ON FINGERING OF CHEMICAL FRONTS:

Facult´ e des Sciences

Service de Chimie Physique et Biologie Th´ eorique

INFLUENCE OF THERMAL EFFECTS AND ELECTRIC FIELDS

ON FINGERING OF CHEMICAL FRONTS:

A THEORETICAL STUDY

Th` ese pr´ esent´ ee en vue de l’obtention du grade l´ egal de

Docteur en Sciences de l’Universit´ e Libre de Bruxelles.

Jessica D’Hernoncourt

D´ ecembre 2007

Contents 3

Remerciements 5

Foreword 7

1 Dynamic instabilities of chemical fronts 11

1.1 Chemical fronts . . . . 11

1.2 Fronts in the IAA reaction-diffusion system . . . . 12

1.3 Diffusive instabilities of fronts . . . . 15

1.4 Influence of electric fields . . . . 17

1.5 Hydrodynamic instabilities in non reactive systems . . . . 17

1.6 Buoyancy-driven instabilities of chemical fronts . . . . 20

1.7 Fingering of chemical fronts in tubes . . . . 23

1.8 Fingering of chemical fronts in Hele-Shaw cells . . . . 27

1.9 Objectives . . . . 33

2 Influence of heat losses on fingering of exothermic chemical fronts 35 2.1 General context . . . . 36

2.2 Model of fingering in non-insulated reactors . . . . 37

2.3 Traveling exothermic fronts . . . . 41

2.3.1 Concentration fronts . . . . 41

2.3.2 Thermal fronts . . . . 42

2.4 Linear stability analysis method . . . . 47

2.5 Stability of chemical fronts . . . . 52

2.5.1 Ascending fronts . . . . 52

2.5.2 Descending fronts . . . . 56

2.6 Nonlinear dynamics . . . . 59

2.6.1 Numerical scheme . . . . 59

2.6.2 Non linear dynamics with and without heat losses . . . . 60

2.6.3 Quantitative measures characterizing nonlinear dynamics 61 2.6.4 Parametric study . . . . 68

2.6.5 Flow field during the fingering phenomena . . . . 75

2.7 Discussion and perspectives . . . . 80

3

3.1.2 Linear stability analysis . . . . 88

3.2 Double diffusive instability of chemical fronts . . . . 92

3.2.1 Model . . . . 92

3.2.2 Base state . . . . 95

3.2.3 Linear stability analysis, numerical method . . . . 96

3.2.4 Stability boundaries for Le = 1 . . . . 97

3.2.5 Stability boundaries for Le

= 1 . . . . 99

3.3 Conclusions and perspectives . . . 108

4 Coupling between diffusive and buoyancy-driven instabilities of fronts 111 4.1 Model . . . 112

4.1.1 Planar travelling waves . . . 115

4.2 Linear stability analysis . . . 116

4.2.1 R

> R

i.e. decreasing the density during the reaction . 116 4.2.2 R

> R

i.e. increasing the density during the reaction . . 123

4.3 Nonlinear simulations . . . 126

4.3.1 Pure diffusive instability . . . 127

4.3.2 Rayleigh-Taylor instability . . . 128

4.3.3 R

> R

i.e the density decreases during the reaction . . . 129

4.3.4 R

< R

i.e. the density increases during the reaction . . 135

4.4 Discussion and perspectives . . . 136

5 Influence of electric fields on the dynamics of fronts 139 5.1 Derivation of RD equations in a constant electric field . . . 140

5.1.1 Impact of the electric field on reaction diffusion fronts . . 143

5.2 Effect of a constant electric field on the diffusive instability . . . 147

5.2.1 Reaction-diffusion model . . . 147

5.2.2 Linear stability analysis . . . 148

5.2.3 Nonlinear simulations . . . 150

5.3 Effect of an electric field on the fingering instability . . . 155

5.3.1 Reaction-diffusion convection model . . . 155

5.3.2 Linear stability analysis . . . 157

5.3.3 Nonlinear simulations . . . 158

5.4 Conclusions and perspectives . . . 166

Discussions and perspectives 169

Appendix 173

Bibliography 176

4

“The Moving Finger writes; and, having writ, Moves on: nor all your Piety nor Wit

Shall lure it back to cancel half a Line, Nor all your Tears wash out a Word of it”

Omar Khayyam

“So much honey, so little time”

Winnie the Pooh

J’aimerais avant toute chose exprimer ma gratitude envers Messieurs les Professeurs G. Nicolis et R. Lefever de m’avoir accueillie dans le Service de Chimie Physique. Je remercie les Professeurs P. Borckmans, C. Buess-Herman, T. Erneux, R. Lefever et D. Salin d’avoir accept´e de faire partie de mon jury de th`ese.

Pendant ces cinq ann´ees, Anne De Wit a toujours ´et´e l` a. Ses qualit´es hu- maines, son enthousiasme sans faille et ses connaissances scientifiques en font quelqu’un d’exceptionnel. Je tiens ` a la remercier pour tout ce que j’ai appris.

I would like to thank all the people I have collaborated with: Marcus Hauser, Serafim Kalliadasis, Lenka ˇ Sebest´ıkov´ a, and late Hana ˇ Sevˇc´ıkov´ a. Many special thanks go to Abdel Zebib and John Merkin for fruitful collaborations, many in- teresting discussions and a great patience trying to teach me all these things. I would also like to thank several people for the different enlightening discussions we have had: Agot´ a T´ oth, Dezs´ o Horv´ ath, Kenneth Showalter, Jacques Boisson- ade, Patrick De Kepper and Stefan C. M¨ uller. Enfin un ´enorme merci ` a Pierre Borckmans pour toutes ses histoires et les diff´erentes discussions scientifiques ou non....

Beaucoup de personnes m´eritent une place sur cette page, que ce soit celles qui sont venues et reparties, ont bri`evement crois´e mon petit chemin ou celles qui ont d´ecid´e de s’attarder un peu plus... Que tous ces gens -qu’ils aient ´et´e un peu ch... ou d’une bonne humeur communicative- soit remerci´es, on est quelque part la somme de toutes ces exp´eriences.

Je tiens en particulier ` a remercier Laurence pour ces ann´ees pass´ees en- semble, pour, en particulier, ses conseils dans les heures plus graves -celles o` u j’allais tout dire-, ses remarques judicieuses, son allemand d´eplorable et son sens de l’orientation encore plus... Bref pour toutes ces choses qui g´en`erent une amiti´e. Que Matthieu trouve ici un remerciement tout particulier pour 8 ann´ees nettement am´elior´ees par sa pr´esence... N’oublie pas que S=bleu!

J’ai pu au cours de ces quasi 5 ann´ees pass´ees dans ce service d´ecouvrir

5

enzovoort.

Que mes parents trouvent ici mes plus vifs remerciements pour leur support constant, leur main d’oeuvre gratuite et leur conseils -parfois- judicieux (y a rien ` a faire mais cette maison ` a Ellezelles...). Mon petit poulet a aussi droit ` a une pens´ee toute particuli`ere pour ˆetre une esp`ece de grande bringue adorable...

Tack ocks˚ a mormor, nu vet jag hur jag vill bli n¨ ar jag blir ¨ aldre. Je remercie

´egalement ma belle-famille pour sa gentillesse et son accueil.

Merci ` a toi mon petit chat qui a chang´e mon existence. Tu es ce que j’ai toujours esp´er´e consciemment ou inconsciemment. Merci d’avoir accept´e de partager ma vie. Un grand merci aussi ` a Kali et Caramel pour tous les poutous et les rourous.

Enfin, je souhaite ` a tous tout le bonheur du monde parce que savoir ce que l’on veut est parfois difficile, l’atteindre encore plus.

Je remercie le F.R.I.A. (Fonds pour la formation la Recherche dans l’Industrie et dans l’Agriculture) ainsi que la Communaut´e fran¸caise de Belgique (pro- gramme ARC-ARCHIMEDE) pour leur soutien financier.

6

Convective motions in fluids are known to be extremely varied and to sometimes follow very complicated dynamics in nature and in man-made applications. If chemical species are present in the flow and no reaction takes place, they can be simply advected by the flow in which case they play only the role of pas- sive tracers. This is the case when the physical properties of the fluid such as its density or viscosity are independent of the concentrations of the chemicals.

Their spatio-temporal dynamics are then merely the one of the hydrodynamical flows. This is the case when markers or pollutants are dispersed in rivers or oceans for instance.

There exist however numerous situations where chemicals (and actually also temperature) are not slaved to the flow but actively influence it. This occurs when the physical properties of the flow are a function of the concentrations of the solutes present in the fluid or of temperature. Spatial gradients in concen- tration or heat are then able to trigger flows in various ways. As an example, dy- namic instabilities related to compositional and thermal density gradients across the interface separating two miscible fluids are known to affect the Earth’s core- mantle boundary, ocean and atmosphere flows as well as CO

sequestration in porous media for instance.

The situation becomes even more complicated if chemical reactions come into play. The evolution in space and time of the concentrations then re- sults from a highly nonlinear coupling between chemical reactions and transport processes such as diffusion and convection. Complex dynamics resulting from such a chemo-hydrodynamic coupling are common in the chemical industry and in applications such as oil biodegradation, bioremediation of pollution zones, petroleum recovery, chromatographic separation, geochemical transformations or combustion processes to name but a few.

From a fundamental point of view, this field is very challenging as several important questions then arise: Do the chemical reactions influence the fluid flow and in what way? How do the convective fluid motions in turn affect the spreading of the reacting species and the yield of the reaction? Are chemical reactions able to induce hydrodynamic instabilities in a fluid that would re- main quiescent under non reactive conditions? Are there possibilities of new instabilities due solely to the interplay between reaction-diffusion and advection processes?

Even if answers to such questions have undoubtedly already been given

7

understanding of such a chemo-hydrodynamical coupling from the more fun- damental perspective of the physico-chemist. Therefore we will theoretically analyze instability scenarios and spatio-temporal dynamics resulting from the coupling between chemical reactions and convection in a model as simple as possible allowing to focus on isolated specific aspects of the problem.

Such one model system consists in conversion fronts. These fronts result from the coupling between autocatalytic reactions and diffusion. They prop- agate in space invading the reactants and leaving the products behind. They allow to create a self-organized diffusive interface between the products and the reactants solutions with different physical properties. The reaction can further- more consume or release heat. Density or viscosity variations through gradients in the chemical concentrations and in temperature can also occur. Such spatial changes in the physical parameters of the solution can then in turn induce dif- ferent types of instability. These conversion fronts thus create both the interface between two miscible fluids and the property changes across this interface which may lead to its hydrodynamic destabilization.

Although several hydrodynamic instabilities may appear at the interface be- tween two miscible fluids we will here mainly focus on two driven by buoyancy.

The first one is the Rayleigh-Taylor instability which is triggered when a denser fluid is placed on top of a lighter one in the gravity field. This leads to the onset of fingers of the denser fluid sinking into the less denser fluid hence its name density fingering. The second one is the double diffusive instability which may appear even when the initial density stratification is statically stable. Convec- tion then arises when a dense faster diffusing component is placed below a less dense and less mobile one. The faster diffusing component (invading the less denser one) locally creates an unstable density stratification leading to an insta- bility. These buoyancy-driven instabilities of chemical fronts have been studied both experimentally and theoretically.

Experimentally hydrodynamical instabilities of such chemical fronts have been investigated in several set-ups. Initially, experiments were conducted in thin tubes allowing for one or two convection rolls to deform the front. More recently, Hele-Shaw cells have been used to allow for more spatially extended dynamics to be analyzed. These set-ups are made of two transparent plates which allow an easy visualization of the front destabilization. Different stud- ies using isothermal reaction diffusion fronts have shown that hydrodynamics induce the destabilization of the front when the denser solution is placed on top of the less denser one in the gravity field. Further studies have analyzed thermal effects due to the exothermicity of the reaction. They have shown the impact of the heat of reaction on the instability and on the possibility to trigger double-diffusive instabilities through the front.

Theoretical studies carried out in parallel have first focused on the behavior of the reaction-diffusion-convection chemical fronts in tubes. The modeling of

8

ever, an experimental set-up is never perfectly insulated and a part of the heat generated during the reaction is therefore lost through the walls of the reactor.

Furthermore, the importance of such heat losses has been highlighted in some experimental studies.

In this framework, the objectives and organization of this work are the fol- lowing. After a first chapter documenting the various dynamic instabilities a front can sustain, we present in the second chapter our theoretical study of the impact of heat losses on the dynamics of a Rayleigh-Taylor instability. We the- oretically study fingering of exothermic chemical fronts taking into account the possibility of thermal leaks through the walls of the experimental set-up. These effects induce drastic changes to the fingering dynamics both in the stability conditions and in the nonlinear dynamics.

As already mentioned, the density changes during a reaction can have two origins: solutal or thermal. These two effects can either reinforce or oppose each other. When they reinforce they lead to simple convection. A heavier fluid placed upon a lighter one will thus be unstable and the interface will be subjected to a Rayleigh-Taylor instability and deform. When they oppose each other, they can, depending on the relative magnitude of the two effects, lead to different instabilities and trigger double diffusion. The third chapter is, in this context, devoted to the classification of the different cases that can be encountered when heat and mass have either reinforcing or opposing effects.

In that chapter we further present a new convective instability of solute light and hot fluid on top of a solute heavier and colder one that can be triggered due to a non intuitive interaction between chemistry and double diffusive effects.

The reaction-diffusion fronts can be subjected, not only to density-driven convection, but to a pure diffusive instability as well. This instability may be triggered when the autocatalyst of the reaction diffuses more slowly than the reactant. In chapter 4, we turn to the study of the coupling between the two instabilities: the diffusive one arising in pure reaction diffusion systems and the hydrodynamic Rayleigh-Taylor one. We show that this interaction can trigger destabilization of otherwise buoyantly stable fronts and lead to complex spatio- temporal dynamics.

As the chemical species involved in autocatalytic fronts usually consist of ions their dynamics can be influenced by applying an electric field to the sys- tem. In chapter 5 we study the influence of such an external constant electric field on both the diffusive and the buoyancy driven instabilities of fronts.

We now start by introducing in the next chapter the properties of reaction-

diffusion fronts and the various instabilities that can affect their progression

before detailing in following chapters the results of our theoretical study of

fingering of chemical fronts.

Dynamic instabilities of chemical fronts

1.1 Chemical fronts

By maintaining a system away from equilibrium multiple types of complex dy- namics can be generated: oscillations, bistability, pulses, waves... [54, 102].

Examples of such self-organization processes can be found in chemistry [44], physics, engineering and biology [98]. In this thesis we will specifically focus on chemical fronts. These fronts result from the coupling between diffusion and autocatalytic kinetic processes where the latter involve feedback loops through which the outcome of the kinetic processes influence their own production.

An example is that of a polymerization front [108, 111] where the monomers are converted into polymers in a narrow reaction zone. Initially the monomer solution is for instance contained in a tube. In order to start the reaction, the tube is heated at one end. This heating triggers the reaction that will then propagate through the tube. Indeed, the polymerization process releases heat which diffuses into the monomers. As the monomers need to be heated up to start polymerizing because of the high activation energy of the reaction, the diffusion of heat will thus trigger the reaction ahead of the front. This interplay between the need to go over the activation energy barrier and the diffusion of the heat released by the reaction allows to create a front which prevents the reaction to start at once everywhere in the tube. The same mechanism is present in combustion where the reaction generates burned products and heat which warm up the fresh reactants ahead of the reaction zone triggering in that way a flame front advancing and burning the gases on its way [18]. Such kinds of front can also be generated by specific chemical reactions in aqueous solutions when a coupling is made between diffusion and autocatalytic reactions.

Autocatalytic chemical reactions are a special class of reactions involving

feedback loops in which one product of the reaction influences (either positively

or negatively) its own production. If they feature a positive feedback loop the

products enhance their own production, this species being then identified as

the autocatalyst. When diffusing ahead of the reaction zone this autocatalyst

can trigger the reaction further away. In a spatially extended medium such as

Properties of such reaction-diffusion fronts have been thoroughly analyzed (see [150] for a review).

Several redox reactions are known to exhibit front behavior. In this thesis we will mainly focus on an autocatalytic oxydoreduction reaction known to generate conversion fronts: the Iodate-Arsenous Acid (IAA) reaction [56, 125, 131]. This reaction is catalyzed by iodide, one of the product of the reaction. In some concentration regime this reaction can be modeled using a simple cubic kinetics making it very useful for theoretical studies.

Let us first recall the properties of the IAA reaction-diffusion system before detailing the different types of destabilization these reaction diffusion fronts can be subject to.

1.2 Fronts in the IAA reaction-diffusion system

The IAA reaction involves iodate IO

and iodide I

as reactants and iodide I

and arsenite acid H

AsO

as products when performed in excess of arsenous acid H

AsO

. The detailed mechanism of the IAA reaction in terms of elementary steps is still unknown but its temporal dynamics is actually quantitatively well described using the Dushman-Roebuck scheme. The first step of the reaction is the Dushman [40] scheme describing the reduction of iodate by iodide to generate iodine:

IO

+ 5I

+ 6H

3I

+ 3H

O. (1.1) The rate law of this reaction has been determined experimentally. The evolution of the iodate concentration has been measured to follow the empirical kinetic speed:

−

d[IO

]

dt = (k

+ k

[I

])[I

][IO

][H

]

(1.2) with k

= 4.5

10

M

s

and k

= 4.5

10

M

s

. The second step is given by the Roebuck reaction [119] which reduces the iodine back to iodide thanks to arsenous acid in a rapid process:

H

AsO

+ I

+ H

O

H

AsO

+ 2I

+ 2H

(1.3) with a measured rate law:

−

d[I

] dt = k

[H

AsO

][I

]

[I

][H

] (1.4)

with k

= 3.2

10

Ms

. If the reaction is conducted in the presence of an

excess of arsenous acid the overall stoechiometry is given by (1.1)+3(1.3):

3H

AsO

+ IO

+ 5I

3H

AsO

+ 6I

(1.5) and we see that indeed there is an autocatalysis in I

. The Roebuck process being much more rapid, the production rate of I

is determined by the Dushman process:

d[I

]

dt = (k

+ k

[I

])[I

][IO

][H

]

= R. (1.6) Experiments on this reaction are often performed in a buffered solution which allows a model simplification as no evolution equation for the protons has to be written. The full model system of this reaction is then a four variable one where the evolution equations for the four species (iodate IO

, arsenous acid H

AsO

, iodide I

and iodine I

) are then taken into account.

The final outcome of the reaction depends on the relative values of the concentrations of IO

and H

AsO

in the reactants [56, 94, 125, 131]. When there is an excess of iodate, arsenous acid is fully consumed and iodine is the final product of the reaction. The system needs then to be described with a four variable model. On the contrary, in the case of an excess of arsenous acid in the reactants, iodide is only produced as an intermediate and iodine is now the product. In this case and if transport by diffusion is also taken into account the reaction-diffusion model can further be reduced from the four variable model to a two variable one [56, 94]:

∂c

∂t = D

c + R, (1.7)

∂c

∂t = D

c

R (1.8)

where c = [I

] and c

= [IO

]. If these two species diffuse at the same speed, which is usually the case, the model can further be simplified to a single variable model using a conservation law for iodine atoms. This conservation relation states that at each time the concentration of iodide atoms is equal at each point of the system to:

c + c

= [I

]

+ [IO

]

(1.9) where [I

]

and [IO

]

are the initial concentrations of iodide and iodate re- spectively. As the concentration of iodide [I

]

in the reactants is negligible this equality (1.9) can be reduced to:

c + c

= [IO

]

= c

. (1.10) The system (1.7)-(1.8) can thus be reduced to a single variable model

∂c

∂t = D

c + (k

+ k

c)c(c

c) (1.11)

The equation (1.11) involving a cubic source term will be the basis of our theoretical analysis as it gives a quantitative description of the IAA reaction in arsenous acid excess and is also a simple generic cubic model to describe the dynamics of chemical fronts in general.

This one variable model is valid only if the two species I

and IO

diffuse at the same speed. If this is not the case the conservation law (1.10) is not correct in all points and the two-variable system (1.7)-(1.8) for the two main species need to be considered.

An analytical front solution to equation (1.11) in one-dimension is given by Hanna, Saul and Showalter [56], who have derived both the analytical from and the propagation speed of the reaction-diffusion front.

The concentration profile can be described using a hyperbolic tangent form:

c(x, t) = [IO

]

1 + B e

(1.12)

where p =

k

/2D [IO

]

and B is an arbitrary constant (see Fig. 1.1). The propagation speed v is given by

v =

k

D

2 [IO

]

[H

] +

2D

k

k

[H

]. (1.13)

Typical values of parameters for experiments on the IAA reaction [15] are [IO

]

= 4.80

10

M, [H

] = 6.45

10

M, D = 2.50

10

cm

/s and k

/k

10

M. We see that, for these values of parameters, we can neglect the second term in (1.13) i.e. we have

v

k

D

2 [IO

]

[H

] (1.14)

which gives a propagation speed of the order of 10

mm/s for the typical con- centrations presented above [56, 58, 125]. The propagation of fronts generated by the IAA reaction has been studied experimentally in Petri dishes (e.g. [58]), in tubes (e.g. [110]) and in gels (e.g. [65]).

Other examples of chemical reactions featuring autocatalytical chemical

fronts are given by the Chlorite-Tetrathionate (CT) reaction to which we will

refer many times [140], the Bromate-Sulfite reaction [78] and the Chlorite-Sulfite

reaction [100] to name a few. If the reaction is exothermic the concentration

reaction-diffusion profile will further be coupled to a thermal front, the hot

products invading the cool reactants as the reaction front passes by. We see in

Fig. 1.1 that the thermal front is more spread out than the concentration front,

because heat diffuses more rapidly than mass.

32500 3300 3350 3400 0.2

0.4 0.6 0.8 1

PSfrag replacements

Hot products Cool reactants

Concen tration, T emp erature

Space

T hermalf ront Concentrationf ront

Figure 1.1: Example of a chemical front of hot products invading cooler reactants at room temperature.

It has been observed experimentally that such reaction diffusion fronts can be prone to different types of instabilities. These instabilities can be either driven only by reaction-diffusion mechanisms or due to convective motion in the fluid. In gels, where convective motions are suppressed, the destabilizations which are observed are due solely to reaction-diffusion processes. In Petri dishes and in tubes without gel convection can come into play and trigger instabilities as well. Let us first detail the pure reaction-diffusion instability mechanisms which can destabilize fronts before turning our attention to the hydrodynamic ones.

1.3 Diffusive instabilities of fronts

When the two key chemical species involved into the dynamics of a front i.e.

the limiting reactant and the autocatalyst diffuse sufficiently differently, the chemical front may become unstable, deforming from a flat interface to a cellular modulation, an instability known as a “diffusive instability”[64, 84, 92]. It was predicted first theoretically [64] then evidenced experimentally first on the IAA reaction [65] and then on the CT reaction [51, 66, 142, 143]. It can be understood on the basis of the instability mechanism depicted in Fig. 1.2.

The direction of front propagation is from left to right, with the reactant ahead of the wave and the autocatalyst behind it. Let us discuss the effects of differential diffusion in the various regions of the perturbed front.

First in the regions that are advanced in the direction of propagation, there

is an enhanced dispersion (compared to the planar front) of the product – the

PSfrag replacements

Autocatalyst

I

IO

Reactant

D

< D

D

> D

Figure 1.2: Mechanism of the diffusive instability

autocatalyst – ahead of the front due to the curvature (see Fig. 1.2 upper part).

This leads to a dilution of the driving force that triggers the autocatalytic reaction ahead. Thus the wave speed in these advanced segments will decrease compared to the wave speed of the planar front. The opposite situation arises in the valleys where the perturbation lies behind the planar front. Here, there is a focusing by diffusion of the autocatalyst ahead of the front (see Fig. 1.2 upper part), leading to a local increase in the wave speed. These local changes in the propagation speed tend to eliminate the local perturbations and the diffusion of the autocatalyst has thus a stabilizing effect.

Applying a similar argument for the diffusion of the reactant (see Fig. 1.2 lower part), we see that the segments which are advanced in the direction of propagation bring an enhanced supply of reactant into the front, tending to en- hance the local speed of these already advanced sections. The retarded sections, on the other hand, witness a dilution of the reactant, tending to decrease the local speed. Thus, the diffusion of the reactant has a destabilizing effect on the planar wave.

From these arguments we see that when the diffusion coefficient of the au- tocatalyst is larger than that of the reactant, the overall tendency will be a stabilization of the planar front. On the contrary, if the diffusion of the re- actant is sufficiently larger than that of the autocatalyst the planar front will loose stability as the destabilizing influence of the reactant diffusion becomes dominant.

As mentioned above, after its theoretical prediction [64] this instability was

found experimentally for the first time using the IAA reaction in a gel [65]. In

this reaction the autocatalyst, iodide and the limiting reactant iodate, diffuse

in normal conditions at the same rate. To observe the diffusive instabilities,

Horv´ ath et al. have resorted to a complexing agent, α-cyclodextrin, to bind

the iodide and decrease its effective diffusivity. They could in this way increase

the difference between the diffusion coefficients of the two species [65]. The

CT reaction can also be subject to diffusive instabilities and which have been studied both experimentally and theoretically using a two variable model of the CT system [51, 66, 72, 73, 142, 143]. In this reaction the activators are the protons and the inhibitor the tetrathionate ions. The protons are known to diffusive more rapidly so they need, in the experiments, to be slowed down by carboxylate groups to observe the diffusive destabilization [66, 142, 143]. Mixed theoretical and experimental studies [66, 72] have shown good agreements.

The time scale for the onset of such diffusive patterns is of the order of hours and produce a cellular deformation of the front with a wavelength of the order of cm.

This diffusive destabilization can be triggered by changing the diffusion co- efficients but can also be influenced by an external applied electric field.

1.4 Influence of electric fields

Electric fields are known to have marked influence on the characteristics of ionic systems and more specifically on reaction-diffusion fronts [41, 45, 107, 127, 128].

This is due to the fact that the different charged species present in the auto- catalytic redox reactions will have their mobility influenced by the electric field depending on their size and charge. This electric field can also induce convec- tion and deformation of a traveling front [130].

Theoretical studies have also shown the influence of an electric field on reaction-diffusion processes [91, 93, 95, 96, 135, 134, 133]. For a strong enough electric field the reaction diffusion fronts can even cease to exist and an elec- trophoretic separation between the products and the reactants can appear [93, 96]. The two species then travel separately along two fronts moving at different speed. As the electric field influences the propagation speed of the ions involved, it also has an impact on diffusive instabilities of fronts. This has been shown theoretically using a general reaction diffusion model involving two species and the influence of an electric field [33, 67, 141]. Experimental results as well as a theoretical analysis of the influence of an electric field on the dynamics of a CT reaction model [161, 162, 163] have also been performed.

Chemical fronts can be subject to diffusive instabilities and be influenced by electric fields. They can also be deformed by convective instabilities which trigger a fluid motion around the front.

1.5 Hydrodynamic instabilities in non reactive systems

Before tackling dynamics due to convection in reactive systems it is first useful

to recall some of the various hydrodynamic instabilities that can be observed

between two miscible non reactive fluids [55, 145]. We will thus first describe

the various types of destabilization that can appear at the interface between

two fluids before turning back in section 1.6 to the hydrodynamic instabilities

generated by chemical reactions.

field of the physical properties of fluids such as viscosity, density, surface ten- sion,... These gradients are able to trigger convection leading to hydrodynamical instabilities.

One well known instability is fingering of an interface which appears as soon as a fluid of higher mobility displaces a less mobile one. The interface between the two fluids looses its stability and deforms into fingers when the faster moving fluid invades the less mobile one. The differences in the mobility of the two phases can be due to viscosity or density changes leading to respectively viscous fingering or density fingering. When a less viscous fluid pushes a more viscous one it gives rise to viscous fingering also referred to as the Saffman- Taylor instability (for a review see [60]). This is typically the case in petroleum recovery, when water is injected in the cavities containing oil to extract it. The less viscous water penetrates the oil and deforms the interface into fingers. If the mobility difference is due to density, it will lead to fingering as soon as a heavy fluid is placed over a lighter one in the gravity field. This gives a Rayleigh- Taylor instability here also referred to as density fingering. This is the case for example when a heavier glycerin-water mixture is poured upon pure water (see Fig. 1.3). The heavier solution sinks into the lighter one deforming the interface into fingers.

Figure 1.3: Rayleigh-Taylor instability between two layers of miscible fluids.

The heavier water-glycerine colored mixture on top sinks into the lighter pure water on bottom. The length of the image is 8 cm. [46, 47]

Such a buoyancy driven convection can be driven not only by solutal effects but also by thermal ones. Indeed, a hotter fluid will usually have a lower density and will thus contribute to the density differences between the fluids. This leads to convective movements such as observed over a heater. The hot, less dense air rises and triggers a movement in the cooler air above the heater.

These two solutal and thermal effects may either reinforce each other or

compete. If the two effects reinforce each other it will lead to an increased

convection compared to the two separate effects. On the contrary, in the case

of opposite thermal and solutal effects the overall density stratification across

the interface is stable in the gravity field if for instance the heat contained in

the upper fluid compensates the density increase due to the solute, yielding an

overall statically stable density stratification with regard to a Rayleigh-Taylor

instability. In that case the interface can nevertheless still be unstable due to a

difference in diffusivity of the solute and of heat generating instabilities known

as double diffusion phenomena. This mechanism is based on the differences

between the diffusion coefficients of the two competing components influencing the density of the solution [4, 103, 104, 146, 147, 148, 149]. Heat and mass are good candidates to trigger such destabilizations as heat is known to diffuse about 100 times faster than mass in water.

Hot and solute denser fluid

Cold and solute less dense fluid Solute

Fluid A

Fluid B Heat

Figure 1.4: Direct double diffusive mechanism which can appear in a statically stable stratified fluid layer.

The mechanism of the double diffusive instability is the following one. Con- sider a stratification of hot and solute denser fluid above a cold and solute lighter one. The upper fluid is hot enough so that its overall density is lighter than that of the lower fluid. The initial density gradient is thus statically stable. Imagine then a perturbation of fluid A into fluid B (see Fig 1.4), the blob of hot solute heavy water which finds itself surrounded by cold solute light fluid rapidly looses its heat while retaining the solute due to the very different diffusion rates. The perturbation becomes colder and hence denser than the surrounding fluid. This tends to make the blob sink further, pulling down more hot solute dense water from above giving rise to sinking fingers of fluid. For the perturbation of fluid B into fluid A, heat penetrates into the parcel quicker than mass and makes it lighter, the perturbation will then rise.

Another instability mechanism based on differential diffusivity between heat and mass is the oscillatory double diffusive instability the mechanism of which is depicted in Fig. 1.5. The density stratification compared to the previous picture is reversed, the hot and denser fluid lies beneath the solute lighter and cooler one, the total density stratification being statically stable. A perturbation of fluid B into fluid A will loose heat more rapidly than solute. This will yield a heavier parcel of fluid which will sink back. As the perturbation has lost heat but retained solute it will overshoot its initial position and will at the bottom of the parcel be heated up and rise again. This mechanism leads to an oscillatory behavior.

The first mention of such types of double-diffusive destabilization was made

by Stommel, Arons and Blanchard in 1956 [138] who made an experiment where

a conducting pipe was plunged in a solution such as depicted in Fig. 1.4. The

solute less dense water from the bottom rises in the tube and gets heated on it’s

way, decreasing its density. This gives rise to a salt fountain, the less dense fluid

being pumped up along the tube. In the opposite case they considered a parcel of

fluid isolated from it’s environment by a conducting shell and which is displaced

Hot and solute denser fluid Solute

Fluid B

Heat

Figure 1.5: Oscillatory double diffusive mechanism which can appear in a stably stratified fluid layer.

upwards. The parcel of fluid looses its heat but not the salt and is driven back down. A few years later in 1960, Stern took a great leap forward pointing out that this instability doesn’t need the physical boundaries to exist, that the diffusivity differences between heat and salt are enough to trigger the instability [137]. These mechanisms are common in oceans where the sun evaporates the top water layer making it more concentrated in salt and hotter than the deeper fresh water even if the total density gradient is stabilizing. The double diffusive mechanism of Fig. 1.4 can lead to long fingers of salt sinking in the ocean which explains why this type of double diffusive convection is also known as “salt fingers”[147]. The oscillatory mechanism is also present in oceans where it is shown to lead at the bottom of the oceans to well mixed layers of fluid separated from each other by sharp interfaces, a phenomenon known as “thermohaline staircases”[147]. These two mechanisms have been shown to be present in 44%

of the oceans of the earth’s surface [172].

Double diffusive convection is susceptible to appear whenever two compo- nents of a fluid (not only mass and heat) are placed in the gravity field and diffuse at different rates and is known to exist in crystal growth and metallurgy.

In a very general way one can point out that compositional and thermal density gradients are able to generate dynamical instabilities. This is the case in a wide range of applications ranging from oceans movements [77, 172], atmo- spheric flows [116], convection in the earth mantle [26, 57], CO

sequestration [69, 126] and combustion [1, 132, 164, 175, 177]. As during a chemical reaction several solution parameters, such as density or viscosity, can vary it may give rise to convection movements [2, 3, 11, 32, 97, 114]. We will thus in the next paragraph start to address possible interactions between chemical reactions and such hydrodynamic instabilities.

1.6 Buoyancy-driven instabilities of chemical fronts

In the already mentioned polymerization front the conversion from monomers to

polymer can lead to instabilities. Indeed, the reaction implies density and/or vis-

cosity differences associated to compositional changes and to the strong exother-

micity of the reaction. These convection motions can destroy the front prop- erties of the polymerization and trigger the reaction in the whole experimental tube, something you want to avoid to get a controlled polymerization [12, 17, 52, 53, 90, 108, 111, 112, 165]. Polymerization liquid to liquid fronts can also be subjected to double diffusive instabilities if the hot denser polymer is placed upon the lighter and cool monomers [108].

Fronts in oxydoreduction autocatalytic systems can also be affected by con- vection motions. To study these convection flows experiments were first per- formed in tubes in absence of gel. When the reaction diffusion front propagates in a thin tube, different experiments have shown a difference in the speed of the front depending on the direction of propagation with regard to the gravity field. It has been shown for the IAA reaction that a front traveling upwards moves several times faster than the downward propagating one [88, 110]. The explanation for such a phenomenon lies in the presence of convection in the fluid for upward moving fronts, the generated flows accelerating the front. The onset of such a fluid flow is due to the density variations across the front during the reaction. This density jump can have two origins. The first origin, a solutal one, is due to changes in the composition between the reactants and the products.

The second, a thermal one, is due to the heat consumption or release by the reaction which influences the density variation during the reaction.

One of our goal is to analyze the influence of chemical reactions on the gen- eral picture of buoyancy-driven convection due to double-diffusion and Rayleigh- Taylor instabilities. As already mentioned, during a chemical reaction the den- sity can vary which can in turn trigger buoyancy driven instabilities. As the density of a solution is function of both composition and temperature, its varia- tion can be expressed in terms of these two contributions. Following Pojman and Epstein [109] let us see what dynamics can be expected. The density changes across a chemical front can be related to the compositional and thermal varia- tions through the thermal α

and the solutal α

expansion coefficients defined as:

α

=

1 ρ

∂ρ

∂T , (1.15)

α

= 1 ρ

∂ρ

∂c

, (1.16)

where T is the temperature and c

the molar concentration of the ith species.

The total density of the solution with regard to a set of reference conditions ρ

, c

and T

is given by:

ρ = ρ

[1

α

(T

T

) +

i

α

(c

c

)]. (1.17)

As a reaction proceeds there will be changes in the composition of the so-

lution and if the reaction is exo- or endothermic it will furthermore lead to an

increase or decrease of the temperature of the solution. Still following [109] the

solutal volume change during the reaction can be expressed as

∆V =

products

n

V

reactants

n

V

= V

V

(1.18) where n

is the number of moles of the ith species and V

its partial molar volume, V

the volume of the products and V

the initial volume of the reaction mixture (i.e. the reactants). The density change ∆ρ

caused by the variation in chemical composition during the reaction is given by:

∆ρ

= ρ

ρ

= m V

m

V

= m( 1

∆V + V

1 V

) (1.19)

where m is the total mass. The density difference ∆ρ

is thus related to ∆V by

∆ρ

=

∆V /V

1 + ∆V /V

ρ

∆V V

ρ

β∆cρ

(1.20)

where ∆V is assumed to be small in front of V

, ρ

is the initial density of the solution, ∆c is the change in concentration of the limiting reagent and β is the mean molar density coefficient. Thus a positive ∆V corresponds to a decrease in density in the course of the reaction. In other words if the reaction leads to an increase of the molar volume of products vs. that of the reactants (∆V > 0), the products are lighter than the reactants.

If the reaction is adiabatic and if the mean coefficient of thermal expansion α

and the heat capacity of the reaction solution c

are known then the density change ∆ρ

due to the heat release during the reaction is

∆ρ

= ∆H ∆c 1 c

α

ρ

. (1.21)

The change in temperature (∆H/c

) and the volume change due to the con- centration can also be measured directly in the experiment. As described by Nagyp´ al et al. [101] the reaction can be performed in a dilatometer where the volume changes between the reactants and the products are measured after let- ting the system cool down after the thermal increase during the reaction. This allows to measure the thermal jump and then the volume difference between the products and the reactants.

It is thus clear that during a chemical reaction the density can change due to

both solutal and thermal contributions. We have also seen that different hydro-

dynamic instabilities can appear at the interface between two miscible fluids due

to density differences between them. The chemical reaction-diffusion conversion

fronts are simple model systems on which to study the influence of chemistry on

hydrodynamical instabilities. Indeed, propagating reaction-diffusion chemical

fronts provide a mechanism for creation of internal concentration and tempera-

ture gradients, the chemical reaction generating not only the interface between

the products and the reactants but the density jump between the solutions as

well.

Let us now detail some of the different experimental studies which have been made on hydrodynamical instabilities of chemical fronts. As no oscillatory dou- ble diffusive phenomena have been observed, the hydrodynamic instabilities of chemical fronts are known as fingering phenomena because the interface between products and reactants is often deformed in a finger like shape.

1.7 Fingering of chemical fronts in tubes

Convective deformation of chemical fronts have been studied experimentally since the 80’s. Some of the first studies were conducted in tubes using various autocatalytic reactions able to yield chemical fronts.

One of the first report on increased front speed due to convection was made by Bazsa and Epstein [10] when they found that a front in the Nitric Acid-Iron (II) reaction travels several times faster going down than up in the gravity field.

This reaction being exothermic this result is at first sight strange as one would expect the upward traveling front to move faster. Indeed, the convective motion could be induced by the hotter products rising in the cooler reactants. In this particular reaction the hypothesis proposed at that time was that the reaction starts with a nucleation phenomenon, the reaction being slowed down in the case of convection because this nucleation is perturbed. The convection generated in the upward propagating front because of the exothermicity was thus supposed to be responsible, through a perturbation in the nucleation, for a decrease of the reaction diffusion speed.

After this first study, Nagypal et al. [101] further analyzed experimentally the Nitric Acid-Iron(II) reaction and carefully measured the density changes due to thermal and solutal effects. They found that although the density decreases due to the exothermicity, it increases due to the solutal contribution. They explained the destabilization of the downward propagating Nitric Acid-Iron (II) front by the diffusivity difference between heat and mass, using a double diffu- sion mechanism such as the one explained earlier in this chapter. The hot but heavy products are, for a downward propagating front, placed upon cool but lighter reactants. The overal density stratification is stable but if a perturbation of products into the reactants is made, the perturbation will loose heat faster than mass becoming heavier than the surroundings driving a destabilization. In the same article they also studied the Chlorite-Thiosulfate front propagation.

This highly exothermic reaction features, although the density increases dur- ing reaction, a front propagating faster upwards than downwards, the thermal effect dominating the solutal one. They were able to reverse the stability by playing with the concentrations of the reactants. They tuned the heat effect by changing the surrounding media around the tubes. They could suppress the thermal contribution by plunging the tube in water. The heat produced dur- ing the reaction being then dissipated through the walls, the front propagating down then becomes the one featuring the destabilization.

These studies lead Pojman and Epstein [109] to do a qualitative classification

of the different effects one can see in these experiments with regard to the

the density difference between the products and the reactants as ∆ρ = ρ

ρ

where ρ

and ρ

stands for the density of the products and reactants respectively.

The density difference will have a negative sign if the overall density decreases during the reaction. This difference is due to the combination between a solutal contribution and a thermal one. As already mentioned, the solutal density jump

∆ρ

is due to the molar volume changes between products vs. reactants, a molar volume increase during the reaction yielding a negative density jump ∆ρ

. The thermal part ∆ρ

is related to the exo- or endothermicity of the reaction. We focus here on exothermic reactions as no endothermic reaction featuring fronts are known up to now, ∆ρ

then always has a negative sign, yielding a density decrease in the products.

When the two effects – the solutal and thermal one – both decrease the density in the products, they are cooperative, ∆ρ

and ∆ρ

are negative. The reaction thus transforms solute heavy reactants at room temperature into solute lighter and hotter products. One then expects an unstable ascending reaction- diffusion front as this corresponds to an unstable Rayleigh-Taylor density strat- ification of heavy on top of light. According to the classification by Pojman and Epstein [109] this is known as simple convection. The IAA reaction (in the Arsenous Acid excess case) falls into this category as the density decreases across the front due to both thermal and solutal contributions. One thus ex- pects, in the IAA system, only simple convection to occur and only the fronts propagating upwards should be unstable above a critical diameter of the tube allowing the presence of a convection roll.

Different phenomena can be triggered if the thermal and solutal effects are antagonists i.e. when ∆ρ

is positive and ∆ρ

is negative as is the case for an exothermic reaction (∆ρ

< 0) with a solutal density increase in the course of the reaction (∆ρ

> 0). In that case both ascending and descending fronts can become unstable depending on the magnitude of both effects. Indeed, consider a front with very weak thermal effects. Ascending fronts are then stable while descending fronts featuring heavy products on top of lighter reactants (∆ρ

> 0) are unstable. This is typically the case for the Chlorite-Tetrathionate reaction [63]. If the thermal effects are increased, it can lead to a reversal of the instability if the products become hot enough to overcome the solutal density jump [8].

Furthermore, as already mentioned, the diffusivity differences between heat and mass can come into play and trigger destabilization of stably stratified fronts.

The Nitric Acid-Iron(II) presented above [10, 101, 113] falls, for example, in this category exhibiting double diffusive instability.

In a tube different factors are affecting the destabilization of a front [109].

The front subject to a density gradient will be less stable if the density gradient is increased. On the contrary a smaller tube radius, a higher transport coefficient of the destabilizing component and a higher viscosity will increase the stability.

All these trends can be summarized into a Rayleigh number quantifying the

stability as a function of the radius r of the tube:

Ra = r

g µT r

dρ dz

(1.22) where T r is the transport coefficient causing the density gradient along the verti- cal direction (z) and µ is the dynamic viscosity. The critical Rayleigh number for a cylinder filled with a non reactive fluid with nonconducting sidewalls and free upper and lower boundaries is Ra

= 67.9, below which the front is stable [24].

The conversion front then remains flat and propagates at the reaction-diffusion speed when the tube is too thin to allow the convection roll too appear. The theory predicts that depending on the diameter of the tube two modes are possi- ble. First, the antisymmetric one which appears at the onset of instability when the diameter of the tube becomes large enough to contain one convection roll.

The fluid rises on one side of the tube and sinks on the other. The fluid is thus

“higher”at one wall of the tube and “lower”on the other wall. This situation is thus antisymmetric with regard to the middle of the tube. Above a theoretical Ra of 452, two convection rolls are predicted for non reactive fluids. The fluid rises then in the middle of the tube and the convection rolls close in dragging the fluid down near the walls. This leads to an axisymmetric situation with regard to the middle of the tube, the wave form having then a parabolic shape.

Experimentally, the antisymmetric situation and its transition to an axisym- metric one have been observed in the IAA reaction [88] but the axisymmetric flow is present at Rayleigh numbers well below the one calculated for a non reactive fluid in a tube [109, 110]. The explanation for this discrepancy is due to the fact that the fluid is reacting. In the antisymmetric configuration the unreacted flow will be dragged down by convection into a reacted area. The dif- fusion of the autocatalytic species into the unreacted zone will cause a reaction decreasing – in the IAA reaction – the density, the descending fluid becomes less dense and rises. Therefore, the axisymmetric situation is more dynamically stable in a reactive fluid than in a non reactive one. Let us note that the hori- zontal configuration is always unstable leading to antisymmetric convection and a front traveling faster [88, 122, 170].

The dynamics of different autocatalytic reactions have experimentally been analyzed in tubes. Experimental studies have focused on different factors influ- encing the onset and the strength of the convection such as the concentrations of the reactants, the radius of the tube and its tilting angle with regard to gravity.

The impact of the concentration variations in the reactants have been stud-

ied in the antagonist category on the Nitric Acid-Iron(II) [10, 101, 113], the

Chlorite-Thiosulfate [101], the Chlorate-Sulfite [100] the Bromate-Sulfite reac-

tion [78] and the 1,4-Cyclohexanedione-Bromate-Sulfuric Acid-Ferroin System

[79]. In the cooperative category the IAA reaction [88, 110], the Iodate-Sulfite

[78, 112] and the Iodate-Nitric Acid reaction [99] have been studied. For these

two last reactions only simple convection is expected according to the Pojman-

Epstein model [109] as a decrease in density should lead to a stable downward

propagating front. This is not always the case and unstable descending fronts

have been observed, this destabilization being then explained using a double-

sion coefficient.

The propagation speed of the front has been studied in different orientation with regard to gravity and is a function of the tilting angle [10, 78, 99, 101, 113].

The experiments of instabilities on horizontal fronts always yield convection and fronts traveling at a higher speed than the reaction-diffusion one [110].

Different experiments have shown that the highest propagation speed is obtained for tubes tilted at an angle of about 40-60 degrees from the horizontal. This was first observed in [101] and then in [78, 99, 113]. This can be explained because the horizontal cross section (elliptical) of a inclined tube is larger than the perpendicular (cylindrical) cross section. The greater the surface area, the greater the convection. The propagation speed distribution as a function of the angle was shown to take a pumping shape [78, 99, 101].

All these studies except [112] have focused on the impact of the radius of the tube on the destabilization. When the diameter becomes large enough the front does not only deform into an axisymmetric form but can feature several fingers [22, 78, 99, 100]. In the IAA reaction, the experiments in thin tubes [110] have considered the reaction as isothermal. This is supported by two arguments: the exothermicity is weak (0.3-0.6 K for initial concentration of the order of 10

M) and the heat produced in the reaction is almost completly evacuated through the walls. An increase in the diameter of the tube leads to the experiments by Rogers et al. [120] who consider the fingering of the ascending front in IAA reaction in a large 3D tank with a localized injection. This leads to the formation of rising buoyant plumes.

Theoretical studies of fingering in tubes have either been based on an eikonal equation giving the speed of the front as a function of its curvature or on a full reaction-diffusion equation. In the first case, the eikonal approach, the front is assumed to have zero thickness, the fluid pressure and velocity are related by jump conditions across the front [43, 71, 70, 153, 158, 154, 160, 151, 166]. This eikonal equation is coupled to Navier-Stokes equation with unbounded [43, 70, 166] or bounded boundary conditions [153]. This eikonal equation is derived from the full reaction-diffusion problem [167] but is still an approximation. In order to model the system more accurately the full reaction-diffusion problem has also been studied for a cubic reaction [156, 159, 160, 157, 170] and for a quadratic one [157].

The model chemical reaction used in these studies have always been based on the IAA reaction which falls in the cooperative class. The density thus decreases behind the front due to both thermal and solutal effects. The first theoretical studies [43, 153, 158] have considered two limit cases for the thermal effects:

the zero thermal diffusivity and the infinite one. The zero thermal diffusivity

(i.e D

= 0) still gives a zero thickness front [43, 153, 158] and thus considers

the density jump across the interface as a simple increase of the solutal density

contribution. The other limit, the infinitely thermal diffusivity (i.e. D

=

),

where the temperature of the products is the same as that of the reactants

yields no thermal density difference, so only the solutal contribution to the

density difference is taken into account [43, 151, 153, 156, 158, 160]. In [43] the

critical wavelength for the onset of convection is shown to vary from 0.99 mm

for D

= 0 to 1.29 mm when D

=

, leading the authors to conclude that heat effects are not so important.

The nonlinear studies performed have shown the flow field around the front to be one or two convection rolls and have studied the transition between the two convection modes [151, 158, 168]. Full 3D simulations have also been performed showing the transition from anti- to axisymmetric flows [169]. Other more accurate models have used a finite thermal diffusivity [154, 166].

The experiments in thin tubes only allow the onset of one or two convection rolls, the front being then only slightly deformed. Increasing the diameter of the tubes leads to more complex patterns such as fingers [22, 78, 99, 100] or buoyant plumes [120]. The 3D systems are pretty complicated and in order to simplify matters one usually resorts to 2D models and experiments. This is for example the case when an experimental study was carried out in a vertical slot [20], two glass plates separated by a thin gapwidth but of small horizontal extent (3.2 cm) which allows some convection rolls to develop. This has also been modeled theoretically [70, 155]. Some studies have also focused on the impact of external imposed advective flows on autocatalytic reaction fronts [42, 81, 136].

In order to study more spatially extended systems than tubes one must resort to the use of Hele-Shaw cells which allows as the system is laterally extended the onset of more convection rolls than in thin tubes. Furthermore, as the plates are transparent, it allows an easy visualization of the fingering phenomena and the fluid flow can -within a certain extent- be described using a 2D formulation.

The experimental set-up, why and to what limits it can be used are explained in the next paragraph.

1.8 Fingering of chemical fronts in Hele-Shaw cells

Hele-Shaw cells consist in two parallel transparent (typically made of glass or plexiglass) plates of length L

and width L

large compared to the gap L

between them (L

< L

, L

). For the experimental studies of Rayleigh-Taylor

instabilities of non-reactive fluids, the heavier fluid is placed upon the lighter

one. An example of density fingering experiment is shown in Fig. 1.3 where a

denser solution of colored water containing glycerine is placed upon pure water

in the gravity field. One sees that the heavier fluid on top sinks into the lighter

one deforming the initially flat interface. One main advantage of the Hele-Shaw

cell is to allow easy visualization because of the transparency of the plates. The

other advantage is that the flow field in the cell can, in some limits, be described

using 2D Darcy’s law. The range of validity of the use of Darcy’s law instead

of the full Navier-Stokes [9] or Brinkman’s [176] equations depends for miscible

fluids on the gap width [60, 85, 124, 176]. We will assume in the following that

the gap width is small enough for Darcy’s law to be used. We can in that way

focus on changes in the dynamics due to chemistry keeping the description of

the hydrodynamics to the same throughout our work. Darcy’s law was first

formulated by Darcy in 1856 [25]. He conducted experiments on fluid transport

in sand columns and formulated the following equation known today as Darcy’s

∇

p =

µ

K u + ρg. (1.23)

This expression states that, in a porous medium, the velocity of the flow u is linearly dependent on the gradient of pressure

p for a given hydrostatic pressure ρg. The mobility of the fluid is given by the coefficient K/µ, the ratio between the permeability of the porous medium K and the viscosity of the fluid µ. This coefficient reflects the fact that for a fixed pressure gradient a less viscous fluid will travel faster than a more viscous one while a given fluid of fixed viscosity travels faster in a more permeable media.

Using this equation the fingering phenomena in porous media can be ex- plained intuitively. This explanation was first introduced by Hill in 1952 [59].

PSfrag replacements µ ₁ , ρ ₁ , p ₁

µ 2 , ρ 2 , p 2

y x

U

∆x g

Figure 1.6: Schematic representation of the fingering destabilization mechanism at an interface.

Consider the flow, presented in Fig. 1.6, injected along x with a mean velocity U and obeying Darcy’s law in one dimension:

dp dx =

µ

K u + ρg. (1.24)

The interface becomes unstable if a small perturbation applied on it grows, this is the case when the driving pressure across the perturbation ∆x is positive i.e. when [60]

p

p

=

µ

µ

K

U ∆x + g(ρ

ρ

)∆x > 0 (1.25)

where the subscripts 1 and 2 refer respectively to the upper displacing fluid

and the lower displaced fluid. If the viscosity of the two fluids is the same

(i.e. µ

= µ

), an instability of the interface will occur when ρ

> ρ

which

corresponds to a denser fluid placed upon a lighter one in the gravity field. This

corresponds to the Rayleigh-Taylor instability or density fingering. The same