Transcript Fingering in reactive systems

Hydrodynamic instabilities of
autocatalytic reaction fronts:
A. De Wit
Unité de Chimie Physique non Linéaire
Université Libre de Bruxelles, Belgium
1
Scientific questions



Can chemical reactions be at the origin
of hydrodynamic instabilities (and not
merely 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 ?
2
Outline
I. Vertical set-ups
•
•
•
•
Convective deformation of chemical fronts
Experiments in Hele-Shaw cells
Model of hydrodynamic instabilities of fronts
Rayleigh-Bénard, Rayleigh-Taylor and doublediffusive instabilities
• Reactive vs non-reactive system
II. Horizontal set-ups
• Marangoni vs buoyancy-driven flows
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Buoyancy-driven instability of a
chemical front in a vertical set-up
Heavy A
Light B
Hydrodynamic RayleighTaylor instability of
autocatalytic IAA fronts
ascending in the gravity
field in a capillary tube
because
reactant A is heavier than
the product B
Bazsa and Epstein, 1985;
Nagypal, Bazsa and Epstein, 1986
Pojman, Epstein, McManus and Showalter, 1991
Quic kTi me ™a nd a
d eco mp resso r
ar e n ee ded to see th is p ictu re.
Heavy A: stable
descending front
QuickTime™ and a
decompressor
are needed to see this picture.
Light B: unstable
ascending front
(Courtesy D. Salin)
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Model system: fingering of chemical
fronts in Hele-Shaw cells
Fresh reactants
Density of reactants
different than
density of products:
∆c=prod-react≠0
Products
Buoyantly unstable
fronts
Carey, Morris and Kolodner, PRE (1996)
Böckmann and Müller, PRL (2000)
Horvath et al. (2002)
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Böckmann and Müller, PRL (2000)
Horvàth et al., JCP (2002)
Chlorite-tetrathionate (CT) reaction
Isothermal system
Density of products (top)
larger than density of
reactants (bottom)
Dc>0
Buoyantly unstable
DESCENDING fronts
Iodate-arsenous acid (IAA) reaction
Reactants heavier than
products Dc<0 :
Buoyantly unstable
ASCENDING fronts
J.A. Pojman and I.R. Epstein, “Convective effects on chemical
waves: 1. Mechanisms and stability criteria”,
J. Phys. Chem., 94, 4966 (1990)
= o[1-aT(T-To)+S ac i(Ci-Cio)]
Across a chemical front: D=products-reactants=Dc+DT
Dc<0: reactants are heavier than products (IAA)
Dc>0: products are heavier than reactants (CT)
DT<0: exothermic reaction, products are hotter
than reactants
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Antagonist solutal and
thermal effects
Cooperative solutal and
thermal effects
Questions
• Which kind of hydrodynamic instabilities can the competition
or cooperative effects between solutal and thermal density
effects generate across a chemical front ?
• Are there new instabilities possible with regard to the
non reactive case ?
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-(C,T)g
Theoretical model
with
Le=DT/D
Rayleigh numbers
Lewis number
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Linear stability analysis of pure
hydrodynamic instabilities
F(C) = 0
Base state : linear concentration and temperature
gradients with a corresponding density gradient
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Hydrodynamic Rayleigh-Bénard instability
Fluid heated from below
RT>0
COLD
HOT
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Hydrodynamic Rayleigh-Taylor instability
Heavy fluid on top of a light one
Rc>0
Fernandez et al., J. Fluid Mech. (2002)
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Double diffusive instability
Salt 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.
RT<0, Rc>0
Salt fingers:
Instability even if light
solution on top of a
heavy solution
(statistically stable
density gradient) !
Pure double diffusion (without chemistry):
Le=20
Heated from below
UNSTABLE
OSCILLATING
Heavy at bottom
Thermal Rayleigh-Bénard
Solutal Rayleigh-Taylor
Salt fingers
STABLE
Light at the
bottom
Cooled from below
Turner
Chemical fronts
F( C ) = - C (C-1) (C+d)
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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
T profile is function of Le
Hot products
v
F( C ) = - C (C-1) (C+d)
Reactants at room temperature
g
Convection with chemistry : Le = 1
Ascending
Exothermic reaction
IAA
CT
Lighter
reactants
Heavier reactants
CT
IAA
Descending
1: light but cold on top of heavy but hot:
unstable if sufficiently exothermic
Thermal plumes
2: heavy and cold on top of light and hot:
always unstable
3: heavy but hot on top of light but cold:
stable descending fronts if sufficiently
exothermic
UNSTABLE
Stable
UNSTABLE
Instability due to thermal
diffusion and chemistry
Descending exothermic front
g
Light and hot
products
F(C1) < F(C2)
T1>T2
C1,T1
C2,T2
Heavy and cold
reactants
Le>1
C1,T2
The little protrusion reaches rapid thermal equilibrium
but still reacts at a rate F(C1) smaller than the rate F(C2)
of its surroundings. It gains thus less heat (the reaction
being exothermic) and it thus continues to sink.
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Properties of this instability
• Because the region with F’(c) >0 is followed by a region
with 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 hot
g
Heavy and cold
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Rayleigh-Taylor
(heavy over light)
Stable
New instability of
light over heavy
Antagonist solutal and
thermal effects:
double diffusive
instabilities
Cooperative solutal and
thermal effects:
candidates for the new
instability for descending
fronts
Conclusions and perspectives
• Classification of the various hydrodynamic instabilities of
exothermic reaction-diffusion fronts in the (RT,Rc) plane
• Double-diffusive instabilities of chemical fronts have some
differences with pure hydrodynamic DD instabilities:
 Different base state
 stability boundaries depend on the kinetics and on Le
 different nonlinear dynamics: frozen fingers
• Uncovering of a new instability due to the coupling between
thermal diffusion and spatial variations in reaction rate: should
be observed in families of exothermic reactions for which
Dc and DT are both negative
Take home message
When chemical reactions are at the core of density
gradients, the possible resulting hydrodynamic instabilities
in the corresponding reaction-diffusion-convection system is
not always the simple addition of the usual buoyancy related
instabilities on a chemical pattern.
New chemically-driven instabilities can arise !
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).
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Front in horizontal set-ups
1
0 ≠ 1
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Equations
where
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3. Boundary conditions
and Marangoni boundary condition at the free surface :
(4)
M>0:C
g
M<0:C
g
with
Marangoni effects
Open surface with no buoyancy effects (Ra=0)
M = 0 : reaction-diffusion
front
QuickTime™ et un
décompresseur GIF
sont requis pour visionner cette image.
M = 500
QuickTime™ et un
décompresseur GIF
sont requis pour visionner cette image.
M = - 500
QuickTime™ et un
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Asymptotic dynamics : focus on the deformed front
surrounded by a stationary asymmetric convection roll
M>0
M<0
Buoyancy effects
Closed surface: no Marangoni effects (Ma=0)
Ra = 0 : reaction-diffusion
front
QuickTime™ et un
décompresseur GIF
sont requis pour visionner cette image.
Ra=100: p  r : products lighter go on top
Quic kTime™ et un
décompr es s eur GIF
s ont r equis pour v is ionner c ett e image.
Ra= - 100: p > r : products heavier sink
Quic kTime™ et un
décompr es s eur GIF
s ont r equis pour v is ionner c ett e image.
Ra = 100
Ra = -100
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Buoyancy effects: Comparison with experiments
Experiments in capillary tubes with the Bromate-Sulfite
reaction : products heavier than reactants => Ra < 0
products
reactants
A. Keresztessy et al., Travelling Waves in the Iodate-Sulfite and
Bromate-Sulfite Systems , J. Phys. Chem. 99, 5379-5384, 1995.
Qualitative agreement between
experiments and theoretical model
Experiments in capillary tubes with the Iodate-Arsenous
Acid reaction : d/d[I-] = -1,7.10-2g/cm3M
Numerical front
velocities (10-3 cm/s)
Vnum = 3.24
Vnum = 4.84
Vnum = 6.44
J. Pojman et al., Convective Effects on Chemical Waves, J. Phys.
Chem. 95, 1299-1306, 1991.
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
Marangoni effects :
Conclusions
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
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
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, 2008
Colby College, Waterville, USA
Chair:
Anne De Wit
Vice Chair:
Oliver Steinbock
http://www.grc.org/programs.aspx?year=2008&program=oscillat