To Chaos

Wang

Manas‐Zloczower

Kaufman

2005

Chemical Engineering Communications

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The flow fields in polymer processing exhibit complex behavior with chaotic characteristics, due in part to the non-linearity of the field equations describing them. In chaotic flows fluid elements are highly sensitive to their initial positions and velocities.A fundamental understanding of such characteristics is essential for optimization and design of equipment used for distributive mixing.In this work we analyze the flow in a twin-flight single screw extruder, obtained through 3-D FEM numerical simulations. We study particle motion and, implicitly, mixing in the extruder. Here, particles are massless points whose presence does not affect the flow field or other particle motion.We visualize chaos through Poincaré sections and calculate Lyapunov exponents as a measure of divergence of initial conditions, signaling chaotic features of flow. We use entropic measures to probe disorder or system homogeneity. The time evolution of the Renyi entropy of β = 1 for the 3-D spatial distribution of particles using different initial conditions are followed. The Kolmogorov-Sinai entropy rate, calculated by the sum of positive Lyapunov exponents, is correlated with the rate of evolution of entropy.In the same context we also examine the eccentric Couette flow.We find that the Renyi entropy dependence on time is logarithmic. To gain further understanding of this numerical observation, we analyze analytically the diffusion with drift entropy and find that it also depends logarithmically on time. Using the logarithmic coefficient of the Shannon entropy (ß = 1), as a measure of the overall rate of 2 mixing, we find that the eccentric Couette device has the highest rate of mixing, followed by the twin-flight single screw extruder, and by the 1-D diffusion with drift.

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Chaotic advection in a Stokes flow

Cited by 253 publications

References 15 publications

Topological chaos in inviscid and viscous mixers

Topological chaos in inviscid and viscous mixers

A magnetohydrodynamic chaotic stirrer

Entropy Time Evolution In A Twin-flight Single-screw Extruder And Its Relationship To Chaos

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