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- By Rose Teteki Abbey, K. C. Abraham, David Tuesday Adamo, LeRoy H. Aden, Efrain Agosto, Victor Aguilan, Gillian T. W. Ahlgren, Charanjit Kaur AjitSingh, Dorothy B E A Akoto, Giuseppe Alberigo, Daniel E. Albrecht, Ruth Albrecht, Daniel O. Aleshire, Urs Altermatt, Anand Amaladass, Michael Amaladoss, James N. Amanze, Lesley G. Anderson, Thomas C. Anderson, Victor Anderson, Hope S. Antone, María Pilar Aquino, Paula Arai, Victorio Araya Guillén, S. Wesley Ariarajah, Ellen T. Armour, Brett Gregory Armstrong, Atsuhiro Asano, Naim Stifan Ateek, Mahmoud Ayoub, John Alembillah Azumah, Mercedes L. García Bachmann, Irena Backus, J. Wayne Baker, Mieke Bal, Lewis V. Baldwin, William Barbieri, António Barbosa da Silva, David Basinger, Bolaji Olukemi Bateye, Oswald Bayer, Daniel H. Bays, Rosalie Beck, Nancy Elizabeth Bedford, Guy-Thomas Bedouelle, Chorbishop Seely Beggiani, Wolfgang Behringer, Christopher M. Bellitto, Byard Bennett, Harold V. Bennett, Teresa Berger, Miguel A. 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Deichmann Edwards, Miguel A. De La Torre, George E. Demacopoulos, Thomas de Mayo, Leah DeVun, Beatriz de Vasconcellos Dias, Dennis C. Dickerson, John M. Dillon, Luis Miguel Donatello, Igor Dorfmann-Lazarev, Susanna Drake, Jonathan A. Draper, N. Dreher Martin, Otto Dreydoppel, Angelyn Dries, A. J. Droge, Francis X. D'Sa, Marilyn Dunn, Nicole Wilkinson Duran, Rifaat Ebied, Mark J. Edwards, William H. Edwards, Leonard H. Ehrlich, Nancy L. Eiesland, Martin Elbel, J. Harold Ellens, Stephen Ellingson, Marvin M. Ellison, Robert Ellsberg, Jean Bethke Elshtain, Eldon Jay Epp, Peter C. Erb, Tassilo Erhardt, Maria Erling, Noel Leo Erskine, Gillian R. Evans, Virginia Fabella, Michael A. Fahey, Edward Farley, Margaret A. Farley, Wendy Farley, Robert Fastiggi, Seena Fazel, Duncan S. Ferguson, Helwar Figueroa, Paul Corby Finney, Kyriaki Karidoyanes FitzGerald, Thomas E. FitzGerald, John R. Fitzmier, Marie Therese Flanagan, Sabina Flanagan, Claude Flipo, Ronald B. 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Yee, Viktor Yelensky, Yeo Khiok-Khng, Gustav K. K. Yeung, Angela Yiu, Amos Yong, Yong Ting Jin, You Bin, Youhanna Nessim Youssef, Eliana Yunes, Robert Michael Zaller, Valarie H. Ziegler, Barbara Brown Zikmund, Joyce Ann Zimmerman, Aurora Zlotnik, Zhuo Xinping
- Edited by Daniel Patte, Vanderbilt University, Tennessee
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- The Cambridge Dictionary of Christianity
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- 05 August 2012
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- 20 September 2010, pp xi-xliv
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Convection in 3He–superfluid-4He mixtures. Part 2. A survey of instabilities
- Guy Metcalfe, R. P. Behringer
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- Journal:
- Journal of Fluid Mechanics / Volume 307 / 25 January 1996
- Published online by Cambridge University Press:
- 26 April 2006, pp. 297-331
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Dilute mixtures of 3He in superfluid 4He have Prandtl numbers easily tunable between those of liquid metals and water: 0.04 < Pr < 2. Moreover, superfluid mixture convection is closely analogous to classical Rayleigh–Bénard convection, i.e. superfluid mixtures convect as if they were classical, single-component fluids. This work has two goals. The first, accomplished in Part 1, is to experimentally validate the superfluid mixture convection analogue to Rayleigh–Bénard convection.
With superfluid effects understood and under control, the second goal is to identify and characterize time-dependence and chaos and to discover new dynamical behaviour in strongly nonlinear convective flows. In this paper, Part 2, we exploit the unique Pr range of superfluid mixtures and the variable aspect ratio (Γ) capabilities of our experiment to survey convective instabilities in the broad, and heretofore largely unexplored, parameter space 0.12 < Pr < 1.4 and 2 < Γ < 95. Within this large parameter space, we have focused on small to moderate Γ and Pr and on large Γ with Pr ≈ 1. The novel behaviour uncovered in the survey includes the following. Changing attractors: at Γ = 6.0 and Pr = 0.3, we observe intermittent bursting destabilizing a fully developed chaotic state. Above the onset of bursting the average length of a burst-free interval and the average length of a burst vary as power laws. At Γ = 4.25 and Pr = 0.12 we observe a particularly novel reversible switching transition involving two chaotic attractors. Instability competition: near the codimension-2 point at the crossing of the skewed-varicose and oscillatory instabilities we find that the effects of instability competition greatly increase the complexity and multiplicity of states. A heat-pulse method allows selection of the active state. Decreasing Γ suppresses the available complexity. Superfluid turbulence: we find that the large-amplitude noisy states, previously believed due to superfluid turbulence, are confined to small values of Γ and Pr and are not consistent with superfluid turbulence. Changing instabilities: at Pr = 0.19 a wavevector detuning changes the type of secondary instability from oscillatory to saddle-node, with an unusual 3/4 exponent time scaling. Very large Γ: at Pr = 1.3 for Γ increasing from 44 to 90, we observe the onset of convection changing from ordered and stationary to disordered and time-dependent. At the beginning of the crossover there are hysteretic transitions to coherent oscillations close to the onset of convection. By the end of the crossover convection is time-dependent and irregular at onset with the fluctuation amplitude correlated with the mean Nusselt number.
Convection in 3He–superfluid-4He mixtures. Part 1. A Boussinesq analogue
- Guy Metcalfe, R. P. Behringer
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- Journal:
- Journal of Fluid Mechanics / Volume 307 / 25 January 1996
- Published online by Cambridge University Press:
- 26 April 2006, pp. 269-296
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Dilute mixtures of 3He in superfluid 4He have Prandtl numbers easily tunable between those of liquid metals and water: 0.04 < Pr < 2. Moreover, owing to the tight coupling of the temperature and concentration fields, superfluid mixture convection is closely analogous to classical Rayleigh–Bénard convection, i.e. superfluid mixtures convect as if they were classical, single-component fluids, well described by the Boussinesq equations. This work has two goals. The first is to put the theory of superfluid mixture convection on a firmer basis. We accomplish this by combining experiment and analysis to measure superfluid effects on the onset of convection. In the process, we demonstrate quantitative control over superfluid effects and, in particular, that deviations from classical convective behaviour can be made small or at worst no larger than finite aspect ratio effects. The size of superfluid effects at convective onset can be less than a few percent for temperatures 1 < T < 2 K. Comparison of the measured properties of superfluid mixture roll instabilities above the onset of convection (e.g. skewed varicose, oscillatory, and particularly near the codimension-2 point) to the properties predicted by Boussinesq calculations further verifies that superfluid mixtures convect as classical fluids.
With superfluid effects understood and under control, the second goal, presented in Part 2, is to exploit the unique Pr range of superfluid mixtures and the variable aspect ratio (Γ) capabilities of our experiment to survey convective instabilities in the broad, and heretofore largely unexplored, parameter space 0.12 < Pr < 1.4 and 2 < Γ < 95. The aim is to identify and characterize time-dependence and chaos, and to discover new dynamical behaviour in strongly nonlinear convective flows.
The onset of convection and turbulence in rectangular layers of normal liquid 4He
- R. W. Motsay, K. E. Anderson, R. P. Behringer
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- Journal of Fluid Mechanics / Volume 189 / April 1988
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- 21 April 2006, pp. 263-286
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We have carried out high-precision measurements of the heat transport in intermediate-size rectangular layers of convecting normal liquid 4He with Prandtl numbers of 0.52 and 0.70. The containers used for these experiments had horizontal dimensions, in units of the height d, of 13.4 × 5.95 (cell I) and 18.2 × 8.12 (cell II). The slopes N1 of the Nusselt curves were 0.56 and 0.70 respectively for cell I and cell II. These values are significantly lower than predictions for N1 for horizontally unbound layers, but comparable with results obtained in cylindrical containers of liquid helium with roughly the same number of convection rolls. For the two containers, the onset of the first instability after the onset of convection occurred at Rayleigh numbers R1 that were in reasonable quantitative agreement with the predictions of Busse and Clever for the skewed-varicose instability. For both containers, the transition at R1 was characterized by long transients ranging from ∼ 102 to ∼ 103 vertical-thermal-diffusion times. A decrease in the Nusselt number was also observed. As the Rayleigh number was increased above R1, a new steady state evolved and then additional transitions were observed. These transitions occurred at Rayleigh numbers labelled R2, R3,…, with a total of five transitions seen in cell I and a total of three transitions seen for cell II. The transition for each cell at R2 can be related quantitatively to the skewed-varicose instability, and the transition at R3 is associated with an oscillatory instability. For cell II, the time-dependence beginning at R3 persisted to the highest Rayleigh number studied, R = 11.7Rc. However, for container I, two more regimes of time-independent flow were observed; the last of these was at an unexpectedly high Rayleigh number of 6.7Rc. This work extends to lower Prandtl number recent studies made on moderate-size rectangular layers of convecting water and alcohol.
Heat-flow experiments in liquid 4He with a variable cylindrical geometry
- H. Gao, G. Metcalfe, T. Jung, R. P. Behringer
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- Journal:
- Journal of Fluid Mechanics / Volume 174 / January 1987
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- 21 April 2006, pp. 209-231
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This paper first describes an apparatus for measuring the Nusselt number N versus the Rayleigh number R of convecting normal liquid 4He layers. The most important feature of the apparatus is its ability to provide layers of different heights d, and hence different aspect ratios [Gcy ]. The horizontal cross-section of each layer is circular, and [Gcy ] is defined by [Gcy ] = D/2d where D is the diameter of the layer. We report results for 2.4 [les ] [Gcy ] [les ] 16 and for Prandtl numbers Pr spanning 0.5 [lsim ] Pr [lsim ] 0.9 These results are presented in terms of the slope N1 = RcdN/dR evaluated just above the onset of convection at Rc. We find that N1 is only a slowly increasing function of [Gcy ] in the range 6 [lsim ] [Gcy ] [lsim ] 16, and that it has a value there which is quite close to 0.72. This value of N1 is in good agreement with variational calcuations by Ahlers et al. (1981) pertinent to parallel convection rolls in cylindrical geometry. Particularly for [Gcy ] [lsim ] 6, we find additional small-scale structure in N1 associated with changes in the number of convection rolls with changing [Gcy ]. An additional test of the linearzied hydrodynamics is given by measurements of Rc. We find good agreement between theory and our data for Rc.
Linear growth rates for the Rayleigh-Bénard instability in cylindrical geometry
- J. N. Shaumeyer, R. P. Behringer, Ralph Baierlein
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- Journal:
- Journal of Fluid Mechanics / Volume 109 / August 1981
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- 20 April 2006, pp. 339-348
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We report theoretical growth rates for the Rayleigh–Bénard instability when the fluid layer is contained by non-slip walls in a cylindrical geometry with diameter D and height L. Our results are for the growth rates of the first two axisymmetric modes as functions of the Prandtl number P and the aspect ratio γ≡D/2L. We have considered the two extreme cases of ideally insulating and ideally conducting side walls, and found that the growth rate is relatively insensitive to the choice of the thermal boundary conditions on the side walls. Our results are useful in understanding recent experimental measurements of the convective time-scale.
Heat transport and temporal evolution of fluid flow near the Rayleigh-Bénard instability in cylindrical containers
- R. P. Behringer, Guenter Ahlers
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- Journal:
- Journal of Fluid Mechanics / Volume 125 / December 1982
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- 20 April 2006, pp. 219-258
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First this paper describes in detail an apparatus for heat-transport measurements in shallow horizontal layers of fluid at low temperatures. Then high-precision results of convective heat transport as a function of the Rayleigh number R are presented for cylindrical cells of aspect ratio L = 2.08,4.72 and 57. The present paper concentrates on the long-time behaviour of Boussinesq systems. Non-Boussinesq effects, transient effects near the convective onset, and time-dependent states are described elsewhere (Walden & Ahlers 1981 Ahlers et al. 1981 Ahlers 1980b and references therein). The measurements show that the convective onset near the critical Rayleigh number Rc is sharp within the experimental resolution of about 0.1 % of the Nusselt number N even in laterally finite containers. Values of R and of the initial slopes of N(R), are obtained and compared with predictions for different flow patterns. Over a wider range of R and for L = 57 and 4.72, N was found within experimental resolution to be a unique, continuous function of R For L = 2.08, hysteretic transitions are revealed by N(R) near R ≈ 3 and R ≈ 10. For L = 4.72, the effect of impulsive heating was studied and revealed complicated, long-lived, but surprisingly repro- ducible transients.
Friction and Flow in Granular Materials
- R. P. Behringer, L. Kondic, G. Metcalfe, D. Schaeffer, Sarath G. K. Tennakoon
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- Journal:
- MRS Online Proceedings Library Archive / Volume 627 / 2000
- Published online by Cambridge University Press:
- 01 February 2011, BB5.4
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- 2000
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We probe the transitions between solid-like and fluid-like granular states in the presence of shaking in the horizontal and vertical directions. These transitions are fundamental to other aspects of granular flow such as avalanche flow, in which there is a free surface. Key control parameters include accelerations in the vertical and horizontal directions, Γi = Aiω2i/g, for shaking of the form si = Ai cos(ωit + φi), i = h, v. Here, g is the acceleration of gravity. Also important is the relative phase between the two modes of shaking. We focus on low to moderate dimensionless accelerations, 0 < Γv,h < 1.6. We consider first the case Γv = 0, i.e. pure horizontal shaking. In this case, there is a hysteretic transition between solid and fluid states, where the fluid state consists of a sloshing layer of material of height H plus additional transverse flow. The hysteresis is lifted in the presence of a modest amount of fluidization by gas flow, or if a slight overburden is provided. We also identify a time scale, τ, for the transition between the phases that diverges inversely as the distance ε = (Γh–Γhc)/Γhc, from the appropriate transition points, i.e. as τ α ε-1. We identify a new convective mechanism, associated with horizontal shearing at the walls, as the mechanism that drives the transverse convective flow. For combined horizontal and vertical shaking, there exist a related set of novel dynamics and stability properties. These include the spontaneous formation of a static heap and a transition to flow, similar to the flow state under horizontal shaking, when the vertical acceleration Γv < 1. A simple friction model provides a good description of the steady states and a reasonably good description of the transition to flow. Horizontal and vertical shaking frequencies that differ by a small amount can lead to a novel switching state, as the relative phase, φh—φv, shifts over time.
Convection and flow in porous media. Part 1. Visualization by magnetic resonance imaging
- M. D. Shattuck, R. P. Behringer, G. A. Johnson, J. G. Georgiadis
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- Journal:
- Journal of Fluid Mechanics / Volume 332 / February 1997
- Published online by Cambridge University Press:
- 10 February 1997, pp. 215-245
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We describe an experimental study of porous media convection (PMC) from onset to 8Rac. The goal of this work is to provide non-invasive imaging and high-precision heat transport measurements to test theories of convection in PMC. We obtain velocity information and visualize the convection patterns using magnetic resonance imaging (MRI). We study both ordered and disordered packings of mono-disperse spheres of diameter d = 3.204 ± 0.029 mm, in circular, rectangular, and hexagonal planforms. In general, the structure of the medium plays a role which is not predicted by theories which assume a homogeneous system. Disordered media are prepared by pouring mono-disperse spheres into the container. Large ordered regions of close packing for the spheres, with grain boundaries and isolated defects, characterize these media. The defects and grain boundaries play an important role in pattern formation in disordered media. Any deviation from close packing produces a region of larger porosity, hence locally larger permeability. The result is spatial variations in the Rayleigh number, Ra. We define the critical Ra, Rac, as the Rayleigh number at the onset of convection in the ordered regions. We find that stable localized convective regions exist around grain boundaries and defects at Ra < Rac. These remain as pinning sites for the convection patterns in the ordered regions as Ra increases above Rac up to 5Rac, the highest Ra studied in the disordered media. In ordered media, spheres are packed such that the only deviations from close packing occur within a thin (<d) region near the vertical walls. Stable localized convection begins at 0.5Rac in the wall regions but appears to play only a weak role in the pattern formation of the interior regions (bulk), since different stable patterns are observed in the bulk at the same Ra after each cycling of Ra below Rac, even for similar patterns of small rolls in the wall regions. The experiments provide a test of the following predictions for PMC: (i) that straight parallel rolls should be linearly stable for Rac < Ra < 5Rac; (ii) that at onset, the rolls should have a dimensionless wavevector qc = π; (iii) that at the upper end of this range rolls should lose stability to cross-rolls; (iv) that the initial slope of the Nusselt curve should be 2; (v) that there should be a rapid decay of vertical vorticity - hence no complex flows, such as those which occur for Rayleigh- Benard convection (RBC) within the nominal regime of stable parallel rolls. These predictions are in partial agreement with our findings for the bulk convection in the ordered media. We observe roll-like structures which relax rapidly to stable patterns between Rac and 5Rac. However we find a wavenumber which is 0.7π compared to π derived from linear stability theory. We find an asymmetry between the size of the upfiowing regions and downfiowing regions as Ra grows above Rac. The ratio of the volume of the upfiowing to the volume of the downfiowing regions decreases as Ra increases and leads to a novel time-dependent state, which does not consist of cross-rolls. This time-dependent state begins at 6Rac and is observed up to 8Rac, the largest Ra which we studied. It seems likely that the occurrence of this state is linked to departures from the Boussinesq approximation at higher Ra. We also find that the slope of the Nusselt curve is 0.7, which does not agree with the predicted value of 2.
Convection and flow in porous media. Part 2. Visualization by shadowgraph
- L. E. Howle, R. P. Behringer, J. G. Georgiadis
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- Journal:
- Journal of Fluid Mechanics / Volume 332 / February 1997
- Published online by Cambridge University Press:
- 10 February 1997, pp. 247-262
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We present results for pattern formation at the onset of convection in fluid-saturated porous media obtained by a novel variation on the shadowgraphic technique (modified shadowgraphic technique). Both ordered and disordered media are used, each exhibiting distinct behaviour. Ordered porous media are constructed from grids of overlapping bars. Convective onset in this type of medium is characterized by a sharp, well-defined bifurcation to straight parallel rolls. The orientation of the convection rolls is determined by the number of bar layers, Nb; odd Nb leads to rolls with axes perpendicular to the direction of the top and bottom bars, and even Nb to rolls at 45° to the bars. Disordered porous layers are produced by stacking randomly drilled disks separated by spacers. In this system, we observe a rounded bifurcation to convection with localized convection near convective onset. More specifically, the flow patterns take on one of several different three-dimensional cellular structures after each cycling through convective onset. These observations may be described by two different mechanisms: random spatial fluctuations in the Rayleigh number (Zimmermann et al. 1993), and/or spatial variation in the thermal conductivity on the length scale of the convection wavelength (Braester & Vadasz 1993).
The Scientist in the Sandbox: Complexity and Dynamics in Granular Flow
- R. P. Behringer
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- Journal:
- MRS Online Proceedings Library Archive / Volume 367 / 1994
- Published online by Cambridge University Press:
- 03 September 2012, 461
- Print publication:
- 1994
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Granular materials exhibit a rich variety of dynamical behavior, much of which is poorly understood. Fractal-like stress chains, convection, a variety of wave dynamics, including waves which resemble capillary waves, and fractional Brownian motion provide examples. Although granular materials consist of collections of interacting particles, there are important differences between the dynamics of a collections of grains and the dynamics of a collections of molecules; in particular, the ergodic hypothesis is generally invalid for granular materials, so that ordinary statistical physics does not apply. Nonlinear Dynamics, Mathematics, Molecular Dynamics, and Condensed Matter Physics as well as traditional Engineering fields have all contributed to recent insights for these phenomena.