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Doan M, Simons JJ, Lilienthal K, Solomon T, Mitchell KA. Barriers to front propagation in laminar, three-dimensional fluid flows. Phys Rev E 2018; 97:033111. [PMID: 29776060 DOI: 10.1103/physreve.97.033111] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/31/2017] [Indexed: 11/07/2022]
Abstract
We present experiments on one-way barriers that block reaction fronts in a fully three-dimensional (3D) fluid flow. Fluorescent Belousov-Zhabotinsky reaction fronts are imaged with laser-scanning in a laminar, overlapping vortex flow. The barriers are analyzed with a 3D extension to burning invariant manifold (BIM) theory that was previously applied to two-dimensional advection-reaction-diffusion processes. We discover tube and sheet barriers that guide the front evolution. The experimentally determined barriers are explained by BIMs calculated from a model of the flow.
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Affiliation(s)
- Minh Doan
- Department of Physics and Astronomy, Bucknell University, Lewisburg, Pennsylvania 17837, USA
| | - J J Simons
- Department of Physics and Astronomy, Bucknell University, Lewisburg, Pennsylvania 17837, USA
| | - Katherine Lilienthal
- Department of Physics and Astronomy, Bucknell University, Lewisburg, Pennsylvania 17837, USA
| | - Tom Solomon
- Department of Physics and Astronomy, Bucknell University, Lewisburg, Pennsylvania 17837, USA
| | - Kevin A Mitchell
- School of Natural Sciences, University of California, Merced, California 95344, USA
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Perlekar P, Pal N, Pandit R. Two-dimensional Turbulence in Symmetric Binary-Fluid Mixtures: Coarsening Arrest by the Inverse Cascade. Sci Rep 2017; 7:44589. [PMID: 28322219 PMCID: PMC5359549 DOI: 10.1038/srep44589] [Citation(s) in RCA: 52] [Impact Index Per Article: 7.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/10/2016] [Accepted: 02/10/2017] [Indexed: 11/28/2022] Open
Abstract
We study two-dimensional (2D) binary-fluid turbulence by carrying out an extensive direct numerical simulation (DNS) of the forced, statistically steady turbulence in the coupled Cahn-Hilliard and Navier-Stokes equations. In the absence of any coupling, we choose parameters that lead (a) to spinodal decomposition and domain growth, which is characterized by the spatiotemporal evolution of the Cahn-Hilliard order parameter ϕ, and (b) the formation of an inverse-energy-cascade regime in the energy spectrum E(k), in which energy cascades towards wave numbers k that are smaller than the energy-injection scale kin j in the turbulent fluid. We show that the Cahn-Hilliard-Navier-Stokes coupling leads to an arrest of phase separation at a length scale Lc, which we evaluate from S(k), the spectrum of the fluctuations of ϕ. We demonstrate that (a) Lc ~ LH, the Hinze scale that follows from balancing inertial and interfacial-tension forces, and (b) Lc is independent, within error bars, of the diffusivity D. We elucidate how this coupling modifies E(k) by blocking the inverse energy cascade at a wavenumber kc, which we show is ≃2π/Lc. We compare our work with earlier studies of this problem.
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Affiliation(s)
- Prasad Perlekar
- TIFR Centre for Interdisciplinary Sciences, 21 Brundavan Colony, Narsingi, Hyderabad 500075, India
| | - Nairita Pal
- Centre for Condensed Matter Theory, Department of Physics, Indian Institute of Science, Bangalore 560012 India
| | - Rahul Pandit
- Centre for Condensed Matter Theory, Department of Physics, Indian Institute of Science, Bangalore 560012 India.,Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore, India
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Mahoney JR, Li J, Boyer C, Solomon T, Mitchell KA. Frozen reaction fronts in steady flows: A burning-invariant-manifold perspective. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:063005. [PMID: 26764802 DOI: 10.1103/physreve.92.063005] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/27/2015] [Indexed: 06/05/2023]
Abstract
The dynamics of fronts, such as chemical reaction fronts, propagating in two-dimensional fluid flows can be remarkably rich and varied. For time-invariant flows, the front dynamics may simplify, settling in to a steady state in which the reacted domain is static, and the front appears "frozen." Our central result is that these frozen fronts in the two-dimensional fluid are composed of segments of burning invariant manifolds, invariant manifolds of front-element dynamics in xyθ space, where θ is the front orientation. Burning invariant manifolds (BIMs) have been identified previously as important local barriers to front propagation in fluid flows. The relevance of BIMs for frozen fronts rests in their ability, under appropriate conditions, to form global barriers, separating reacted domains from nonreacted domains for all time. The second main result of this paper is an understanding of bifurcations that lead from a nonfrozen state to a frozen state, as well as bifurcations that change the topological structure of the frozen front. Although the primary results of this study apply to general fluid flows, our analysis focuses on a chain of vortices in a channel flow with an imposed wind. For this system, we present both experimental and numerical studies that support the theoretical analysis developed here.
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Affiliation(s)
| | - John Li
- University of California, Merced, California 95344, USA
- University of Southern California, Los Angeles, California 90089, USA
| | - Carleen Boyer
- Bucknell University, Lewisburg, Pennsylvania 17837, USA
| | - Tom Solomon
- Bucknell University, Lewisburg, Pennsylvania 17837, USA
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Baskan O, Speetjens MFM, Metcalfe G, Clercx HJH. Direct experimental visualization of the global Hamiltonian progression of two-dimensional Lagrangian flow topologies from integrable to chaotic state. CHAOS (WOODBURY, N.Y.) 2015; 25:103106. [PMID: 26520072 DOI: 10.1063/1.4930837] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
Countless theoretical/numerical studies on transport and mixing in two-dimensional (2D) unsteady flows lean on the assumption that Hamiltonian mechanisms govern the Lagrangian dynamics of passive tracers. However, experimental studies specifically investigating said mechanisms are rare. Moreover, they typically concern local behavior in specific states (usually far away from the integrable state) and generally expose this indirectly by dye visualization. Laboratory experiments explicitly addressing the global Hamiltonian progression of the Lagrangian flow topology entirely from integrable to chaotic state, i.e., the fundamental route to efficient transport by chaotic advection, appear non-existent. This motivates our study on experimental visualization of this progression by direct measurement of Poincaré sections of passive tracer particles in a representative 2D time-periodic flow. This admits (i) accurate replication of the experimental initial conditions, facilitating true one-to-one comparison of simulated and measured behavior, and (ii) direct experimental investigation of the ensuing Lagrangian dynamics. The analysis reveals a close agreement between computations and observations and thus experimentally validates the full global Hamiltonian progression at a great level of detail.
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Affiliation(s)
- O Baskan
- Fluid Dynamics Laboratory, Department of Applied Physics, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
| | - M F M Speetjens
- Energy Technology Laboratory, Department of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
| | - G Metcalfe
- Commonwealth Scientific and Industrial Research Organisation, Melbourne, Victoria 3190, Australia; and Swinburne University of Technology, Department of Mechanical Engineering, Hawthorn VIC 3122, Australia
| | - H J H Clercx
- Fluid Dynamics Laboratory, Department of Applied Physics, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
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Gowen S, Solomon T. Experimental studies of coherent structures in an advection-reaction-diffusion system. CHAOS (WOODBURY, N.Y.) 2015; 25:087403. [PMID: 26328574 DOI: 10.1063/1.4918594] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
We present experimental studies of reaction front propagation in a single vortex flow with an imposed external wind. The fronts are produced by the excitable, ferroin-catalyzed Belousov-Zhabotinsky chemical reaction. The flow is generated using an electromagnetic forcing technique: an almost-radial electrical current interacts with a magnetic field from a magnet below the fluid layer to produce the vortex. The magnet is mounted on crossed translation stages allowing for movement of the vortex through the flow. Reaction fronts triggered in or in front of the moving vortex form persistent structures that are seen experimentally for time-independent (constant motion), time-periodic, and time-aperiodic flows. These results are examined with the use of burning invariant manifolds that act as one-way barriers to front motion in the flows. We also explore the usefulness of finite-time Lyapunov exponent fields as an instrument for analyzing front propagation behavior in a fluid flow.
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Affiliation(s)
- Savannah Gowen
- Department of Physics and Astronomy, Bucknell University, Lewisburg, Pennsylvania 17837, USA
| | - Tom Solomon
- Department of Physics and Astronomy, Bucknell University, Lewisburg, Pennsylvania 17837, USA
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Mitchell KA, Mahoney JR. Invariant manifolds and the geometry of front propagation in fluid flows. CHAOS (WOODBURY, N.Y.) 2012; 22:037104. [PMID: 23020495 DOI: 10.1063/1.4746039] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/01/2023]
Abstract
Recent theoretical and experimental work has demonstrated the existence of one-sided, invariant barriers to the propagation of reaction-diffusion fronts in quasi-two-dimensional periodically driven fluid flows. These barriers were called burning invariant manifolds (BIMs). We provide a detailed theoretical analysis of BIMs, providing criteria for their existence, a classification of their stability, a formalization of their barrier property, and mechanisms by which the barriers can be circumvented. This analysis assumes the sharp front limit and negligible feedback of the front on the fluid velocity. A low-dimensional dynamical systems analysis provides the core of our results.
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Affiliation(s)
- Kevin A Mitchell
- School of Natural Sciences, University of California, Merced, California 95344, USA.
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Bargteil D, Solomon T. Barriers to front propagation in ordered and disordered vortex flows. CHAOS (WOODBURY, N.Y.) 2012; 22:037103. [PMID: 23020494 DOI: 10.1063/1.4746764] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/01/2023]
Abstract
We present experiments on reactive front propagation in a two-dimensional (2D) vortex chain flow (both time-independent and time-periodic) and a 2D spatially disordered (time-independent) vortex-dominated flow. The flows are generated using magnetohydrodynamic forcing techniques, and the fronts are produced using the excitable, ferroin-catalyzed Belousov-Zhabotinsky chemical reaction. In both of these flows, front propagation is dominated by the presence of burning invariant manifolds (BIMs) that act as barriers, similar to invariant manifolds that dominate the transport of passive impurities. Convergence of the fronts onto these BIMs is shown experimentally for all of the flows studied. The BIMs are also shown to collapse onto the invariant manifolds for passive transport in the limit of large flow velocities. For the disordered flow, the measured BIMs are compared to those predicted using a measured velocity field and a three-dimensional set of ordinary differential equations that describe the dynamics of front propagation in advection-reaction-diffusion systems.
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Affiliation(s)
- Dylan Bargteil
- Department of Physics and Astronomy, Bucknell University, Lewisburg, Pennsylvania 17837, USA
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Burke K, Mitchell KA, Wyker B, Ye S, Dunning FB. Demonstration of turnstiles as a chaotic ionization mechanism in Rydberg atoms. PHYSICAL REVIEW LETTERS 2011; 107:113002. [PMID: 22026660 DOI: 10.1103/physrevlett.107.113002] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/17/2011] [Indexed: 05/31/2023]
Abstract
We present an experimental and theoretical study of the chaotic ionization of quasi-one-dimensional potassium Rydberg wave packets via a classical phase-space turnstile mechanism. Turnstiles form a general transport mechanism for numerous chaotic systems, and this study explicitly illuminates their relevance to atomic ionization. We create time-dependent Rydberg wave packets, subject them to alternating applied electric-field pulses, and measure the electron survival probability. Ionization depends not only on the initial electron energy, but also on the classical phase-space position of the electron with respect to the turnstile--that part of the electron packet inside the turnstile ionizes after the applied ionization sequence, while that part outside the turnstile does not. The survival data thus encode information on the shape and location of the turnstile, in good agreement with theoretical predictions.
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Affiliation(s)
- Korana Burke
- School of Natural Sciences, University of California, Merced, California 95344, USA
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Balasuriya S. Optimal frequency for microfluidic mixing across a fluid interface. PHYSICAL REVIEW LETTERS 2010; 105:064501. [PMID: 20867982 DOI: 10.1103/physrevlett.105.064501] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/06/2010] [Revised: 06/23/2010] [Indexed: 05/29/2023]
Abstract
A new analytical tool for determining the optimum frequency for a micromixing strategy to mix two fluids across their interface is presented. The frequency dependence of the flux is characterized in terms of a Fourier transform related to the apparatus geometry. Illustrative microfluidic mixing examples based on electromagnetic forcing and fluid pumping strategies are presented.
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Affiliation(s)
- Sanjeeva Balasuriya
- School of Mathematical Sciences, University of Adelaide, SA 5005, Australia.
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Paoletti MS, Solomon TH. Front propagation and mode-locking in an advection-reaction-diffusion system. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 72:046204. [PMID: 16383509 DOI: 10.1103/physreve.72.046204] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/22/2005] [Indexed: 05/05/2023]
Abstract
Experiments are presented on chemical front propagation in an oscillating chain of vortices in which the mixing of passive impurities is chaotic. The excitable ruthenium-catalyzed Belousov-Zhabotinsky reaction is used in these studies. Velocities of the propagating fronts are measured as a function of the frequency and amplitude of external forcing. Mode locking is observed where the front propagates an integer number of vortices in an integer number of drive periods. Arnol'd tongues are mapped out for two of the locking regimes. These two tongues are shown to form a region of overlap where the velocity of the propagating front switches erratically between two locked values. The experimental results agree with numerical predictions of mode locking in a simplified model of the flow.
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Affiliation(s)
- M S Paoletti
- Department of Physics, Bucknell University, Lewisburg, Pennsylvania 17837, USA
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Nugent CR, Quarles WM, Solomon TH. Experimental studies of pattern formation in a reaction-advection-diffusion system. PHYSICAL REVIEW LETTERS 2004; 93:218301. [PMID: 15601066 DOI: 10.1103/physrevlett.93.218301] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/20/2004] [Indexed: 05/24/2023]
Abstract
Experiments are presented on pattern formation in the Belousov-Zhabotinsky (BZ) reaction in a blinking vortex flow. Mixing in this flow is chaotic, with nearby tracers separating exponentially with time. The patterns that form in this flow with the BZ reaction mimic chaotic mixing structures seen in passive transport. The behavior is analyzed in terms of a mixing time taum and a characteristic decorrelation time TBZ for the BZ system. Flows with taum comparable to or smaller than TBZ generate large-scale patterns whose features are captured by simulations of mixing fields for the flow.
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Affiliation(s)
- C R Nugent
- Department of Physics, Bucknell University, Lewisburg, Pennsylvania 17837, USA
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Solomon TH, Mezić I. Uniform resonant chaotic mixing in fluid flows. Nature 2003; 425:376-80. [PMID: 14508482 DOI: 10.1038/nature01993] [Citation(s) in RCA: 97] [Impact Index Per Article: 4.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/30/2003] [Accepted: 08/11/2003] [Indexed: 11/09/2022]
Abstract
Laminar flows can produce particle trajectories that are chaotic, with nearby tracers separating exponentially in time. For time-periodic, two-dimensional flows and steady three-dimensional (3D) flows, enhancements in mixing due to chaotic advection are typically limited by impenetrable transport barriers that form at the boundaries between ordered and chaotic mixing regions. However, for time-dependent 3D flows, it has been proposed theoretically that completely uniform mixing is possible through a resonant mechanism called singularity-induced diffusion; this is thought to be the case even if the time-dependent and 3D perturbations are infinitesimally small. It is important to establish the conditions for which uniform mixing is possible and whether or not those conditions are met in flows that typically occur in nature. Here we report experimental and numerical studies of mixing in a laminar vortex flow that is weakly 3D and weakly time-periodic. The system is an oscillating horizontal vortex chain (produced by a magnetohydrodynamic technique) with a weak vertical secondary flow that is forced spontaneously by Ekman pumping--a mechanism common in vortical flows with rigid boundaries, occurring in many geophysical, industrial and biophysical flows. We observe completely uniform mixing, as predicted by singularity-induced diffusion, but only for oscillation periods close to typical circulation times.
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Affiliation(s)
- T H Solomon
- Department of Physics, Bucknell University, Lewisburg, Pennsylvania 17837, USA.
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Armstead DN, Hunt BR, Ott E. Anomalous diffusion in infinite horizon billiards. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 67:021110. [PMID: 12636656 DOI: 10.1103/physreve.67.021110] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/15/2002] [Indexed: 05/24/2023]
Abstract
We consider the long time dependence for the moments of displacement <|r|(q)> of infinite horizon billiards, given a bounded initial distribution of particles. For a variety of billiard models we find <|r|(q)> approximately t(gamma(q)) (up to factors of ln t). The time exponent, gamma(q), is piecewise linear and equal to q/2 for q<2 and q-1 for q>2. We discuss the lack of dependence of this result on the initial distribution of particles and resolve apparent discrepancies between this time dependence and a prior result. The lack of dependence on initial distribution follows from a remarkable scaling result that we obtain for the time evolution of the distribution function of the angle of a particle's velocity vector.
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Affiliation(s)
- Douglas N Armstead
- Department of Physics and Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, Maryland 20904, USA.
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Fogleman MA, Fawcett MJ, Solomon TH. Lagrangian chaos and correlated Lévy flights in a non-Beltrami flow: transient versus long-term transport. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 63:020101. [PMID: 11308448 DOI: 10.1103/physreve.63.020101] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/19/2000] [Indexed: 05/23/2023]
Abstract
Long-range transport is studied numerically in a time-independent, three-dimensional (3D) fluid flow composed of the superposition of two chains of alternating vortices, one horizontal and the other vertical. Tracers in this flow follow chaotic trajectories composed of correlated Lévy flights with varying velocities. Locations of the chaotic regimes in the flow are compared with recent theories of chaos in non-Beltrami 3D flows. Growth of the variance of a distribution of tracers is divided into transient and long-term regimes, each with different growth exponents.
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Affiliation(s)
- M A Fogleman
- Department of Physics, Bucknell University, Lewisburg, Pennsylvania 17837, USA
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Solomon TH, Hartley RR, Lee AT. Aggregation and chimney formation during the solidification of ammonium chloride. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1999; 60:3063-71. [PMID: 11970113 DOI: 10.1103/physreve.60.3063] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/06/1998] [Revised: 05/18/1999] [Indexed: 11/07/2022]
Abstract
Experiments study large-scale pattern formation during the growth of ammonium chloride (NH4Cl) from solution in a thin (Hele-Shaw) geometry. In particular a solid-liquid mixture ("mushy layer") forms in which growing solid NH4Cl crystals form a solid network interspersed with liquid. There are different ways that the mushy layer can be formed, however. If the cell is heated from below and cooled from above, thermal convection generates large-scale recirculating flows that carry seed crystals from the upper (cold) boundary to the (warmer) side and bottom boundaries. Ballistic deposition of these seed crystals leads to aggregation patterns with significant voids (filled with liquid) with a wide range of length scales. If the cell is cooled from below with a warm environment, the solid NH4Cl grows dendritically without deposition, resulting in a compact mushy layer. Plume convection within this mushy layer produces one or two well-defined "chimneys." If the environment is cool (comparable to the liquidus temperature of the solution), the mushy layer forms by a combination of dendritic growth and ballistic deposition, resulting in a more permeable mushy layer and enhanced chimney formation. The effects of ballistic deposition are enhanced if the cell is tipped, in which case the voids reappear. Plume convection and chimney formation are dramatically enhanced in this case. Additional experiments are done in which fluid flows in the system are enhanced artificially to verify that enhancements in chimney formation are due primarily to the aggregation process, and not to the increases in fluid flows due to thermal and compositional convection.
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Affiliation(s)
- T H Solomon
- Department of Physics, Bucknell University, Lewisburg, Pennsylvania 17837, USA.
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