1
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Pushenko V, Schumacher J. Connecting finite-time Lyapunov exponents with supersaturation and droplet dynamics in a turbulent bulk flow. Phys Rev E 2024; 109:045101. [PMID: 38755806 DOI: 10.1103/physreve.109.045101] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/25/2024] [Accepted: 02/26/2024] [Indexed: 05/18/2024]
Abstract
The impact of turbulent mixing on an ensemble of initially monodisperse water droplets is studied in a turbulent bulk that serves as a simplified setup for the interior of a turbulent ice-free cloud. A mixing model was implemented that summarizes the balance equations of water vapor mixing ratio and temperature to an effective advection-diffusion equation for the supersaturation field s(x,t). Our three-dimensional direct numerical simulations connect the velocity and scalar supersaturation fields in the Eulerian frame of reference to an ensemble of cloud droplets in the Lagrangian frame of reference. The droplets are modeled as point particles with and without effects due to inertia. The droplet radius is subject to growth by vapor diffusion. We report the dependence of the droplet size distribution on the box size, initial droplet radius, and the strength of the updraft, with and without gravitational settling. In addition, the three finite-time Lyapunov exponents λ_{1}≥λ_{2}≥λ_{3} are monitored which probe the local stretching properties along the particle tracks. In this way, we can relate regions of higher compressive strain to those of high local supersaturation amplitudes. For the present parameter range, the mixing process in terms of the droplet evaporation is always homogeneous, while it is inhomogeneous with respect to the relaxation of the supersaturation field. The probability density function of the third finite-time Lyapunov exponent, λ_{3}<0, is related to the one of the supersaturation s by a simple one-dimensional aggregation model. The probability density function (PDF) of λ_{3} and the droplet radius r are found to be Gaussian, while the PDF of the supersaturation field shows sub-Gaussian tails.
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Affiliation(s)
- Vladyslav Pushenko
- Institut für Thermo- und Fluiddynamik, Technische Universität Ilmenau, Postfach 100565, D-98684 Ilmenau, Germany
| | - Jörg Schumacher
- Institut für Thermo- und Fluiddynamik, Technische Universität Ilmenau, Postfach 100565, D-98684 Ilmenau, Germany
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2
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Markale I, Cimmarusti GM, Britton MM, Jiménez-Martínez J. Phase Saturation Control on Mixing-Driven Reactions in 3D Porous Media. ENVIRONMENTAL SCIENCE & TECHNOLOGY 2021; 55:8742-8752. [PMID: 34106702 DOI: 10.1021/acs.est.1c01288] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/12/2023]
Abstract
Transported chemical reactions in unsaturated porous media are relevant to environmental and industrial applications. Continuum scale models are based on equivalent parameters derived from analogy with saturated conditions and cannot appropriately account for incomplete mixing. It is also unclear how the third dimension controls mixing and reactions. We obtain three-dimensional (3D) images by magnetic resonance imaging using an immiscible nonwetting liquid as a second phase and a fast irreversible bimolecular reaction. We study the impact of phase saturation on the dynamics of mixing and the reaction front. We quantify the temporally resolved effective reaction rate and describe it using the lamellar theory of mixing, which explains faster than Fickian (t0.5) rate of product formation by accounting for the deformation of the mixing interface between the two reacting fluids. For a given Péclet, although stretching and folding of the reactive front enhance as saturation decreases, enhancing the product formation, the product formation is larger as saturation increases. After breakthrough, the extinction of the reaction takes longer as saturation decreases because of the larger nonmixed volume behind the front. These results are the basis for a general model to better predict reactive transport in unsaturated porous media not achievable by the current continuum paradigm.
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Affiliation(s)
- Ishaan Markale
- Eawag, Swiss Federal Institute of Aquatic Science and Technology, 8600 Dübendorf, Switzerland
- Department of Civil, Environmental and Geomatic Engineering, ETH Zürich, Stefano-Franscini-Platz 5, 8093 Zürich, Switzerland
| | | | - Melanie M Britton
- School of Chemistry, University of Birmingham, Edgbaston, Birmingham B15 2TT, U.K
| | - Joaquín Jiménez-Martínez
- Eawag, Swiss Federal Institute of Aquatic Science and Technology, 8600 Dübendorf, Switzerland
- Department of Civil, Environmental and Geomatic Engineering, ETH Zürich, Stefano-Franscini-Platz 5, 8093 Zürich, Switzerland
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3
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Heyman J, Lester DR, Le Borgne T. Scalar Signatures of Chaotic Mixing in Porous Media. PHYSICAL REVIEW LETTERS 2021; 126:034505. [PMID: 33543984 DOI: 10.1103/physrevlett.126.034505] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/04/2020] [Revised: 11/13/2020] [Accepted: 12/09/2020] [Indexed: 06/12/2023]
Abstract
Steady laminar flows through porous media spontaneously generate Lagrangian chaos at pore scale, with qualitative implications for a range of transport, reactive, and biological processes. The characterization and understanding of mixing dynamics in these opaque environments is an outstanding challenge. We address this issue by developing a novel technique based upon high-resolution imaging of the scalar signature produced by push-pull flows through porous media samples. Owing to the rapid decorrelation of particle trajectories in chaotic flows, the scalar image measured outside the porous material is representative of in situ mixing dynamics. We present a theoretical framework for estimation of the Lyapunov exponent based on extension of Lagrangian stretching theories to correlated aggregation. This method provides a full characterization of chaotic mixing dynamics in a large class of porous materials.
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Affiliation(s)
- J Heyman
- Geosciences Rennes, UMR 6118, Université de Rennes 1, CNRS, 35042 Rennes, France
| | - D R Lester
- School of Engineering, RMIT University, 3000 Melbourne, Australia
| | - T Le Borgne
- Geosciences Rennes, UMR 6118, Université de Rennes 1, CNRS, 35042 Rennes, France
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4
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Arrieta J, Cartwright JHE, Gouillart E, Piro N, Piro O, Tuval I. Geometric mixing. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2020; 378:20200168. [PMID: 32762434 PMCID: PMC7422868 DOI: 10.1098/rsta.2020.0168] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 06/08/2020] [Indexed: 06/11/2023]
Abstract
Mixing fluids often involves a periodic action, like stirring one's tea. But reciprocating motions in fluids at low Reynolds number, in Stokes flows where inertia is negligible, lead to periodic cycles of mixing and unmixing, because the physics, molecular diffusion excepted, is time reversible. So how can fluid be mixed in such circumstances? The answer involves a geometric phase. Geometric phases are found everywhere in physics as anholonomies, where after a closed circuit in the parameters, some system variables do not return to their original values. We discuss the geometric phase in fluid mixing: geometric mixing. This article is part of the theme issue 'Stokes at 200 (part 2)'.
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Affiliation(s)
- Jorge Arrieta
- Institut Mediterrani d’Estudis Avançats, CSIC–Universitat de les Illes Balears, 07190 Esporles, Spain
| | - Julyan H. E. Cartwright
- Instituto Andaluz de Ciencias de la Tierra, CSIC–Universidad de Granada, 18100 Armilla, Granada, Spain
- Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, 18071 Granada, Spain
| | | | - Nicolas Piro
- École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
| | - Oreste Piro
- Departament de Física, Universitat de les Illes Balears, 07071 Palma de Mallorca, Spain
| | - Idan Tuval
- Institut Mediterrani d’Estudis Avançats, CSIC–Universitat de les Illes Balears, 07190 Esporles, Spain
- Departament de Física, Universitat de les Illes Balears, 07071 Palma de Mallorca, Spain
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5
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Cassiani M, Bertagni MB, Marro M, Salizzoni P. Concentration Fluctuations from Localized Atmospheric Releases. BOUNDARY-LAYER METEOROLOGY 2020; 177:461-510. [PMID: 33184516 PMCID: PMC7596001 DOI: 10.1007/s10546-020-00547-4] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 01/21/2020] [Accepted: 07/07/2020] [Indexed: 06/11/2023]
Abstract
We review the efforts made by the scientific community in more than seventy years to elucidate the behaviour of concentration fluctuations arising from localized atmospheric releases of dynamically passive and non-reactive scalars. Concentration fluctuations are relevant in many fields including the evaluation of toxicity, flammability, and odour nuisance. Characterizing concentration fluctuations requires not just the mean concentration but also at least the variance of the concentration in the location of interest. However, for most purposes the characterization of the concentration fluctuations requires knowledge of the concentration probability density function (PDF) in the point of interest and even the time evolution of the concentration. We firstly review the experimental works made both in the field and in the laboratory, and cover both point sources and line sources. Regarding modelling approaches, we cover analytical, semi-analytical, and numerical methods. For clarity of presentation we subdivide the models in two groups, models linked to a transport equation, which usually require a numerical resolution, and models mainly based on phenomenological aspects of dispersion, often providing analytical or semi-analytical relations. The former group includes: large-eddy simulations, Reynolds-averaged Navier-Stokes methods, two-particle Lagrangian stochastic models, PDF transport equation methods, and heuristic Lagrangian single-particle methods. The latter group includes: fluctuating plume models, semi-empirical models for the concentration moments, analytical models for the concentration PDF, and concentration time-series models. We close the review with a brief discussion highlighting possible useful additions to experiments and improvements to models.
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Affiliation(s)
- Massimo Cassiani
- NILU - Norwegian Institue for Air Research, 2027 Kjeller, Norway
| | - Matteo B. Bertagni
- Department of Land, Infrastructure and Environmental Engineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10124 Turin, Italy
| | - Massimo Marro
- Laboratoire de Mécanique des Fluides et d’Acoustique, University of Lyon, CNRS UMR 5509 Ecole Centrale de Lyon, INSA Lyon, Université Claude Bernard, 36, avenue Guy de Collongue, 69134 Ecully, France
| | - Pietro Salizzoni
- Laboratoire de Mécanique des Fluides et d’Acoustique, University of Lyon, CNRS UMR 5509 Ecole Centrale de Lyon, INSA Lyon, Université Claude Bernard, 36, avenue Guy de Collongue, 69134 Ecully, France
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6
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Abstract
In this paper, a review of the Lagrangian stochastic models developed in the last decades for the simulation of the concentration–fluctuation dispersion is presented. The main approaches available in the literature are described and their ability in reproducing the higher order moments of the probability density function is discussed. Then, the Lagrangian approaches for evaluating of the odor annoyance are presented. It is worth to notice that, while Lagrangian stochastic models for mean concentrations are well-known and their ability in correctly reproducing the observation is well assessed, concerning concentration fluctuations the approaches are often new and unknown for most of the scientific community.
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7
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Götzfried P, Emran MS, Villermaux E, Schumacher J. Comparison of Lagrangian and Eulerian frames of passive scalar turbulent mixing. PHYSICAL REVIEW FLUIDS 2019; 4:044607. [DOI: 10.1103/physrevfluids.4.044607] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 09/01/2023]
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8
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Dentz M, Lester DR, Le Borgne T, de Barros FPJ. Coupled continuous-time random walks for fluid stretching in two-dimensional heterogeneous media. Phys Rev E 2016; 94:061102. [PMID: 28085355 DOI: 10.1103/physreve.94.061102] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/15/2016] [Indexed: 06/06/2023]
Abstract
We study the relation between flow structure and fluid deformation in steady flows through two-dimensional heterogeneous media, which are characterized by a broad spectrum of stretching behaviors, ranging from sub- to superlinear. We analyze these behaviors from first principles, which uncovers intermittent shear events to be at the origin of subexponential stretching. We derive explicit expressions for Lagrangian deformation and demonstrate that stretching obeys a coupled continuous-time random walk, which for broad distributions of flow velocities becomes a Lévy walk. The derived model provides a direct link between the flow and deformation statistics, and a natural way to quantify the impact of intermittent shear events on the stretching behavior.
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Affiliation(s)
- Marco Dentz
- Spanish National Research Council (IDAEA-CSIC), 08034 Barcelona, Spain
| | - Daniel R Lester
- School of Engineering, RMIT University, 3000 Melbourne, Victoria, Australia
| | - Tanguy Le Borgne
- Geosciences Rennes, UMR 6118, Université de Rennes 1, CNRS, Rennes, France
| | - Felipe P J de Barros
- Sonny Astani Department of Civil and Environmental Engineering, University of Southern California, 3620 South Vermont Avenue, KAP 224B, Los Angeles, California 90089, USA
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9
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Hyman JD, Jiménez-Martínez J, Viswanathan HS, Carey JW, Porter ML, Rougier E, Karra S, Kang Q, Frash L, Chen L, Lei Z, O'Malley D, Makedonska N. Understanding hydraulic fracturing: a multi-scale problem. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2016; 374:rsta.2015.0426. [PMID: 27597789 PMCID: PMC5014299 DOI: 10.1098/rsta.2015.0426] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 04/28/2016] [Indexed: 06/06/2023]
Abstract
Despite the impact that hydraulic fracturing has had on the energy sector, the physical mechanisms that control its efficiency and environmental impacts remain poorly understood in part because the length scales involved range from nanometres to kilometres. We characterize flow and transport in shale formations across and between these scales using integrated computational, theoretical and experimental efforts/methods. At the field scale, we use discrete fracture network modelling to simulate production of a hydraulically fractured well from a fracture network that is based on the site characterization of a shale gas reservoir. At the core scale, we use triaxial fracture experiments and a finite-discrete element model to study dynamic fracture/crack propagation in low permeability shale. We use lattice Boltzmann pore-scale simulations and microfluidic experiments in both synthetic and shale rock micromodels to study pore-scale flow and transport phenomena, including multi-phase flow and fluids mixing. A mechanistic description and integration of these multiple scales is required for accurate predictions of production and the eventual optimization of hydrocarbon extraction from unconventional reservoirs. Finally, we discuss the potential of CO2 as an alternative working fluid, both in fracturing and re-stimulating activities, beyond its environmental advantages.This article is part of the themed issue 'Energy and the subsurface'.
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Affiliation(s)
- J D Hyman
- Earth and Environmental Sciences, Los Alamos National Laboratory, Los Alamos, NM, USA
| | - J Jiménez-Martínez
- Earth and Environmental Sciences, Los Alamos National Laboratory, Los Alamos, NM, USA
| | - H S Viswanathan
- Earth and Environmental Sciences, Los Alamos National Laboratory, Los Alamos, NM, USA
| | - J W Carey
- Earth and Environmental Sciences, Los Alamos National Laboratory, Los Alamos, NM, USA
| | - M L Porter
- Earth and Environmental Sciences, Los Alamos National Laboratory, Los Alamos, NM, USA
| | - E Rougier
- Earth and Environmental Sciences, Los Alamos National Laboratory, Los Alamos, NM, USA
| | - S Karra
- Earth and Environmental Sciences, Los Alamos National Laboratory, Los Alamos, NM, USA
| | - Q Kang
- Earth and Environmental Sciences, Los Alamos National Laboratory, Los Alamos, NM, USA
| | - L Frash
- Earth and Environmental Sciences, Los Alamos National Laboratory, Los Alamos, NM, USA
| | - L Chen
- Earth and Environmental Sciences, Los Alamos National Laboratory, Los Alamos, NM, USA
| | - Z Lei
- Earth and Environmental Sciences, Los Alamos National Laboratory, Los Alamos, NM, USA
| | - D O'Malley
- Earth and Environmental Sciences, Los Alamos National Laboratory, Los Alamos, NM, USA
| | - N Makedonska
- Earth and Environmental Sciences, Los Alamos National Laboratory, Los Alamos, NM, USA
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10
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Keshavarz B, Houze EC, Moore JR, Koerner MR, McKinley GH. Ligament Mediated Fragmentation of Viscoelastic Liquids. PHYSICAL REVIEW LETTERS 2016; 117:154502. [PMID: 27768340 DOI: 10.1103/physrevlett.117.154502] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/06/2015] [Indexed: 05/11/2023]
Abstract
The breakup and atomization of complex fluids can be markedly different than the analogous processes in a simple Newtonian fluid. Atomization of paint, combustion of fuels containing antimisting agents, as well as physiological processes such as sneezing are common examples in which the atomized liquid contains synthetic or biological macromolecules that result in viscoelastic fluid characteristics. Here, we investigate the ligament-mediated fragmentation dynamics of viscoelastic fluids in three different canonical flows. The size distributions measured in each viscoelastic fragmentation process show a systematic broadening from the Newtonian solvent. In each case, the droplet sizes are well described by Gamma distributions which correspond to a fragmentation-coalescence scenario. We use a prototypical axial step strain experiment together with high-speed video imaging to show that this broadening results from the pronounced change in the corrugated shape of viscoelastic ligaments as they separate from the liquid core. These corrugations saturate in amplitude and the measured distributions for viscoelastic liquids in each process are given by a universal probability density function, corresponding to a Gamma distribution with n_{min}=4. The breadth of this size distribution for viscoelastic filaments is shown to be constrained by a geometrical limit which can not be exceeded in ligament-mediated fragmentation phenomena.
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Affiliation(s)
- Bavand Keshavarz
- Hatsopoulos Microfluids Laboratory, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | - Eric C Houze
- Axalta Coating Systems, Two Commerce Square, 2001 Market Street, Suite 3600, Philadelphia, Pennsylvania 19103, USA
| | - John R Moore
- Axalta Coating Systems, Two Commerce Square, 2001 Market Street, Suite 3600, Philadelphia, Pennsylvania 19103, USA
| | - Michael R Koerner
- Axalta Coating Systems, Two Commerce Square, 2001 Market Street, Suite 3600, Philadelphia, Pennsylvania 19103, USA
| | - Gareth H McKinley
- Hatsopoulos Microfluids Laboratory, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
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11
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Engdahl NB, Benson DA, Bolster D. Predicting the enhancement of mixing-driven reactions in nonuniform flows using measures of flow topology. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:051001. [PMID: 25493728 DOI: 10.1103/physreve.90.051001] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/07/2014] [Indexed: 06/04/2023]
Abstract
The ability for reactive constituents to mix is often the key limiting factor for the completion of reactions across a huge range of scales in a variety of media. In flowing systems, deformation and shear enhance mixing by bringing constituents into closer proximity, thus increasing reaction potential. Accurately quantifying this enhanced mixing is key to predicting reactions and typically is done by observing or simulating scalar transport. To eliminate this computationally expensive step, we use a Lagrangian stochastic framework to derive the enhancement to reaction potential by calculating the collocation probability of particle pairs in a heterogeneous flow field accounting for deformations. We relate the enhanced reaction potential to three well known flow topology metrics and demonstrate that it is best correlated to (and asymptotically linear with) one: the largest eigenvalue of the (right) Cauchy-Green tensor.
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Affiliation(s)
- Nicholas B Engdahl
- Department of Civil and Environmental Engineering, Washington State University, Pullman, Washington 99164, USA
| | - David A Benson
- Hydrologic Science and Engineering, Colorado School of Mines, Golden, Colorado 80401, USA
| | - Diogo Bolster
- Department of Civil and Environmental Engineering and Earth Sciences, University of Notre Dame, Notre Dame, Indiana 46556, USA
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12
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Crimaldi JP, Zimmer RK. The physics of broadcast spawning in benthic invertebrates. ANNUAL REVIEW OF MARINE SCIENCE 2013; 6:141-165. [PMID: 23957600 DOI: 10.1146/annurev-marine-010213-135119] [Citation(s) in RCA: 23] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/02/2023]
Abstract
Most benthic invertebrates broadcast their gametes into the sea, whereupon successful fertilization relies on the complex interaction between the physics of the surrounding fluid flow and the biological properties and behavior of eggs and sperm. We present a holistic overview of the impact of instantaneous flow processes on fertilization across a range of scales. At large scales, transport and stirring by the flow control the distribution of gametes. Although mean dilution of gametes by turbulence is deleterious to fertilization, a variety of instantaneous flow phenomena can aggregate gametes before dilution occurs. We argue that these instantaneous flow processes are key to fertilization efficiency. At small scales, sperm motility and taxis enhance contact rates between sperm and chemoattractant-releasing eggs. We argue that sperm motility is a biological adaptation that replaces molecular diffusion in conventional mixing processes and enables gametes to bridge the gap that remains after aggregation by the flow.
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Affiliation(s)
- John P Crimaldi
- Department of Civil, Environmental, and Architectural Engineering, University of Colorado, Boulder, Colorado 80309-0428;
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13
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Wendell DM, Pigeonneau F, Gouillart E, Jop P. Intermittent flow in yield-stress fluids slows down chaotic mixing. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:023024. [PMID: 24032940 DOI: 10.1103/physreve.88.023024] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/25/2013] [Indexed: 06/02/2023]
Abstract
We present experimental results of chaotic mixing of Newtonian fluids and yield-stress fluids using a rod-stirring protocol with a rotating vessel. We show how the mixing of yield-stress fluids by chaotic advection is reduced compared to the mixing of Newtonian fluids and explain our results, bringing to light the relevant mechanisms: the presence of fluid that only flows intermittently, a phenomenon enhanced by the yield stress, and the importance of the peripheral region. This finding is confirmed via numerical simulations. Anomalously slow mixing is observed when the synchronization of different stirring elements leads to the repetition of slow stretching for the same fluid particles.
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Affiliation(s)
- D M Wendell
- Surface du Verre et Interfaces, UMR 125 CNRS/Saint-Gobain, 39, quai Lucien Lefranc, F-93303 Aubervilliers Cedex, France
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14
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Le Borgne T, Dentz M, Villermaux E. Stretching, coalescence, and mixing in porous media. PHYSICAL REVIEW LETTERS 2013; 110:204501. [PMID: 25167417 DOI: 10.1103/physrevlett.110.204501] [Citation(s) in RCA: 27] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/18/2013] [Indexed: 06/03/2023]
Abstract
We study scalar mixing in heterogeneous conductivity fields, whose structural disorder varies from weak to strong. A range of stretching regimes is observed, depending on the level of structural heterogeneity, measured by the log-conductivity field variance. We propose a unified framework to quantify the overall concentration distribution predicting its shape and rate of deformation as it progresses toward uniformity in the medium. The scalar mixture is represented by a set of stretched lamellae whose rate of diffusive smoothing is locally enhanced by kinematic stretching. Overlap between the lamellae is enforced by confinement of the scalar line support within the dispersion area. Based on these elementary processes, we derive analytical expressions for the concentration distribution, resulting from the interplay between stretching, diffusion, and random overlaps, holding for all field heterogeneities, residence times, and Péclet numbers.
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Affiliation(s)
- Tanguy Le Borgne
- Geosciences Rennes, UMR 6118, Université de Rennes 1, CNRS, 35042 Rennes, France
| | - Marco Dentz
- Spanish National Research Council (IDAEA-CSIC), 08034 Barcelona, Spain
| | - Emmanuel Villermaux
- Institut de Recherche sur les Phénomènes Hors Equilibre, Aix-Marseille Université, 13384 Marseille Cedex 13, France
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15
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Deboeuf S, Katzav E, Boudaoud A, Bonn D, Adda-Bedia M. Comparative study of crumpling and folding of thin sheets. PHYSICAL REVIEW LETTERS 2013; 110:104301. [PMID: 23521261 DOI: 10.1103/physrevlett.110.104301] [Citation(s) in RCA: 21] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/07/2012] [Indexed: 06/01/2023]
Abstract
Crumpling and folding of paper are at first sight very different ways of confining thin sheets in a small volume: the former one is random and stochastic whereas the latest one is regular and deterministic. Nevertheless, certain similarities exist. Crumpling is surprisingly inefficient: a typical crumpled paper ball in a waste-bin consists of as much as 80% air. Similarly, if one folds a sheet of paper repeatedly in two, the necessary force becomes so large that it is impossible to fold it more than six or seven times. Here we show that the stiffness that builds up in the two processes is of the same nature, and therefore simple folding models allow us to capture also the main features of crumpling. An original geometrical approach shows that crumpling is hierarchical, just as the repeated folding. For both processes the number of layers increases with the degree of compaction. We find that for both processes the crumpling force increases as a power law with the number of folded layers, and that the dimensionality of the compaction process (crumpling or folding) controls the exponent of the scaling law between the force and the compaction ratio.
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Affiliation(s)
- S Deboeuf
- Université Paris-Est, Laboratoire Navier, UMR 8205, CNRS, ENPC, IFSTTAR, F-77420 Marne-la-Vallée, France
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16
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Rossi L, Doorly D, Kustrin D. Lamination and mixing in three fundamental flow sequences driven by electromagnetic body forces. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:026313. [PMID: 23005860 DOI: 10.1103/physreve.86.026313] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/19/2011] [Revised: 02/16/2012] [Indexed: 06/01/2023]
Abstract
This article pursues the idea that the degree of striations, called lamination, could be engineered to complement stretching and to design new sequential mixers. It explores lamination and mixing in three new mixing sequences experimentally driven by electromagnetic body forces. To generate these three mixing sequences, Lorentz body forces are dynamically controlled to vary the flow geometry produced by a pair of local jets. The first two sequences are inspired from the "tendril and whorl" and "blinking vortex" flows. The third novel sequence is called the "cat's eyes flip." These three mixing sequences exponentially stretch and laminate material lines representing the interface between two domains to be mixed. Moreover, the mixing coefficient (defined as 1-σ(2)/σ(0)(2) where σ(2)/σ(0)(2) is the rescaled variance) and its rate grow exponentially before saturation. This saturation of the mixing process is related to the departure of the mixing rate from an exponential growth when the striations' thicknesses reach the diffusive length scale of the measurements or species and dyes. Incidentally, in our experiments, for the same energy or forcing input, the cat's eyes flip sequence has higher lamination, stretching, and mixing rates than the tendril and whorl and the blinking vortex sequences. These features show that bakerlike in situ mixers can be conceived by dynamically controlling a pair of local jets and by integrating lamination during stirring stages with persistent geometries. Combined with novel insights provided by the quantification of the lamination, this paper should offer perspectives for the development of new sequential mixers, possibly on all scales.
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Affiliation(s)
- L Rossi
- Department of Aeronautics, Imperial College London, London, United Kingdom
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17
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Thiffeault JL, Gouillart E, Dauchot O. Moving walls accelerate mixing. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:036313. [PMID: 22060498 DOI: 10.1103/physreve.84.036313] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/20/2011] [Indexed: 05/31/2023]
Abstract
Mixing in viscous fluids is challenging, but chaotic advection in principle allows efficient mixing. In the best possible scenario, the decay rate of the concentration profile of a passive scalar should be exponential in time. In practice, several authors have found that the no-slip boundary condition at the walls of a vessel can slow down mixing considerably, turning an exponential decay into a power law. This slowdown affects the whole mixing region, and not just the vicinity of the wall. The reason is that when the chaotic mixing region extends to the wall, a separatrix connects to it. The approach to the wall along that separatrix is polynomial in time and dominates the long-time decay. However, if the walls are moved or rotated, closed orbits appear, separated from the central mixing region by a hyperbolic fixed point with a homoclinic orbit. The long-time approach to the fixed point is exponential, so an overall exponential decay is recovered, albeit with a thin unmixed region near the wall.
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Affiliation(s)
- Jean-Luc Thiffeault
- Department of Mathematics, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA
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Yee E, Skvortsov A. Scalar fluctuations from a point source in a turbulent boundary layer. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:036306. [PMID: 22060491 DOI: 10.1103/physreve.84.036306] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/09/2011] [Revised: 07/14/2011] [Indexed: 05/31/2023]
Abstract
The downstream development of the concentration probability distribution along the mean-plume centerline of a dispersing plume in the wake of a ground-level continuous point source in a neutrally stratified wall-shear layer is studied. It is shown that the concentration distribution is well described by a family of one-parameter gamma distributions, as first suggested by Villermaux and Duplat [Phys. Rev. Lett. 91, 184501 (2003)] in the context of confined mixing. A prediction of the downstream evolution of the parameter k (which specifies the gamma distribution) is obtained. This prediction includes explicitly the effects of mean shear on the mean-square concentration.
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Affiliation(s)
- Eugene Yee
- Defence R&D Canada Suffield, P.O. Box 4000 Station Main, Medicine Hat, Alberta, Canada T1A 8K6
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Skvortsov A, Yee E. Scaling laws of peripheral mixing of passive scalar in a wall-shear layer. Phys Rev E 2011; 83:036303. [PMID: 21517583 DOI: 10.1103/physreve.83.036303] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/08/2010] [Indexed: 11/07/2022]
Abstract
The scaling laws governing the concentration moments of a passive scalar released from a ground-level localized source in a neutrally stratified wall-shear layer are investigated using a theoretical framework recently formulated by Lebedev and Turitsyn [Phys. Rev. E 69, 036301 (2004)]. For the current application, this theoretical framework is generalized from the smooth random velocity field applicable in the viscous sublayer to the nonsmooth random velocity field that applies to the bulk of the wall-shear layer. Theoretical relationships for the passive scalar concentration moments are compared to a water-channel simulation of turbulent diffusion from a ground-level source in a wall-shear layer. The diffusion measurements in the wall-shear layer are shown to be consistent with the theoretical description and also imply the robustness of the identified scaling laws for the scalar concentration moments.
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Affiliation(s)
- Alex Skvortsov
- Defence Science and Technology Organisation, 506 Lorimer Street, Fishermans Bend, Victoria 3207, Australia.
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Pan L, Scannapieco E. Passive scalar structures in supersonic turbulence. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:045302. [PMID: 21599230 DOI: 10.1103/physreve.83.045302] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/13/2010] [Revised: 03/16/2011] [Indexed: 05/30/2023]
Abstract
We conduct a systematic numerical study of passive scalar structures in supersonic turbulent flows. We find that the degree of intermittency in the scalar structures increases only slightly as the flow changes from transonic to highly supersonic, while the velocity structures become significantly more intermittent. This difference is due to the absence of shocklike discontinuities in the scalar field. The structure functions of the scalar field are well described by the intermittency model of She and Lévêque [Phys. Rev. Lett 72, 336 (1994)], and the most intense scalar structures are found to be sheetlike at all Mach numbers.
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Affiliation(s)
- Liubin Pan
- School of Earth and Space Exploration, Arizona State University, P.O. Box 871404, Tempe, Arizona 85287-1404, USA
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Duplat J, Jouary A, Villermaux E. Entanglement rules for random mixtures. PHYSICAL REVIEW LETTERS 2010; 105:034504. [PMID: 20867769 DOI: 10.1103/physrevlett.105.034504] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/14/2010] [Indexed: 05/29/2023]
Abstract
We discuss how two subparts of a randomly stirred scalar mixture interact to form the overall concentration distribution. We derive, in particular, the appropriate composition laws in the absence and presence of a strong correlation between the fields. The resulting concentration distributions compare favorably with several distinct experiments, illustrating the two limits. The initial relative spatial position of the subparts plays a crucial role on the nature of their subsequent entanglement.
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Affiliation(s)
- J Duplat
- Aix-Marseille Université, IUSTI, 13453 Marseille Cedex 13, France
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Eidelman A, Elperin T, Kleeorin N, Hazak G, Rogachevskii I, Sadot O, Sapir-Katiraie I. Mixing at the external boundary of a submerged turbulent jet. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:026311. [PMID: 19391844 DOI: 10.1103/physreve.79.026311] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/02/2008] [Revised: 10/03/2008] [Indexed: 05/27/2023]
Abstract
We study experimentally and theoretically mixing at the external boundary of a submerged turbulent jet. In the experimental study we use particle image velocimetry and an image processing technique based on the analysis of the intensity of the Mie scattering to determine the spatial distribution of tracer particles. An air jet is seeded with the incense smoke particles, which are characterized by a large Schmidt number and a small Stokes number. We determine the spatial distributions of the jet fluid characterized by a high concentration of the particles and of the ambient fluid characterized by a low concentration of the tracer particles. In the data analysis we use two approaches, whereby one approach is based on the measured phase function for the study of the mixed state of two fluids. The other approach is based on the analysis of the two-point second-order correlation function of the particle number density fluctuations generated by tangling of the gradient of the mean particle number density by the turbulent velocity field. This gradient is formed at the external boundary of a submerged turbulent jet. We demonstrate that probability density function of the phase function of a jet fluid penetrating into an external flow and the two-point second-order correlation function of the particle number density do not have universal scaling and cannot be described by a power-law function. The theoretical predictions made in this study are in qualitative agreement with the obtained experimental results.
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Affiliation(s)
- A Eidelman
- The Pearlstone Center for Aeronautical Engineering Studies, Department of Mechanical Engineering, Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva 84105, Israel.
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Gouillart E, Dauchot O, Dubrulle B, Roux S, Thiffeault JL. Slow decay of concentration variance due to no-slip walls in chaotic mixing. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:026211. [PMID: 18850925 DOI: 10.1103/physreve.78.026211] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/04/2008] [Revised: 05/08/2008] [Indexed: 05/26/2023]
Abstract
Chaotic mixing in a closed vessel is studied experimentally and numerically in different two-dimensional (2D) flow configurations. For a purely hyperbolic phase space, it is well known that concentration fluctuations converge to an eigenmode of the advection-diffusion operator and decay exponentially with time. We illustrate how the unstable manifold of hyperbolic periodic points dominates the resulting persistent pattern. We show for different physical viscous flows that, in the case of a fully chaotic Poincaré section, parabolic periodic points at the walls lead to slower (algebraic) decay. A persistent pattern, the backbone of which is the unstable manifold of parabolic points, can be observed. However, slow stretching at the wall forbids the rapid propagation of stretched filaments throughout the whole domain, and hence delays the formation of an eigenmode until it is no longer experimentally observable. Inspired by the baker's map, we introduce a 1D model with a parabolic point that gives a good account of the slow decay observed in experiments. We derive a universal decay law for such systems parametrized by the rate at which a particle approaches the no-slip wall.
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Affiliation(s)
- E Gouillart
- Surface du Verre et Interfaces, UMR 125 CNRS/Saint-Gobain, 93303 Aubervilliers, France
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Venaille A, Sommeria J. Is turbulent mixing a self-convolution process? PHYSICAL REVIEW LETTERS 2008; 100:234506. [PMID: 18643510 DOI: 10.1103/physrevlett.100.234506] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/28/2007] [Indexed: 05/26/2023]
Abstract
Experimental results for the evolution of the probability distribution function (PDF) of a scalar mixed by a turbulent flow in a channel are presented. The sequence of PDF from an initial skewed distribution to a sharp Gaussian is found to be nonuniversal. The route toward homogeneization depends on the ratio between the cross sections of the dye injector and the channel. In connection with this observation, advantages, shortcomings, and applicability of models for the PDF evolution based on a self-convolution mechanism are discussed.
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Affiliation(s)
- Antoine Venaille
- Laboratoire des Ecoulements Géophysiques et Industriels (LEGI) CNRS-UJF-INPG, Coriolis, BP53, Grenoble, France.
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Villermaux E, Stroock AD, Stone HA. Bridging kinematics and concentration content in a chaotic micromixer. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 77:015301. [PMID: 18351904 DOI: 10.1103/physreve.77.015301] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/11/2007] [Indexed: 05/15/2023]
Abstract
We analyze the mixing properties of the microfluidic herringbone configuration introduced to mix scalar substances in a narrow channel at low Reynolds but large Péclet numbers. Because of the grooves sculpted on the channel floor, substantial transverse motions are superimposed onto the usual longitudinal Poiseuille dispersion along the channel, whose impact on both the mixing rate and mixture content is quantified. We demonstrate the direct link between the flow kinematics and the deformation rate of the mixture's concentration distribution, whose overall shape is also determined.
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Affiliation(s)
- E Villermaux
- Aix-Marseille Université, IRPHE, 13384 Marseille Cedex 13, France.
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Boudaoud A, Bico J, Roman B. Elastocapillary coalescence: aggregation and fragmentation with a maximal size. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:060102. [PMID: 18233803 DOI: 10.1103/physreve.76.060102] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/17/2007] [Indexed: 05/25/2023]
Abstract
Aggregation processes generally lead to broad distributions of sizes involving exponential tails. Here, experiments on the capillary-driven coalescence of regularly spaced flexible structures yields a self-similar distribution of sizes with no tail. At a given step, the physical process imposes a maximal size for the aggregates, which appears as the relevant scale for the distribution. A simple toy model involving the aggregation of nearest neighbors exhibits the same statistics. A mean-field theory accounting for a maximal size is in agreement with both experiments and numerics. This approach is extended to iterative fragmentation processes where the largest object is broken at each step.
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Affiliation(s)
- Arezki Boudaoud
- Laboratoire de Physique Statistique, UMR 8550 du CNRS/ENS/Paris 6/Paris 7, 24 rue Lhomond, Paris Cedex 5, France
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Károlyi G, Tél T. Effective dimensions and chemical reactions in fluid flows. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:046315. [PMID: 17995114 DOI: 10.1103/physreve.76.046315] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/22/2007] [Indexed: 05/25/2023]
Abstract
We show that chemical activity in hydrodynamical flows can be understood as the outcome of three basic effects: the stirring protocol of the flow, the local properties of the reaction, and the global folding dynamics which also depends on the geometry of the container. The essence of each of these components can be described by simple functional relations. In an ordinary differential equation approach, they determine a new chemical rate equation for the concentration, which turns out to be coupled to the dynamics of an effective fractal dimension. The theory predicts an exponential convergence to the asymptotic chemical state. This holds even in flows characterized by a linear stirring protocol where transient fractal patterns are shown to exist despite the lack of any chaotic set of the advection dynamics. In the exponential case the theory applies to flows of chaotic time dependence (chaotic flows) as well.
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Affiliation(s)
- György Károlyi
- Centre for Applied Dynamics Research, School of Engineering and Physical Sciences, University of Aberdeen, King's College, Aberdeen AB24 3UE, Scotland, United Kingdom
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Bachelard R, Benzekri T, Chandre C, Leoncini X, Vittot M. Targeted mixing in an array of alternating vortices. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:046217. [PMID: 17995092 DOI: 10.1103/physreve.76.046217] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/16/2007] [Indexed: 05/25/2023]
Abstract
Transport and mixing properties of passive particles advected by an array of vortices are investigated. Starting from the integrable case, it is shown that a special class of perturbations allows one to preserve separatrices which act as effective transport barriers, while triggering chaotic advection. In this setting, mixing within the two dynamical barriers is enhanced while long range transport is prevented. A numerical analysis of mixing properties depending on parameter values is performed; regions for which optimal mixing is achieved are proposed. Robustness of the targeted mixing properties regarding errors in the applied perturbation are considered, as well as slip/no-slip and/or boundary conditions for the flow.
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Affiliation(s)
- R Bachelard
- Centre de Physique Théorique, CNRS Aix-Marseille Universités, campus de Luminy, Case 907, F-13288 Marseille cedex 9, France
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29
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Gouillart E, Kuncio N, Dauchot O, Dubrulle B, Roux S, Thiffeault JL. Walls inhibit chaotic mixing. PHYSICAL REVIEW LETTERS 2007; 99:114501. [PMID: 17930441 DOI: 10.1103/physrevlett.99.114501] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/21/2006] [Revised: 04/20/2007] [Indexed: 05/25/2023]
Abstract
We report on experiments of chaotic mixing in a closed vessel, in which a highly viscous fluid is stirred by a moving rod. We analyze quantitatively how the concentration field of a low-diffusivity dye relaxes towards homogeneity, and we observe a slow algebraic decay of the inhomogeneity, at odds with the exponential decay predicted by most previous studies. Visual observations reveal the dominant role of the vessel wall, which strongly influences the concentration field in the entire domain and causes the anomalous scaling. A simplified 1D model supports our experimental results. Quantitative analysis of the concentration pattern leads to scalings for the distributions and the variance of the concentration field consistent with experimental and numerical results.
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Affiliation(s)
- E Gouillart
- Service de Physique de l'Etat Condensé, DSM, CEA Saclay, URA2464, 91191 Gif-sur-Yvette Cedex, France
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30
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Vergassola M, Villermaux E, Shraiman BI. 'Infotaxis' as a strategy for searching without gradients. Nature 2007; 445:406-9. [PMID: 17251974 DOI: 10.1038/nature05464] [Citation(s) in RCA: 269] [Impact Index Per Article: 15.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/01/2006] [Accepted: 11/14/2006] [Indexed: 11/08/2022]
Abstract
Chemotactic bacteria rely on local concentration gradients to guide them towards the source of a nutrient. Such local cues pointing towards the location of the source are not always available at macroscopic scales because mixing in a flowing medium breaks up regions of high concentration into random and disconnected patches. Thus, animals sensing odours in air or water detect them only intermittently as patches sweep by on the wind or currents. A macroscopic searcher must devise a strategy of movement based on sporadic cues and partial information. Here we propose a search algorithm, which we call 'infotaxis', designed to work under such conditions. Any search process can be thought of as acquisition of information on source location; for infotaxis, information plays a role similar to concentration in chemotaxis. The infotaxis strategy locally maximizes the expected rate of information gain. We demonstrate its efficiency using a computational model of odour plume propagation and experimental data on mixing flows. Infotactic trajectories feature 'zigzagging' and 'casting' paths similar to those observed in the flight of moths. The proposed search algorithm is relevant to the design of olfactory robots, but the general idea of infotaxis can be applied more broadly in the context of searching with sparse information.
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Affiliation(s)
- Massimo Vergassola
- CNRS URA 2171, Institut Pasteur, In Silico Genetics, 75724 Paris Cedex 15, France
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31
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Villermaux E, Duplat J. Coarse grained scale of turbulent mixtures. PHYSICAL REVIEW LETTERS 2006; 97:144506. [PMID: 17155260 DOI: 10.1103/physrevlett.97.144506] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/09/2006] [Indexed: 05/12/2023]
Abstract
What is the physical length scale which supports the concentration content in a stirred mixture? Among the length scales familiar in stirred mixtures is the dissipation scale which equilibrates substrate deformation and diffusive smearing rates. That scale is a decreasing function of the deformation rate and is, thus, a decreasing function of the Reynolds number in turbulent flows. Experiments show that the mixture concentration content is defined on a support whose elementary brick eta=LSc(-2/5) is much larger. It scales like the stirring scale L, depends on the Schmidt number Sc, and is independent of the Reynolds number. The above law is supported by measurements covering two decades in L and three decades in Sc. We suggest that this scale results from the aggregation of bundles of elementary stretched scalar sheets merging under large-scale substrate deformation.
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Affiliation(s)
- E Villermaux
- IRPHE, Université de Provence, 13384 Marseille Cedex 13, France
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Kushnir D, Schumacher J, Brandt A. Geometry of intensive scalar dissipation events in turbulence. PHYSICAL REVIEW LETTERS 2006; 97:124502. [PMID: 17025974 DOI: 10.1103/physrevlett.97.124502] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/11/2006] [Indexed: 05/12/2023]
Abstract
The maxima of the scalar dissipation rate in turbulence appear in the form of sheets and correspond to the potentially most intensive scalar mixing events. Their cross section extension determines a locally varying diffusion scale of the mixing process and extends the classical Batchelor picture of one mean diffusion scale. The distribution of the local diffusion scales is analyzed for different Reynolds and Schmidt numbers with a fast multiscale technique applied to very high-resolution simulation data. The scales always take values across the whole Batchelor range and beyond. Furthermore, their distribution is traced back to the distribution of the contractive short-time Lyapunov exponent of the flow.
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Affiliation(s)
- Dan Kushnir
- Department of Computer Science and Applied Mathematics, The Weizmann Institute of Science, 76100 Rehovot, Israel
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33
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Sultan E, Boudaoud A. Statistics of crumpled paper. PHYSICAL REVIEW LETTERS 2006; 96:136103. [PMID: 16712006 DOI: 10.1103/physrevlett.96.136103] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/05/2005] [Indexed: 05/09/2023]
Abstract
A statistical study of crumpled paper is allowed by a minimal 1D model: a self-avoiding line bent at sharp angles--in which the elastic energy resides--put in a confining potential. Many independent equilibrium configurations are generated numerically and their properties are investigated. At small confinement, the distribution of segment lengths is log-normal in agreement with previous predictions and experiments. At high confinement, the system approaches a jammed state with a critical behavior, whereas the length distribution follows a gamma law in which the parameter is predicted as a function of the number of layers in the system.
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Affiliation(s)
- Eric Sultan
- Laboratoire de Physique Statistique, UMR 8550 du CNRS/Paris VI/Paris VII, Ecole normale supérieure, 24 rue Lhomond, F-75231 Paris Cedex 05, France.
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Doshi P, Cohen I, Zhang WW, Siegel M, Howell P, Basaran OA, Nagel SR. Persistence of Memory in Drop Breakup: The Breakdown of Universality. Science 2003; 302:1185-8. [PMID: 14615531 DOI: 10.1126/science.1089272] [Citation(s) in RCA: 123] [Impact Index Per Article: 5.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/03/2022]
Abstract
A low-viscosity drop breaking apart inside a viscous fluid is encountered when air bubbles, entrained in thick syrup or honey, rise and break apart. Experiments, simulations, and theory show that the breakup under conditions in which the interior viscosity can be neglected produces an exceptional form of singularity. In contrast to previous studies of drop breakup, universality is violated so that the final shape at breakup retains an imprint of the initial and boundary conditions. A finite interior viscosity, no matter how small, cuts off this form of singularity and produces an unexpectedly long and slender thread. If exterior viscosity is large enough, however, the cutoff does not occur because the minimum drop radius reaches subatomic dimensions first.
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Affiliation(s)
- Pankaj Doshi
- School of Chemical Engineering, Purdue University, West Lafayette, IN 47907, USA
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