Bulletin of the American Physical Society
2005 58th Annual Meeting of the Division of Fluid Dynamics
Sunday–Tuesday, November 20–22, 2005; Chicago, IL
Session NK: Chaos |
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Chair: Thomas Solomon, Bucknell University Room: Hilton Chicago Joliet |
Tuesday, November 22, 2005 11:01AM - 11:14AM |
NK.00001: Measuring stretching to predict the progress of diffusively limited chemical reactions P.E. Arratia, J.P. Gollub We investigate an acid-base reaction in the presence of chaotic advection and diffusion using experimentally measured stretching fields and fluorescent monitoring of the product. Both the flow symmetry and the Reynolds number (\textit{Re}) affect the spatial distribution and time dependence of the reaction product, which grows more slowly than expected, possibly as a result of highly non-uniform stretching.~ A single parameter, the product of the mean Lyapunov exponent $\bar {\lambda }$ and the number of cycles $N$, can be used to predict the spatial average time-dependent product concentration for flows possessing different degrees of spatial symmetry and various \textit{Re}. Unexpected oscillations occur on a time-scale much slower than the basic flow period, a phenomenon that is not reproduced by the usual model for fast reactions.~ This work was supported by National Science Foundation grant DMR-0405187. [Preview Abstract] |
Tuesday, November 22, 2005 11:14AM - 11:27AM |
NK.00002: Chaos and threshold for irreversibility in sheared suspensions Jerry Gollub, David Pine, John Brady, Alex Leshansky Slowly sheared suspensions of solid particles are governed by time-reversible equations of motion. Here we report a precise experimental test showing that time-reversibility fails for slowly sheared suspensions. We study a dense suspension of PMMA particles (index and density matched to the fluid), at low Reynolds number in a Couette cell using oscillatory strain. We find that there is a concentration-dependent threshold strain amplitude beyond which particles do not return to their starting configurations after one or more cycles. Instead, their displacements follow the statistics of an anisotropic random walk. We determine the dependence of the effective diffusivities on strain amplitude, and the concentration dependence of the threshold. The experimental results are compared to numerical simulations, which demonstrate that the threshold strain amplitude is associated with a pronounced growth in the Lyapunov exponent for chaotic particle interactions. The comparison illuminates the connections between chaos, reversibility, and predictability. [Preview Abstract] |
Tuesday, November 22, 2005 11:27AM - 11:40AM |
NK.00003: Chaotic advection and mixing inside drops subject to transient electric fields Xiumei Xu, G.M. Homsy We numerically study the 3D chaotic trajectories inside a neutrally buoyant drop driven by periodically switching the orientation of a uniform electric field. The extent of the chaotic trajectories is related to two parameters: the modulation frequency and the change in the orientation angle $\alpha$ of the electric field. When $\alpha$ equals $\pi$/2, although the axisymmetry of the streamlines is broken, Poincare maps show that for all modulation frequencies, particle trajectories are confined by certain KAM surfaces determined by initial positions. The curvilinear coordinates of the KAM surfaces are formed by the rings of fixed points at the vortex centers which are orthogonal when $\alpha$ equals $\pi$/2. For $\alpha$ is different than $\pi$/2, the Poincare sections depend on the modulation frequency, and there is a range of intermediate frequencies where the volume of the chaotic region is a maximum. Typically a single trajectory can fill most portion of the drop, and there are ordered islands near the crossings of the rings of center fixed points. These results suggest there are optional protocols that maximize mixing. [Preview Abstract] |
Tuesday, November 22, 2005 11:40AM - 11:53AM |
NK.00004: Stirring Generated by a Pair of Elliptic Vortex Patches Luca Cortelezzi, Igor Mezic In this two-dimensional study we analyze the stirring generated by two elliptical vortex patches. The domain is infinite and the fluid is inviscid and incompressible. We consider the condition of pure advection, where diffusion and chemical reactions are neglected. Vorticity of the same sign and magnitude is uniformly distributed over the patches. The resulting flow is laminar and quasi-periodic. However, it is a finite-time type of flow because the vortices would eventually merge. The time-evolution of the patches is computed using a contour dynamic algorithm which solves Euler's equations. We consider as initial unstirred field the case where the fluid is colored differently in the top and bottom half-planes. The stirring efficiency is intimately related to the motion of co-rotation of the vortex pair and the rotation of each vortex about itself. We characterize stirring in terms of interface stretching, patchiness and mixnorm. We show that vortices with lower aspect-ratio (= minor-axis/major-axis) have better stirring efficiency although they are more prone to deform and merge. We also show that the period of rotation modulates the time evolution of the mixnorm and interface stretching. We use computer animations to discuss the stirring mechanisms. [Preview Abstract] |
Tuesday, November 22, 2005 11:53AM - 12:06PM |
NK.00005: Periodic Solutions and Chaos in a Nonlinear Model for the Delayed Immune Response Askery Canabarro, Iram Gleria, Marcelo Lyra We model the cellular immune response using a set of non- Newtonian delayed nonlinear differential equations. The production of defense cells is taken to be proportional to the abundance of pathogenic particles in a previous time. We observe that the stationary solution becomes unstable above a critical immune response time $\tau_c$. In the periodic regime, the minimum virus load is substantially reduced with respect to the stationary solution. Further increasing the delay time, the dynamics display a series of bifurcations evolving to a chaotic regime characterized by a set of 2D portraits. Time series data of the immune state of patients look rather irregular, pointing out to the possibility of a chaotic dynamics. [Preview Abstract] |
Tuesday, November 22, 2005 12:06PM - 12:19PM |
NK.00006: Influence of Diffusion on Optimal Mixing Protocols Alessandra Adrover, Stefano Cerbelli, Massimiliano Giona, Luca Cortelezzi In recent years several studies have addressed the optimization of mixing protocols. These studies can be divided in two classes depending on the treatment of molecular diffusivity. In the first class of studies, diffusivity is neglected, and the characterization of the flow is, in general, purely kinematic. In the second class of studies, diffusivity is included, and the flow is characterized by analyzing and solving the advection-diffusion equation or performing an experiment. In general, it is not obvious if an optimal protocol derived for a given purely advective flow is still optimal for the corresponding flow with nonzero diffusivity and vice versa. We consider the optimization of mixing protocols for a prototypal flow: the sine flow. We choose this flow because its dynamics is quite realistic and has been extensively studied. We consider a fluid with and without diffusivity and derive finite-horizon optimal mixing protocols for a prescribed initial data. We obtain insight about the role played by diffusivity by applying protocols derived in the diffusive case to the purely advective case and vice versa. Results are discussed emphasizing the role of diffusivity and horizon length. [Preview Abstract] |
Tuesday, November 22, 2005 12:19PM - 12:32PM |
NK.00007: Chemical fronts and waves in a chain of alternating vortices Tom Solomon, Matt Paoletti, Carolyn Nugent We present results of experimental studies of advection-reaction-diffusion phenomena in a flow composed of an oscillating and/or drifting chain of vortices in an annulus. The oscillating/drifting vortex chain flow has been shown to exhibit chaotic mixing and (in some cases) superdiffusive transport. We investigate the behavior of the Belousov-Zhabotinsky reaction in this system. We consider both the propagation of a reaction front in this system, as well as wave behavior observed for oscillatory reactions. For the front propagation, the role of coherent vortices in the flow is discussed. In particular, the front is shown to mode-lock to the external stimulus if forced periodically. We extend this result to cases in which transport is superdiffusive. For the oscillatory reaction, source-sink waves form in the diffusive regime, although the behavior of the waves continually changes during the experiments. [Preview Abstract] |
Tuesday, November 22, 2005 12:32PM - 12:45PM |
NK.00008: Synchronization via superdiffusive mixing in an extended, advection-reaction-diffusion system Matt Paoletti, Carolyn Nugent, Tom Solomon We study synchronization of the Belousov-Zhabotinsky (BZ) chemical reaction in an annular chain of alternating vortices. The vortex chain can (a) oscillate, in which case chaotic advection enhances mixing between adjacent vortices, and/or (b) drift, in which case a jet region forms allowing tracers to travel rapidly around the annulus. If the chain both oscillates and drifts, the long-range transport is diffusive for drift velocity $v_d <$ oscillation velocity $v_o$ and superdiffusive for $v_d > v_o$. We map out the regimes in parameter space ($v_o$ versus $v_d$) where the BZ reaction synchronizes. We find that synchronization is much more prevalent for the regimes in which transport is superdiffusive. The results are interpreted by considering Levy flights -- tracer trajectories characterized by long jumps -- associated with superdiffusive transport as ``short-cuts'' connecting distant parts of the system, similar to those proposed for discrete ``small world'' networks. [Preview Abstract] |
Tuesday, November 22, 2005 12:45PM - 12:58PM |
NK.00009: Active mixing and self-consistent chaos in a vortex chain Amanda Kinney, Tom Solomon We present results of experimental studies of the destabilization of a chain of alternating vortices due to mixing in the flow of an active impurity, i.e., an impurity whose mixing can change the flow itself. The flow is driven by magnetohydrodynamic forcing -- an electrical current passing through a layer of salt water interacts with a spatially-periodic magnetic field produced by permanent magnets below the fluid. The active impurity is a salt-water solution with larger salt concentration; consequently, regions with higher impurity concentration have higher electrical conductivity and a higher current density, resulting in a change in the local forcing of the flow. Instabilities are found in which blobs of higher and lower salt concentration form and advect around the vortices, bumping the vortex boundaries periodically and causing chaotic mixing between adjacent vortices. An instability is also found in which the vortices breath in the lateral direction, as well as an instability to aperiodic time dependence. The various instabilities are mapped out in parameter space as a function of driving current and concentration difference. [Preview Abstract] |
Tuesday, November 22, 2005 12:58PM - 1:11PM |
NK.00010: Topological chaos in a lid-driven cavity flow Jie Chen, Mark Stremler Periodic motion of three or more stirrers in a two-dimensional flow can lead to exponential stretching and folding of the surrounding fluid. For certain stirrer motions, the generation of chaos is guaranteed solely by the topology of the stirrer motion and continuity of the fluid. Appropriate stirrer motions are determined using the Thurston--Nielsen classification theorem, which also predicts a lower bound on the fluid stretching rate. Most of the work in this area has focused on using physical rods as stirrers, but the theory applies equally well when the `stirrers' are passive fluid particles. We demonstrate the occurrence of topological chaos in Stokes flow in a two-dimensional lid-driven cavity without internal rods for periodic operation of piecewise constant boundary velocities. For appropriate choices of boundary velocity, there exist three periodic points in the flow that produce a chaos-generating stirrer motion. These points are found using a numerical solution of the biharmonic equation for Stokes flow in a rectangular cavity. [Preview Abstract] |
Tuesday, November 22, 2005 1:11PM - 1:24PM |
NK.00011: A maximum entropy approach to optimal mixing in a 2D pulsed source-sink flow Mark Stremler, Baratunde Cola Fluid mixing in a Hele--Shaw cell can be accomplished by periodically pulsing pairs of sources and sinks. The mixing efficiency of this system depends largely on the volume of fluid that is injected (and extracted) during each pulse. Here a two-dimensional potential flow model is used to find the pulse volumes that optimize mixing in a rectangular domain containing two source--sink pairs, a system of current interest for DNA microarray analysis. Optimal mixing protocols are identified by determining maximum entropy using an analysis of chaotic advection. [Preview Abstract] |
Tuesday, November 22, 2005 1:24PM - 1:37PM |
NK.00012: Reaction-Diffusion Simulations for Multiply-Waisted Hourglass Geometries Thomas Olsen, Yu Hou, Adam Kowalski, Richard Wiener In previous work, the Reaction-Diffusion model \footnote{H. Riecke and H.-G. Paap, Europhys. Lett. \textbf{14}, 1235 (1991).} correctly predicted a period doubling cascade to chaos in Taylor-Couette flow with hourglass geometry\footnote{Richard J. Wiener \textit{et al}, Phys. Rev. E \textbf{55}, 5489 (1997).}. Our current calculations apply the model to Taylor-Couette flow in a cylindrical geometry with multiple waists of super-critical flow connected by regions of barely super-critical flow. We compare our results to the findings of an ongoing experimental program. [Preview Abstract] |
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