1

Hydrodynamic instabilities of autocatalytic reaction fronts:

A. De Wit

Unité de Chimie Physique non Linéaire

Université Libre de Bruxelles, Belgium

2

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 ?

3

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

4

Buoyancy-driven instability of a chemical front in a vertical set-up

QuickTime™ and a decompressor

are needed to see this picture.QuickTime™ and a decompressorare needed to see this picture.

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

5

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

7

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

9

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 ?

10

Theoretical model

with

Le=DLe=DTT/D/D

Rayleigh numbersRayleigh numbers Lewis numberLewis number

--C,TC,Tgg

11

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

12

Hydrodynamic Rayleigh-Bénard instability

Fluid heated from belowFluid heated from below

HOTHOT

COLDCOLD

RRTT>0>0

13

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

16

Chemical fronts

F( C ) = - C (C-1) (C+d)F( C ) = - C (C-1) (C+d)

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

StableStable

UNSTABLEUNSTABLE

UNSTABLEUNSTABLE

20

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

21

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

25

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).

26

Front in horizontal set-ups

1 0 ≠1

27

where

Equations

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)

QuickTime™ et undécompresseur GIF

sont requis pour visionner cette image.

M = 0 : reaction-diffusion front

M = 500

QuickTime™ et undécompresseur GIF

sont requis pour visionner cette image.

Marangoni effects

QuickTime™ et undécompresseur GIF

sont requis pour visionner cette image.

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

QuickTime™ et undécompresseur GIF

sont requis pour visionner cette image.

Buoyancy effects

Ra= - 100: pr : products heavier sink

QuickTime™ et undécompresseur GIF

sont requis pour visionner cette image.

QuickTime™ et undécompresseur GIF

sont requis pour visionner cette image.

32

Ra = 100

Ra = -100

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

Asymptotic mixing length

Asymmetric Marangoni effects Symmetric buoyancy effects

Constant propagation speed

Asymmetric Marangoni effects Symmetric buoyancy effects

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).

2008 Gordon conference“Oscillations and dynamic instabilities in chemical systems”

July 13-18, 2008Colby College, Waterville, USA

Chair: Vice Chair:Anne De Wit Oliver Steinbock

http://www.grc.org/programs.aspx?year=2008&program=oscillat