1 hydrodynamic instabilities of autocatalytic reaction fronts: a. de wit unité de chimie physique...
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Hydrodynamic instabilities of autocatalytic reaction fronts:
A. De Wit
Unité de Chimie Physique non Linéaire
Université Libre de Bruxelles, Belgium
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Scientific questions
Can chemical reactions be at the originof hydrodynamic instabilities (and notmerely be passively advected by the flow) ?
What are the properties of the new patterns that can then arise ?
Influence on transport and yield of reaction ?
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Outline
• Convective deformation of chemical frontsConvective deformation of chemical fronts• Experiments in Hele-Shaw cellsExperiments in Hele-Shaw cells• Model of hydrodynamic instabilities of frontsModel of hydrodynamic instabilities of fronts• Rayleigh-Bénard, Rayleigh-Taylor and double-Rayleigh-Bénard, Rayleigh-Taylor and double-
diffusive instabilitiesdiffusive instabilities• Reactive vs non-reactive systemReactive vs non-reactive system
I. Vertical set-upsI. Vertical set-ups
II. Horizontal set-upsII. Horizontal set-ups
• Marangoni vs buoyancy-driven flowsMarangoni vs buoyancy-driven flows
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Buoyancy-driven instability of a chemical front in a vertical set-up
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Hydrodynamic Rayleigh-Taylor instability of autocatalytic IAA fronts ascending in the gravity field in a capillary tube because reactant A is heavier than the product B
Light B
Heavy A: stabledescending front
Heavy A
Light B: unstableascending front
(Courtesy D. Salin)
Bazsa and Epstein, 1985; Bazsa and Epstein, 1985; Nagypal, Bazsa and Epstein, 1986Nagypal, Bazsa and Epstein, 1986Pojman, Epstein, McManus and Showalter, 1991Pojman, Epstein, McManus and Showalter, 1991
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Model system: fingering of chemical fronts in Hele-Shaw cells
Fresh reactants
Products
Density of reactantsdifferent than
density of products:∆c=prod-react≠0
Buoyantly unstablefronts
Carey, Morris and Kolodner, PRE (1996)Böckmann and Müller, PRL (2000)Horvath et al. (2002)
Density of products (top) larger than density of
reactants (bottom)cc>0
Buoyantly unstableDESCENDING fronts
Horvàth et al., JCP (2002)
Chlorite-tetrathionate (CT) reaction
Böckmann and Müller, PRL (2000)
Iodate-arsenous acid (IAA) reaction
Reactants heavier than products cc<0 :Buoyantly unstableASCENDING fronts
Isothermal systemIsothermal system
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J.A. Pojman and I.R. Epstein, “Convective effects on chemical waves: 1. Mechanisms and stability criteria”, J. Phys. Chem., 94, 4966 (1990)
= = oo[1-[1-(T-T(T-Too)+)+ccii(C(Cii-C-Cioio)])]
Across a chemical front: Across a chemical front: productsproductsreactantsreactantscc
cc<0: reactants are heavier than products (IAA) cc>0: products are heavier than reactants (CT)
TT<0: exothermic reaction, products are hotterthan reactants
Antagonist solutal and Antagonist solutal and thermal effectsthermal effects
Cooperative solutal andCooperative solutal andthermal effectsthermal effects
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QuestionsQuestions
• Which kind of hydrodynamic instabilities can the competitionWhich kind of hydrodynamic instabilities can the competitionor cooperative effects between solutal and thermal density or cooperative effects between solutal and thermal density effects generate across a chemical front ? effects generate across a chemical front ?
• Are there new instabilities possible with regard to theAre there new instabilities possible with regard to thenon reactive case ? non reactive case ?
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Theoretical model
with
Le=DLe=DTT/D/D
Rayleigh numbersRayleigh numbers Lewis numberLewis number
--C,TC,Tgg
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Linear stability analysis of pure Linear stability analysis of pure hydrodynamic instabilitieshydrodynamic instabilities F(C) = 0F(C) = 0
Base state :Base state : linearr concentration and temperature concentration and temperature gradients with a corresponding density gradientgradients with a corresponding density gradient
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Hydrodynamic Rayleigh-Bénard instability
Fluid heated from belowFluid heated from below
HOTHOT
COLDCOLD
RRTT>0>0
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Hydrodynamic Rayleigh-Taylor instability
Fernandez et al., J. Fluid Mech. (2002)
Heavy fluid on top of a light oneHeavy fluid on top of a light one RRcc>0>0
Double diffusive instabilitySalt fingers: Hot salty water lies over cold fresh water of a higher density. The stratification is kept gravitationally stable. The key to the instability is the fact that heat diffuses much more rapidly than salt (hence the term double-diffusion). A downward moving finger of warm saline water cools off via quick diffusion of heat, and therefore becomes more dense. This provides the downward buoyancy force that drives the finger. Similarly, an upward-moving finger gains heat from the surrounding fingers, becomes lighter, and rises.
Salt fingersSalt fingers::Instability even if light Instability even if light solution on top of a solution on top of a heavy solution heavy solution (statistically stable (statistically stable density gradient) !density gradient) !
RRTT<0, R<0, Rcc>0>0
Pure double diffusion (without chemistry):Le=20Le=20
Thermal Rayleigh-BénardThermal Rayleigh-Bénard
Solutal Rayleigh-TaylorSolutal Rayleigh-Taylor
UNSTABLEUNSTABLE
STABLESTABLE
OSCILLATINGOSCILLATING
Heated from belowHeated from below
Cooled from belowCooled from below
Light at theLight at thebottombottom
Heavy at bottomHeavy at bottom
Salt fingersSalt fingers
TurnerTurner
Base state for the linear stability analysis: reaction-diffusion fronts for both concentration and temperature, connecting the reactants where (C,T)=(0,0) to the products for which (C,T)=(1,1)and traveling at a speed v
vv
Reactants at room temperatureReactants at room temperature
Hot productsHot products
gg
T profile is function of LeT profile is function of Le
F( C ) = - C (C-1) (C+d)F( C ) = - C (C-1) (C+d)
Convection with chemistry : Le = 1
1: light but cold on top of heavy but hot:1: light but cold on top of heavy but hot:unstable if sufficiently exothermicunstable if sufficiently exothermicThermal plumesThermal plumes2: heavy and cold on top of light and hot:2: heavy and cold on top of light and hot:always unstablealways unstable3: heavy but hot on top of light but cold:3: heavy but hot on top of light but cold:stable descending fronts if sufficiently stable descending fronts if sufficiently exothermicexothermic
AscendingAscending
DescendingDescending
Heavier reactantsHeavier reactantsLighter Lighter reactantsreactants
Exothermic reactionExothermic reactionIAAIAA
IAAIAA
CTCT
CTCT
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Instability due to thermal diffusion and chemistry
Descending exothermic frontDescending exothermic front
Light and hot Light and hot productsproducts
Heavy and cold Heavy and cold reactantsreactants
F(CF(C11) < F(C) < F(C22))
CC11,T,T11
The little protrusion reaches rapid thermal equilibriumThe little protrusion reaches rapid thermal equilibriumbut still reacts at a rate F(Cbut still reacts at a rate F(C11) smaller than the rate F(C) smaller than the rate F(C22))
of its surroundings. It gains thus less heat (the reactionof its surroundings. It gains thus less heat (the reactionbeing exothermic) and it thus continues to sink. being exothermic) and it thus continues to sink.
CC22,T,T22CC11,,TT22
TT11>>TT22
Le>1Le>1
gg
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Properties of this instability• Because the region with F’(c) >0 is followed by a regionwith F’(c) <0 , the local instability is constrained by the region of local stability. • This unusual instability magnifies with a larger negative RT
and larger Le since (x) = -RT T -Rc C
Light and hotLight and hot
Heavy and coldHeavy and cold
gg
StableStable
Rayleigh-TaylorRayleigh-Taylor(heavy over light)(heavy over light)
New instability of New instability of light over heavylight over heavy
Antagonist solutal and Antagonist solutal and thermal effects:thermal effects:double diffusivedouble diffusiveinstabilitiesinstabilities
Cooperative solutal andCooperative solutal andthermal effects: thermal effects: candidates for the newcandidates for the newinstability for descendinginstability for descendingfrontsfronts
Conclusions and perspectivesConclusions and perspectives
• Classification of the various hydrodynamic instabilities ofClassification of the various hydrodynamic instabilities ofexothermic reaction-diffusion fronts in the (Rexothermic reaction-diffusion fronts in the (RTT,R,Rcc) plane) plane
• Double-diffusive instabilities of chemical fronts have someDouble-diffusive instabilities of chemical fronts have somedifferences with pure hydrodynamic DD instabilities:differences with pure hydrodynamic DD instabilities:
Different base stateDifferent base state stability boundaries depend on the kinetics and on Lestability boundaries depend on the kinetics and on Le different nonlinear dynamics: frozen fingersdifferent nonlinear dynamics: frozen fingers
• Uncovering of a new instability due to the coupling betweenUncovering of a new instability due to the coupling betweenthermal diffusion and spatial variations in reaction rate: should thermal diffusion and spatial variations in reaction rate: should be observed in families of exothermic reactions for which be observed in families of exothermic reactions for which cc and and TT are both negative are both negative
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Take home messageTake home message
When chemical reactions are at the core of density When chemical reactions are at the core of density gradients, the possible resulting hydrodynamic instabilities gradients, the possible resulting hydrodynamic instabilities in the corresponding reaction-diffusion-convection system isin the corresponding reaction-diffusion-convection system isnot always the simple addition of the usual buoyancy relatednot always the simple addition of the usual buoyancy relatedinstabilities on a chemical pattern. instabilities on a chemical pattern.
New chemically-driven instabilities can arise !New chemically-driven instabilities can arise !
References:References:J. D'Hernoncourt, A. Zebib and A. De Wit, Phys. Rev. Lett. 96, 154501 (2006).J. D'Hernoncourt, A. De Wit and A. Zebib, J. Fluid Mech. 576, 445-456 (2007).J. D'Hernoncourt, A. Zebib and A. De Wit, Chaos, 17, 013109 (2007).
3. Boundary conditions
and Marangoni boundary condition at the free surface :
(4)
withM > 0 : C
M < 0 : C
Open surface with no buoyancy effects (Ra=0)
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M = 0 : reaction-diffusion front
M = 500
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Marangoni effects
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M = - 500
Asymptotic dynamics : focus on the deformed front surrounded by a stationary asymmetric convection roll
M>0
M<0
Closed surface: no Marangoni effects (Ma=0)
Ra = 0 : reaction-diffusion front
Ra=100: pr : products lighter go on top
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Buoyancy effects
Ra= - 100: pr : products heavier sink
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Buoyancy effects: Comparison with experiments
Experiments in capillary tubes with the Bromate-Sulfite reaction : products heavier than reactants => Ra < 0
Qualitative agreement between experiments and theoretical model
A. Keresztessy et al., Travelling Waves in the Iodate-Sulfite and Bromate-Sulfite Systems , J. Phys. Chem. 99, 5379-5384, 1995.
products reactants
Experiments in capillary tubes with the Iodate-Arsenous Acid reaction : d/d[I-] = -1,7.10-2g/cm3M
J. Pojman et al., Convective Effects on Chemical Waves, J. Phys. Chem. 95, 1299-1306, 1991.
Vnum = 3.24Vnum = 4.84Vnum = 6.44
Numerical front velocities (10-3 cm/s)
Quantitative agreement between experiments and theoretical model
Buoyancy effects :
Symmetry between the results for Ra > 0 and Ra < 0
Increase of the front deformation, the propagation speed and the convective motions with Ra and Lz
ConclusionsMarangoni effects :
Asymmetry between the results for M > 0 and M < 0
Increase of the front deformation, the propagation speed and the convective motions with M and Lz
References
Marangoni effects:- L. Rongy and A. De Wit, ``Steady Marangoni flow traveling with a chemical front", J. Chem. Phys. 124, 164705 (2006).
- L. Rongy and A. De Wit, ``Marangoni flow around chemical fronts traveling in thin solution layers: influence of the liquid depth", J. Eng. Math. 59,221-227 (2007).
Buoyancy effects:- L. Rongy, N. Goyal, E. Meiburg and A. De Wit, ``Buoyancy-driven convection around chemical fronts traveling in covered horizontal solution layers", J. Chem. Phys. 127, 114710 (2007).