Th`ese pr´esent´ee en vue de l’obtention du grade d’Agr´eg´ee de l’Enseignement Sup´erieur de la Facult´e des Sciences de l’Universit´e Libre de Bruxelles.
Bruxelles, d´ecembre 2003
Contents i
Acknowledgment iii
1 Introduction 1
2 Fingering phenomena 5
2.1 Fingering in porous media . . . . 6
2.2 Hele-Shaw cells . . . . 8
2.3 Viscous fingering . . . . 12
2.4 Density fingering . . . . 14
2.5 Boussinesq approximation . . . . 16
3 Density fingering of isothermal iodate-arsenous acid reaction fronts 19 3.1 Fingering in reactive systems . . . . 19
3.2 The iodate-arsenous acid (IAA) system . . . . 23
3.3 Chemical fronts in the IAA reaction . . . . 25
3.4 Experimental results on the fingering of IAA fronts . . . . 27
3.5 Model for the fingering of the IAA reaction . . . . 30
3.5.1 Reaction-diffusion front . . . . 32
3.6 Linear stability analysis . . . . 34
3.7 Nonlinear simulations . . . . 38
3.7.1 Numerical scheme and validation studies . . . . 39
3.7.2 Density fingering of a chemical front: nonlinear dynamics 41 3.7.3 Parametric study . . . . 50
3.8 Discussion . . . . 60
4 Density fingering of isothermal chlorite-tetrathionate reaction fronts 63 4.1 The chlorite-tetrathionate (CT) system . . . . 64
4.2 Experimental results on the fingering of CT fronts . . . . 65
4.3 Model for the fingering of the CT reaction . . . . 67
4.4 Chemical fronts in the CT reaction . . . . 70
4.5 Linear stability analysis . . . . 74
4.5.1 Dispersion relations . . . . 77
4.5.2 Comparison to experiments . . . . 79
4.6 Nonlinear simulations . . . . 82
4.7 Discussion . . . . 84
5 Density fingering of exothermic reaction fronts 89 5.1 Model incorporating the heat equation . . . . 92
5.2 Traveling concentration and temperature fronts . . . . 95
5.3 Linear stability . . . . 98
5.3.1 Eigenvalue problem . . . . 98
5.3.2 Dispersion curves and stability characteristics . . . 101
5.4 Experimental results . . . 108
5.5 Nonlinear simulations . . . 109
5.6 Discussion . . . 115
6 Fingering of spatially extended bistable chemical systems 119 6.1 Viscous fingering of bistable extended systems . . . 120
6.1.1 Basic equations . . . 121
6.1.2 Choice of the rate expressionf′(c) . . . 124
6.1.3 Numerical methods . . . 125
6.2 Nonlinear viscous fingering with bistable chemical reactions . . . 126
6.2.1 Standard viscous fingering . . . 126
6.2.2 Viscous fingering of bistable chemical fronts . . . 128
6.2.3 Droplet formation mechanism . . . 130
6.2.4 Single droplet studies . . . 133
6.2.5 Chemistry-induced tip splitting . . . 136
6.2.6 Parametric study . . . 138
6.2.7 Summary . . . 146
6.3 Density fingering of bistable extended systems . . . 146
6.3.1 Model system . . . 149
6.3.2 Droplet formation . . . 152
6.3.3 Spatial bistability in extended open reactors . . . 156
6.4 Discussion . . . 161
7 Discussion and perspectives 163
8 Appendix: Numerical pseudo-spectral scheme 167
Index 171
S.C. M¨uller, G. Nicolis and J.-W. Turner for having kindly accepted to be the members of the jury of this thesis.
I wish to express my sincere gratitude to Professor G.M. Homsy who has welcomed me in his laboratory during my postdoctoral stay in the Chemical En- gineering Department of Stanford University (USA) in 1993-1994. Since then, we have developed a very fruitful collaboration over the years. This collabo- ration has been decisive in the development of my current research activities as I have learned most I know about fingering phenomena from Bud Homsy.
I thank Bud for his constant support and interest in our discussions through- out the years and for initiating me patiently to the fascinating world of fluid dynamics.
Two other persons have always been closely associated to my scientific ac- tivities: Guy Dewel and Pierre Borckmans. The loss of Guy has been deeply felt in our group. Let me say here how much I miss him, his scientific advices as well as his humor. I owe him and Pierre most of my knowledge of nonlin- ear physico-chemical systems. I wish also to express my gratitude to Pierre for learning me so many things in our every-day discussions.
These three persons, Pierre Borckmans, Guy Dewel and Bud Homsy are really my scientific ”fathers” thanks to whom I have been able to develop the research subject presented in this work, at the frontier between hydrodynamics and nonlinear chemistry.
I wish also especially to thank Professor G. Nicolis for guiding me to this field of research thanks to his courses and his constant support and interest throughout all these years.
This work has benefited from discussions with many, many persons.
First of all, my theoretical contributions to the field of fingering of chemical fronts has been crucially motivated by experiments. I thank therefore Profes- sor Stefan C. M¨uller, Martin B¨ockmann and Osamu Inomoto from Magdeburg University; Agot´a T´oth and Dezs´o Horv´ath from Szeged University as well as Kerstin Eckert, Ying Shi, Margret Acker and A. Grahn from Dresden University
of Technology for their friendly collaboration and for sharing their experimental results with me prior to their publication. I thank them above all for having tested in experiments some of my crazy ideas of theoretician !
Some of the results presented here have been obtained in collaboration with Serafim Kalliadasis and Jianbo Yang from Leeds University. It is a pleasure to acknowledge the interesting and friendly discussions we always have had together.
Let me also particularly thank Yann Bertho, Dmitry Bratsun, Jessica D’Her- noncourt, Alejandro D’Onofrio and Diana Lima with whom I have worked on fingering phenomena at the Universit´e Libre de Bruxelles.
I have benefitted from discussions with K. Benyaich, Jacques Boissonade, Jean-Pierre Boon, Pierre Colinet, Patrick De Kepper, Henk Dijkstra, Etiennette Dulos, Fabienne Gauffre, Patrick Grosfils, Marcus Hauser, Ren´e Lefever, J´erˆome Martin, Michel Martin, Eckart Meiburg, John Merkin, John Pojman, Felix Otto, Nicole Rakotomalala, Dominique Salin, Benoit Scheid, Hana ˇSevˇc´ıkov´a, Ken- neth Showalter, Desiderio Vasquez, Andrei Vedernikov, Vitaly Volpert, Daniel Walgraef and Abdel Zebib. Many thanks to all of them.
Finally, many thanks also to all members of the Service de Chimie Physique at ULB for every day discussions, not all scientific...
Je pense sinc`erement que je n’aurais pu effectuer une telle carri`ere scien- tifique sans le soutien constant de mes parents. Je les remercie du fond du coeur pour leur confiance sans faille. Enfin, Merci `a F´elix, Tania et Daniele `a la source de mon bonheur.
Chemical reactions interplay with fluid motions in numerous situations of every day’s life. As an example, flow induced spreading of chemical pollutants and their interaction with the environment is a problem of major concern in our society. Norms on the authorized concentration of chemicals in the atmosphere or in rivers depend on a good description of the concentration dependence on time and space which in turn depends on knowledge of the chemical kinetics and on the velocity of the fluid carrying the chemical species. In that respect, there is a vast literature studying the influence of fluid flows on the transport of passive chemical species. The situation becomes more complex if the chemical reactions affect properties of the fluid such as its density for instance. In that case, the chemistry can be at the origin of hydrodynamic instabilities of the fluid inducing motions unexpected in non reactive systems. As a corollary, the hydrodynamic flows can in turn affect the yield of the reactions and an interesting feedback between chemistry and hydrodynamics sets then in. Such a feedback is observed in numerous applications as, for instance, in ocean and atmospheric flows, spreading of spills in undergrounds and of course in chemical reactors at all scales where transport of reactive species in solution is most common. The spatiotemporal dynamics resulting from the interplay between hydrodynamics and chemical reactions can easily become rather complex as a number of effects such as viscosity or density variations, surface tension or heat effects may come into play. In that respect, the fundamental understanding of the dynamics of reactive fluids can benefit from simple model systems in which only some of the possible sources of instability are present.
In that context, the present work aims at studying the coupling between hydrodynamic fingering instabilities and chemical reactions at the interface be- tween two miscible solutions. Hydrodynamic deformations of interfaces between two reactive fluids as well as flows induced by chemical reactions at the front between two initially steady fluids are encountered frequently in combustion,
petroleum, chemical and pharmaceutical engineering. Most of the time, concrete applications imply a very large number of variables so that an understanding of the fundamental processes of chemo-hydrodynamic coupling is out of reach.
Our goal is here to analyze a much simpler model system in which only one mechanism of hydrodynamic instability is at play and for which the chemical reactions can be modelled by a one or two-variable model.
Buoyantly unstable, autocatalytic chemical fronts, are one such model sys- tem which can be used as a prototype to study the effects of the coupling between chemical reactions and hydrodynamic fingering instabilities. Fingering processes occur whenever a fluid of high mobility displaces a less mobile one in a porous medium. The initially planar interface looses then stability and a cellular fingering deformation of the interface is observed. Such an instability has been observed, for instance, in the iodate-arsenous acid and chlorite-tetrathionate re- actions, redox reactions known to produce a change of density during the course of the reaction. Fingering happens there when the heavier solution lies on top of the lighter one in the gravity field.
As such, the study of fingering of chemical autocatalytic fronts participates also to the development of new research paths in nonlinear chemistry. The ex- perimental and theoretical studies of clock reactions in batch reactors as well as of bistability, chemical oscillations and chaos in continuously stirred tank re- actors have witnessed a major development since the sixties. Numerous efforts have been invested in kinetic studies aiming at deciphering the autocatalytic and feedback processes at the heart of the complex temporal dynamics. In the eighties, the construction of open constantly fed gel reactors has allowed to add the diffusive spatial component to the problem and beautiful spatio-temporal patterns such as controlled waves, spirals and Turing patterns have then been successively observed and studied in detail. The present work contributes to the next step in increasing spatial complexity which consists in adding a flow on top of nonlinear chemistry and molecular diffusion. In that regard, numerous works on new spatio-temporal flow induced instabilities in nonlinear chemical systems have been devoted to situations where the chemical species are only advected by the flow without influencing the properties of the fluid. We address here the situation where the chemical reactions are at the heart of the hydrodynamic motions as chemistry modifies the density or viscosity of the fluid inducing fluid instabilities that would not occur in a non reactive system. The buoyantly un- stable autocatalytic fronts feature then a new class of spatio-temporal dynamics in spatially extended systems. As such this subject is genuinely interdisciplinary as it adds hydrodynamics to the field of nonlinear chemistry and in parallel intro- duces the influence of chemical reactions on pure fingering phenomena yielding insight into several engineering applications.
chlorite-tetrathionate reaction in order to address the influence of the chemical kinetics on the dynamics observed. The influence of the exothermicity of the reaction will be presented in chapter 5. Eventually, chapter 6 will analyze what happens if the kinetics is now bistable and will further compare the situation of both viscous and density fingering of bistable fronts. We will then conclude and present suggestions for future work in this subject at the frontier between chemistry, physics and engineering.
Fingering of an interface is a common phenomenon encountered in fields as diverse as petroleum recovery (Araktingi & Orr, Jr., 1990; Wooding & Morel- Seytoux, 1976; Homsy, 1987), combustion (Zik et al., 1998), chromatography (Norton & Fernandez, 1996), and electrochemical deposition (de Bruyn, 1995) for instance. This hydrodynamical instability occurs as soon as a higher mobility fluid displaces a low mobility fluid. The interface between the two fluids is then no longer planar but deforms into “fingers” of the more mobile phase that invade the other phase. This is usually undesirable as it leads to a mixing of the two phases, a phenomenon that engineers want to avoid in porous media applications where fingering is observed. Mobility variations are usually related to differences in viscosity and/or density of the two fluids considered. In that respect, fingering typically occurs when a less viscous fluid displaces a more viscous one giving rise to the so-called Saffman-Taylor instability, orviscous fingering instability (for a review, see Homsy (1987)). Similarly, a morphological instability of the horizontal interface between liquids of different densities occurs when the heavier fluid is placed on top of the lighter one in the gravity field. One refers then to the Rayleigh-Taylor instability, ordensity fingering(Fernandezet al., 2002).
Fingering affects both immiscible and miscible systems. In both cases, the viscous and/or density difference across the interface is the driving mechanism of the instability which, in spatially extended geometries, destabilizes a wide spectrum of length scales. For immiscible fluids, surface tension tends to favor a planar interface and is hence the competing stabilizing mechanism. For miscible fluids, the diffusive mixing of the two solutions tends to decrease in time the gradient of mobility and diffusion is thus the stabilizing process. The wavelength of the fingering pattern appearing at onset is thus set-up by a balance between the destabilizing mobility ratio and the stabilizing surface tension or diffusion leading, for immiscible and miscible fluids respectively, to cut-off of small scales.
Because of the influence of such fingers on the physical and chemical pro-
cesses, numerous studies have analyzed to what extent the fingering instability can be affected for instance by heterogeneities of the medium (Tan & Homsy, 1992; Tchelepiet al., 1993; Fernandezet al., 1995; Chen & Meiburg, 1998b; De Wit & Homsy, 1997a; Camhiet al., 2000), the non-Newtonian character of flows (Nittmann et al., 1985; Makinoet al., 1995; Corvera Poire & Ben Amar, 1998;
Azaiez & Singh, 2002), particle accretion (Tang et al., 2000), the presence of a magnetic field (Flamentet al., 1998) , etc. There is a huge literature on finger- ing phenomena and it is not our purpose here to review it. Indeed, depending on the geometry, the important driving forces and the applications considered, many different articles have been devoted to the problem so that the existing review articles usually already chose a specific point of view (Wooding & Morel- Seytoux, 1976; Homsy, 1987; Bensimon et al., 1986). Our focus here will be specifically on fingering phenomena between two miscible fluids in the presence of chemical reactions. Fingering interacts with chemical reactions in applica- tions ranging from chromatographic separations (Czok et al., 1991; Ilg et al., 1992; Planteet al., 1994; Dicksonet al., 1997), fixed bed chemical processing and regeneration (Hill, 1952), reactive infiltration in geochemical settings (Ortoleva, 1994) to chemical treatment of oil-bearing formations (Homsy, 1987).
The influence of chemical reactions on fingering will be addressed in the next chapters. Before introducing explicitly chemistry, we will in this chapter review the important characteristics of fingering phenomena for non reactive miscible fluids. We will first recall the mechanisms at the origin of both viscous and density fingering in non reactive flows in porous media. We will then introduce the Hele-Shaw set up as a model system used to study fingering phenomena in linear geometries and discuss the conditions in which they can be used to have insight into porous media flows. We will contrast the specificities of both viscous and density fingering and discuss the equations used to model these instabilities. Theoretical modeling of density fingering often resorts to the so- called Boussinesq approximation which we introduce next.
2.1 Fingering in porous media
The majority of applications in which fingering phenomena are observed (petro- leum recovery, spreading of spills in aquifers, chromatography, ...) imply fluid flow in porous media. In porous media, the velocity of the flowuhas been shown experimentally by Darcy to vary linearly with the gradient of pressure applied at the boundaries of the system (see appendix D of Darcy (1856)). Henry Philibert Gaspard Darcy (1803-1858) was a french engineering in charge of the fountains of the city of Dijon. He carried extensive experimental work on transport of fluids in sand columns and stated the following evolution equation which now
µ2, ρ2, p2
Figure 2.1: Schematic of an unstable displacement carries his name i.e. Darcy’s law:
∇p=−µ
Ku+ρg. (2.1)
This expression shows that, in a porous medium, the velocity of the flow uis linearly related to the gradient of pressure∇pfor a given hydrostatic pressure ρg. The coefficientK/µis the mobility of the fluid defined as the ratio between the permeability of the porous mediumKand the viscosity of the fluidµ. Hence for a fixed gradient of pressure, less viscous fluids travel faster than more viscous ones while a given fluid of fixed viscosity travels faster in systems with higher permeability.
Hill (1952) was the first to conduct experiments on fingering in porous me- dia in the context of regeneration processes analyzing the displacement of sugar liquors by water from columns of granular media. Hill gave a qualitative ex- planation of fingering as follows. Consider the displacement shown on Fig.2.1 and assume that the flow field, injected with a mean velocity U along the x direction, obeys Darcy’s law in one dimension:
dp dx =−µ
Ku+ρg (2.2)
The interface will be unstable if a small perturbation applied to it is growing which occurs if the driving pressure across the perturbation ∆xis positive i.e.
if (Homsy, 1987)
p1−p2=−µ1−µ2
K
U∆x+g(ρ1−ρ2)∆x >0 (2.3)
where the subscripts 1 and 2 refer to the displacing and displaced fluids respec- tively. If the density of the two fluids across the interface is the same, fingering then occurs if µ1 < µ2 i.e. if a less viscous fluid displaces a more viscous one leading to viscous fingering. For fluids of same viscosity, instability of the in- terface takes place whenρ1> ρ2 which corresponds to a heavier solution lying on top of a lighter one in the gravity field. The corresponding instability is called Rayleigh-Taylor instability or also density fingering. If both viscosities and densities are different across the interface, there exists a critical velocity
Uc=(ρ2−ρ1)gK
(µ1−µ2) (2.4)
across which the instability sets in if the two effects are competitive (Homsy, 1987). The two basic forces responsible for the instability are thus viscosity and/or buoyancy differences across the interface. Surface tension and diffusion (or more generally dispersion in porous media) are the stabilizing mechanisms that saturate the instability in immiscible and miscible fluids respectively as can be shown in more refined linear stability analysis (Chuokeet al., 1959; Saffman
& Taylor, 1958; Tan & Homsy, 1986; Homsy, 1987).
Imaging of viscous fingering in real porous media has been first provided by Slobod and Thomas (1963) and Perkins et al. (1965) using X-ray studies.
Magnetic resonance imaging has also been used for experimental visualization of fingering in real porous media and in chromatographic columns (Davieset al., 1992; Planteet al., 1994; Fernandezet al., 1995). Acoustic techniques have fur- thermore allowed tracking of the properties of fingering in packed bed columns (Bacri et al., 1991, 1992; Tchelepi et al., 1993). Analysis of fingering in real porous media has received particular interest in the petroleum engineering lit- erature due to its important impact on the efficiency of oil recovery. It is however generally difficult to get detailed information about initial steps and later non- linear dynamics of fingering in such porous media which are by nature opaque.
Some experimental studies have used glass beads with the same refraction index as the displacing fluid to overcome this difficulty and obtain images of fingering in chromatographic columns (Broyleset al., 1998; Shallikeret al., 1999). Such works remain however an exception. Instead, physicists often resort to the use of Hele-Shaw cells for quantitative analysis of fingering phenomena. Let us now explain this experimental set-up and underline to what extent it can be used to model real porous media systems.
2.2 Hele-Shaw cells
Hele-Shaw cells consist in two parallel transparent plates (typically made of glass or plexiglas) of large extentLx×Lyseparated by a thin gap widthLz<< Lx, Ly
z
y
Ly
Figure 2.2: Schematic of a Hele-Shaw cell and axis definitions.
(see Fig.2.2). For viscous fingering studies, the system is filled by one fluid which is then displaced by another fluid injected either at a point source leading to radial displacement or along one side giving rise to linear displacement. For density fingering experiments, the heavier fluid is placed on top of a lighter one in a vertically oriented cell. The main advantages of Hele-Shaw cells are the easy optical tracking of the interface evolution inside the plates and a quasi- two dimensional dynamics in some conditions. Most experimental studies on fingering phenomena have been performed in such a geometry. As an example of viscous and density fingering experiments conducted in Hele-Shaw cells, see Figs.2.3 and 2.4 respectively.
The evolution equation for the gap-averaged flow velocity inside the Hele- Shaw set up can be correctly modeled by Darcy’s law in some limits. Consider indeed a Hele-Shaw cell filled with one given fluid. Inside the cavity, the velocity field obeys Navier-Stokes equation
∂v
∂t +v.∇v=−1
ρ ∇p+ν∇2v+g , (2.5) where v = (vx, vy, vz) is a three-dimensional velocity field and ν = µ/ρis the kinematic viscosity. If the gap-width is small enough, the fluid flow is quasi two-dimensional. The equation for this two-dimensional gap-averaged velocity is obtained by applying the following averaging operator to the Navier-Stokes equation (2.5):
h· · · i ≡ 1 2d
Z +d
−d
dz· · · , (2.6) where d = Lz/2. In order to derive an explicit equation, a profile along the z direction has to be assumed for the velocity. In the limit of thin gap, it is reasonable to assume a local laminar Poiseuille profile constrained by the no-slip conditionv= 0 atz=±d:
vx(x, y, z, t) = F(z)ux(x, y, t)
vy(x, y, z, t) = F(z)uy(x, y, t) (2.7) vz(x, y, z, t) = 0
where F(z) = 32(1−zd22) andu= (ux, uy) is the two-dimensional velocity field.
The amplitude ofF(z) has been chosen so thathFi= 1. With this assumption, the three-dimensional continuity equation∇.v= 0 reduces immediately to:
∂ux
∂x +∂uy
∂y = 0 (2.8)
which will be written as∇⊥.u= 0. Let us now consider the application of the op- erator (2.6) on the various terms appearing in the Navier-Stokes equation (2.5).
The simplest is the time derivative which yields for the first component vx: h∂vx
∂t i = ∂ux
∂t 1 2d
Z +d
−d
dzF(z)
= ∂ux
∂t . (2.9)
A similar result is readily obtained forvy. The non-linear term is treated simi- larly and yields for instance:
hvx
∂vx
∂xi = ux
∂ux
∂x 1 2d
Z +d
−d
dzF(z)2
= 6
5 ux
∂ux
∂x . (2.10)
∂u
∂t +6
5u.∇⊥ u=−1
ρ ∇⊥p+ν∇2⊥u−3ν
d2u+g . (2.12) The new friction term−3νd2uoriginates from the wall resistance imposed by the no-slip condition. The effect of the wall is indeed to decrease the velocity in the Hele-Shaw cell. We now consider the respective importance of the various terms.
First, assuming that the velocityuscales likeUand thatLis the length scale characterizing the velocity variations, the ratio between the nonlinear convective termu.∇⊥ uand the viscous termν∇2⊥uis given by the Reynolds number of the flow:
Re=U L
ν (2.13)
This number is expected to be very small for small gap width. This is also consistent with the fact that we have assumed a laminar Poiseuille profile along z for the velocity. The non-linear term is thus negligible in a Hele-Shaw cell.
Secondly, we consider the stabilizing friction term−3νd2uwhich can be re-written as follows
−3ν
d2u=−τf−1u , (2.14)
whereτf = d3ν2 is the characteristic friction time. This friction time scales like d2 and is thus assumed to be very small in the Hele-Shaw limit. Hence, the evolution of the velocity field is expected to be very rapid and consequently the velocity field immediately reaches the stationary solution and adjusts to the pressure gradient and the gravitation term. Such a quasi-static approximation, together with the low Reynolds number limit thus lead to
∇⊥p≈µ∇2⊥u−3µ
d2u+ρ g , (2.15)
whereµ=ρ ν. This equation is usually referred to as Brinkman’s approxima- tion. It gives a static differential equation between the two-dimensional velocity field in the Hele-Shaw cell and the pressure gradient. It can be further simpli- fied by noting that, in many systems, the characteristic length scaleLis much
Figure 2.3: Viscous fingering instability observed in the linear miscible displace- ment from left to right of a solution of glycerin by less viscous colored water in a Hele-Shaw cell. Length of the image: 20 cm. (Courtesy A.A. Vedernikov, B.
Scheid, E. Istasse, and J.-C. Legros (2001)).
larger than the gap-width a = 2d. Hence, the ratio between the viscous and the friction term, which scales like a/L is usually very small. In that limit, Brinkman’s approximation reduces to Darcy’s law, which reads:
∇⊥p≈ −µ
κu+ρ g , (2.16)
where κ=a2/12 is the permeability of the Hele-Shaw cell. Thus, we see that single-phase Hele-Shaw flow is analogous to two dimensional incompressible flow in porous media.
2.3 Viscous fingering
Viscous fingering is observed when a less viscous fluid is displacing a more viscous one in a porous medium (typically water displacing petroleum in un- derground reservoirs). If the two fluids are immiscible, the fingering is often called Saffman-Taylor instability following the work by P.G. Saffman and G.I.
Taylor (1958). Many studies have been devoted to analyze to what extent such viscous fingering is analog to other free-boundary problems that exhibit similar pattern formation as in directional solidification, diffusion limited aggregation or electrodeposition for instance. Introducing the specific mathematics of the Saffman-Taylor problem as well as its analogies with various applications is be- yond the scope of this work. Hence, such surface-tension dependent fingering will not be addressed here. In miscible fluids, viscous fingering is also driven by viscosity gradients which now result from spatial variations in the concentration
∂tc+u· ∇c=D∇2c (2.19) where the buoyancy hydrostatic term has been incorporated in the pressure gra- dient. The problem is governed by two dimensionless parameters. The first one is the Peclet numberP e=U L/D where U is the injection velocity, La char- acteristic length and D the dispersion or molecular diffusion coefficient. The second one is the log mobility ratio defined asR=ln[µ2/µ1]. Viscous fingering occurs as soon asR >0 for a non-homogeneous initial concentration distribu- tion. The higherRthe more unstable the system. As dispersion is the stabilizing mechanism, it is the Peclet number which is the main parameter fixing the scale of the pattern with increased stabilization and smaller wavenumbers whenP eis decreased. For fixedD, fingering is more active whenP eis large i.e. when the injection velocityU is large. These trends have been experimentally observed in experiments related to viscous fingering of miscible fluids in porous media (Hill, 1952; Perkins et al., 1965; Brock & Orr, Jr., 1991; Slobod & Thomas, 1963;
Bacriet al., 1991, 1992; Vedernikovet al., 2001).
Theoretically, time and space scales of the fingering pattern at onset can quantitatively be summarized by dispersion curves computed by the linear sta- bility analysis of the base state interface. Such dispersion curves give the growth rate of the perturbations as a function of their wavenumbers. The linear stabil- ity analysis of viscous fingering for miscible fluids is complicated by the fact that the base-state solution is time-dependent. Indeed, diffusion leads to a mixing of the two fluids in time and hence to a time-dependent viscosity gradient across the interface of growing thickness. A review of the stability analysis including diffusive widening of the zone of viscosity variation is given in Homsy (1987).
Let us here mention the first work by Chuoke et al. (1959) as well as the articles by Tan and Homsy (1986) and Yortsos and Zeybek (1988) that study miscible viscous fingering for both isotropic and anisotropic dispersion in rectilinear flow in unbounded domains.
In the course of time, diffusion or dispersion act to spread the viscosity profile leading thus to an overall stabilization of the system and a shift to longer
Figure 2.4: Rayleigh-Taylor instability between two layers of miscible fluids in a Hele-Shaw cell. The fluids consist in a heavier glycerin-water colored mixture lying above pure water. Length of the image: 8cm. (Courtesy of J. Fernandez, P. Kurowski , P. Petitjeans and E. Meiburg (2002)).
wavelength fingers. This leads thus to coarseningof the fingers. The nonlinear dynamics of the fingers is also characterized by spreading i.e. widening of the width of one finger by diffusion, and by shielding, i.e. the fact that one finger stops growing because its neighboring finger is growing more effectively and screens it. In addition, if a finger becomes very wide, its front can become steep because of shear flows. The front of the finger becomes then unstable again because of fingering andtip-splittingoccurs (Homsy, 1987, 1988; Tan & Homsy, 1988).
Both linear stability properties and nonlinear dynamics of viscous fingering have been shown theoretically to depend on anisotropic dispersion (Zimmer- man & Homsy, 1991, 1992b), velocity dependent dispersion (Yortsos & Zeybek, 1988; Zimmerman & Homsy, 1992b), effect of non-monotonic viscosity profiles (Manickam & Homsy, 1993, 1994, 1995) and of shearing velocities (Rogerson &
Meiburg, 1993b,a). In addition, several works have also focused on the influ- ence on fingering of permeability heterogeneities (Tan & Homsy, 1992; Tchelepi et al., 1993; Fernandezet al., 1995; Chen & Meiburg, 1998b; De Wit & Homsy, 1997a,b; Camhi et al., 2000) and of the non-Newtonian character of the flow (Azaiez & Singh, 2002). Finally let us note that some studies of viscous fin- gering have also been performed in three dimensions (Bacri et al., 1991, 1992;
Zimmerman & Homsy, 1992a; Christieet al., 1993; Tchelepi & Orr, Jr., 1994;
Lajeunesseet al., 1997; Riaz & Meiburg, 2003).
2.4 Density fingering
Rayleigh-Taylor instabilities occur whenever fluids of different density are sub- ject to acceleration in a direction opposite to that of the density gradient (Chan- drasekhar, 1961). Such a situation is encountered when a denser solution lies on top of a lighter one in the gravity field (see Fig.2.4). This instability is very
centration fieldc(x, y, t) is then the solution of a convection-diffusion equation.
If one assumes, as we do for the remain of this work, that the fluid flow in the Hele-Shaw cell is well described by Darcy’s law because no three-dimensional effects occur along the gap, then the equations governing the problem can be written as:
∇ ·u= 0, (2.20)
∇p=−µ
Ku+ρ(c)g, (2.21)
∂tc+u· ∇c=D∇2c. (2.22) In writing such equations, one resorts to the Boussinesq approximation (see below) assuming that density changes are so small that they affect only the buoyancy term ρg and that the fluid can still be considered as incompress- ible. The important dimensionless parameter of density-driven instabilities is the Rayleigh numberRa =U L/D where nowU is the characteristic velocity U = ∆ρgL2/ν which varies with the relative density difference ∆ρacross the interface,ν being the kinematic viscosity.
In Hele-Shaw cells, experiments on density-driven instabilities between mis- cible fluids of constant viscosity have been first performed by Wooding (1969) who has analyzed in particular the nonlinear regime. Wooding’s photographs clearly show that diffusion plays the same role for miscible fluids as surface ten- sion for immiscible ones in the sense that diffusion sets the initial length scale of the pattern. In the nonlinear regime, diffusion produces lateral spreading of fingers which can lead to tip-splitting when the width of the finger exceeds the characteristic wavelength of the fingering instability. Wooding has shown experimentally that the mean wavelength and amplitude of the fingers grow respectively as√
t and t. In addition, he has shown that tip-splittings appear only above a critical value of the Rayleigh number. More recently, Fernandez et al. (2002) have studied experimentally an analog system focusing on the linear regime. They have provided detailed information regarding the growth rates and
most unstable wavenumbers of the pattern as a function of a Rayleigh number Ra= ∆ρga3/νD constructed on the basis of the gap widthaof the Hele-Shaw cell. They have shown that for Ra≤10 the instability is essentially 2D while above that value, 3D effects come into play. Indeed, these twoRaranges feature different scaling laws for the dependence of the wavelength of the pattern as a function ofRa. In particular, for lowerRa, the wavelength is inversely propor- tional toRa while the high Raregime is characterized by an instability across the gap and a wavelength of about 5 times the gapwidth. This shows that the Hele-Shaw averaged equations such as Darcy or Brinkman equations must then be replaced by full 3D Navier-Stokes equation in order to fully describe the prob- lem. Comparison with linear stability analysis based on 2D Brinkman equation (Fernandezet al., 2001, 2002) and on the full 3D Stokes equation (Grafet al., 2002) confirm this result. In this framework, numerous studies have provided evidences of the limit of applicability of the 2D averaged Hele-Shaw equations versus a full 3D description for instabilities of miscible fluids (see for instance Lajeunesseet al. (1997, 1999, 2001); Petitjeanset al. (1999); Chen & Meiburg (1998a); Martin et al. (2001); Fernandez et al. (2001); Grafet al. (2002) and references therein). The exact stability parameters above which the Hele-Shaw equations are no longer applicable are at present not clearly defined because the switch from 2D to 3D dynamics depends on viscosity ratio and the nature of the two fluids, on diffusion and on the characteristic mixing time for instance. In that context, we will assume throughout this work that the Hele-Shaw cells are always thin enough (without however expliciting this condition quantitatively) so that Darcy’s law is applicable. This point of view will allow us to keep the hydrodynamic conditions constant, and use the above model (2.20-2.22) to focus on the specific role of chemistry on fingering instabilities.
To end this short survey, let us mention that the situation where both vis- cosity and density variations are simultaneously operative has been addressed by several authors (Bacriet al., 1992; Rogerson & Meiburg, 1993a,b; Manickam
& Homsy, 1995; Loggiaet al., 1995; Lajeunesseet al., 1997; Ruith & Meiburg, 2000; Riaz & Meiburg, 2003; Goyal & Meiburg, 2004).
2.5 Boussinesq approximation
Concentration and/or temperature gradients are at the origin of the fluid mo- tions considered here because they induce a gradient of density or viscosity. The theoretical description of fingering phenomena benefits from the simplification due to the Boussinesq approximation (Vidal et al., 1994; Turner, 1973). This approximation introduced by Boussinesq (1903) consists in neglecting variations of density in all terms of the equations giving the velocity field evolution except
solutal and thermal density changes. While solutal expansion coefficients are not measured systematically for the chemical species we will be dealing with, it is however known that ∆ρstypically amounts to values of the order of 10−4 for the redox reactions we will consider. For water, the thermal expansion coefficientβ = 2.57 10−4 K−1 so that for temperature changes of the order of 2 to 3 K as involved in the chemical fronts we will address (Edwards et al., 1991), the relative thermal density jump is also of the order of 10−4. For such temperature variations, it is reasonable to assume that the viscosity of the fluids, diffusion coefficients of the chemical species and kinetic constants do not depend on the temperature. As a consequence, Boussinesq approximation consists then in retaining buoyancy effects only in the ρg term of the flow equation. In particular, the fluid is still considered incompressible (∇.u = 0) as the local pressure variations due to the slight relative density changes are negligible. This means that in the continuity condition
∂ρ
∂t +u.∇ρ=−ρ∇.u (2.24)
we consider that the relative fluctuations of the densityδρ/ρo are much smaller than the relative fluctuations of the velocity|δu|/|u|. This approximation does not hold for extreme conditions as sound or shock waves for instance. The fingering of chemical fronts that, as said above, will be studied here typically involves relative solutal and thermal density jumps of the order of 10−4 while macroscopic flow movements triggered by this slight density changes are ob- served. Hence, it is legitimate to assume the fluid as being incompressible and to use Boussineq’s approximation throughout the remainder of the work.
Having recalled the general properties of fingering phenomena and the con- ditions of validity of Darcy’s law and of Boussinesq approximation, let us now turn to the analysis of the interplay between fingering and autocatalytic reac- tions providing chemical fronts.
Density fingering of
isothermal iodate-arsenous acid reaction fronts
Chemical reactions can interact with hydrodynamic fingering instabilities and affect the stability properties as well as the nonlinear spatiotemporal dynamics of the system. Let us first review the literature on this subject.
3.1 Fingering in reactive systems
Several works have focused on the interplay between viscous fingering and chem- ical reactions in porous media or Hele-Shaw cells. The first detailed study of chemically induced viscous fingering was made by Ortoleva and coworkers (Or- toleva, 1994) in a body of literature involving so-called “reactive infiltration instabilities”. In this instability, the invading fluid reacts with the solid ma- trix of the porous media, leading to chemical dissolution and an increase in porosity. As a result of the further coupling between the porosity and the flow resistance (permeability), the dissolution of matrix material leads to an increase in the mobility of the fluid in those regions. The resulting convection of reac- tant through these higher mobility regions results in further dissolution and a positive feedback, leading to instability. The dispersion of reactant away from such regions in which the flow focuses is the only factor mitigating against the creation of clear fluid channels, and if dispersion is sufficiently strong, it cre- ates the possibility of the equilibration of travelling wave patterns of porosity.
Linear stability studies have been performed which show the usual features of instability at long wavelength and a cut-off at small scales due to dispersion (Chadamet al., 1986; Crompton & Grindrod, 1996). Further linear analyses have taken the change in viscosity as a function of reactant concentration into
account, providing an understanding of the coupling between the viscous and reaction driven instabilities (Chadam et al., 1991). In general, they enhance each other when the dissolution results in an increase in viscosity. Weakly nonlinear analyses have considered the issue of equilibration of the patterns (Crompton & Grindrod, 1996; Chadamet al., 1988). Numerical solutions have shown equilibration (Crompton & Grindrod, 1996) as well as dynamically un- stable nonlinear states which exhibit secondary instabilities and tip-splitting in a fashion analogous to viscous fingering (Meiburg & Homsy, 1988b).
Fingering is also relevant in chromatography, in which the viscosity and density of the solution being separated is a function of concentration and hence of the degree of adsorption of different components of the mixture. As shown in a series of visualization and tracer experiments, preferential adsorption of the more viscous components leads to a viscous driven instability, deleterious mixing of the chemical fronts, and long tails in the elution profiles (Czok et al., 1991;
Plante et al., 1994; Fernandezet al., 1995, 1996; Dickson et al., 1997; Broyles et al., 1998; Shallikeret al., 1999). Simulations which include sample properties have provided further quantitative understanding (Norton & Fernandez, 1996).
Fixed bed regeneration is the context in which fingering was first analyzed (see Homsy (1987) for a historical note as well as Hill (1952)). In most treat- ments the chemistry is considered slow relative to any physical processes and the focus remains on the fingering of pure non reactive solutions (Tchelepiet al., 1993; Tan & Homsy, 1988; Zimmerman & Homsy, 1992b). Another application where chemical reactions induce possible unstable viscosity gradients is frontal polymerization (Pojmanet al., 1996, 1998). This polymerization technique con- verts monomers into polymer through a localized reaction zone that propagates.
Viscosity and/or density gradients caused by thermal and solutal gradients in the reaction zone may lead to fingering phenomena that can affect the front shape and velocity.
In Hele-Shaw cells, the influence of chemistry on immiscible viscous fingering patterns has experimentally been strikingly evidenced by Hornof and co-workers (Nasr-El-Dinet al., 1990; Hornof & Bernard, 1992; Hornofet al., 1995; Hornof
& Baig, 1995; Hornof et al., 2000) and more recently by Sastry et al. (2001) and by Fernandez and Homsy (2003). In the latter case, surface tension effects are dominant.
For miscible solutions, experiments on viscous fingering in reactive systems have addressed the effect of the variation of reactant concentrations and of the finger pattern on the spatial distribution of chemical species showing that the flow can drastically influence the reaction (Nagatsu & Ueda, 2001, 2003).
However, in these systems, the chemical reaction does not feed back on the
solution and the polymer matrix can have quite strong differences in viscosity (Pojmanet al., 1996).
Chemical reactions are more prone to provide density differences that can drive buoyantly unstable situations as soon as a heavier fluid lies on top of a lighter one. In the chemical and pharmaceutical industries, separation tech- niques sometimes involve two-layers systems in which two immiscible fluids each containing separate reactants are brought into contact. When one chemical species diffuses through the interface, chemical reactions start in the bulk of the lower phase. If the reactants and products are such that an unstable density stratification sets in, density fingering instabilities occur leading to fingers de- taching from the interface and invading the bulk of the lower phase (Eckert &
Grahn, 1999; Eckert et al., 2004). Theoretical analysis of such configurations (Bratsun & De Wit, 2004) indicates that the resulting hydrodynamic motions can increase the mass transfer rate of the chemicals showing that fingering can here have a positive influence on the yield of the separation process.
A large body of literature on density fingering induced by chemical reactions is devoted to the case of interest in this thesis i.e. that of Rayleigh-Taylor in- stability of autocatalytic fronts. In this case, the coupling between nonlinear chemical kinetics and molecular diffusion provides the interface between two different miscible zones of the fluid i.e. the reactants and the products respec- tively in front and behind the interface. If the molar volume of reactants and products is different, a density difference across the front may be at the origin of a hydrodynamic instability of the front in miscible solutions. Experimental evidences (Nagyp´alet al., 1986; Pojmanet al., 1991a; Nagyet al., 1994; Masere et al., 1994; Chinake & Simoyi, 1994; Carey et al., 1996; B¨ockmann & M¨uller, 2000; Martinet al., 2001; Horv´athet al., 2002; B¨ockmannet al., 2003; B´ans´agi Jr. et al., 2003a,b, 2004) and theoretical studies (Pojman & Epstein, 1990; Ed- wardset al., 1991; Vasquezet al., 1993; Huanget al., 1993; Zhang et al., 1995;
Huang & Edwards, 1996; Vasquezet al., 1996; Vasquez & Lengacher, 1998; De
Wit, 2001; Martin et al., 2001, 2002b; Yang et al., 2002; Demuth & Meiburg, 2003; B´ans´agi Jr. et al., 2003a; De Wit, 2004) on density driven instabilities of autocatalytic fronts have shown that the coupling between buoyancy driven flows and chemical reactions can strongly affect the properties of the fingering instability. The coupling with chemical reactions has been addressed mainly for the study of the stability of planar autocatalytic fronts with regard to Rayleigh- Taylor fingering. A prototype system in that respect is the iodate-arsenous acid (IAA) reaction. In this redox reaction involving the oxidation of arsenous acid by iodate, traveling fronts between fresh reactants and products can eas- ily be triggered in capillary tube, Hele-Shaw cells or porous media. The main advantage of the IAA system is that a simple one-variable model describing the evolution of iodide concentration is sufficient to quantitatively account for experimental observations for the concentration ranges considered (Merkin &
Sevˇc´ıkov´a, 1999). As the density of the solution is decreasing in the course ofˇ reaction, ascending fronts are buoyantly unstable in the gravity field as they correspond to heavier reactants lying on top of lighter products. Experiments in capillary tubes (Nagyp´al et al., 1986; Pojman et al., 1991a; Masere et al., 1994) have shown that curved ascending fronts traveling with a speed higher than that of descending stable planar fronts are then observed. Theoretical work for such narrow geometry provide the critical radius of the tube above which the hydrodynamic instability can develop and a discussion on whether axi or non axisymmetric modes should be observed at onset (Edwards et al., 1991; Vasquez et al., 1993; Huang et al., 1993; Vasquez et al., 1996; Huang &
Edwards, 1996; Vasquez & Lengacher, 1998).
More recently, experiments with the IAA reaction in Hele-Shaw cells have shown that several fingers can develop in this geometry where the front is later- ally extended (Nagyet al., 1994; Careyet al., 1996; B¨ockmann & M¨uller, 2000;
Martin et al., 2001; B¨ockmannet al., 2003). Carey et al. (1996) were the first to show that density fingering of IAA fronts lead to cellular patterns with a well defined wavelength and amplitude. Dispersion curves providing the growth rate of the IAA fingers as a function of their wavenumber have then been measured experimentally for the first time by B¨ockmann and M¨uller (2000). These disper- sion curves characterizing the early stages of the instability are well reproduced theoretically by linear stability studies of models coupling the evolution equa- tion for the flow to that of the concentration of the species ruling the density of the solution (De Wit, 2001; Martin et al., 2001, 2002b; Demuth & Meiburg, 2003). Little is known however on the long time dynamics of the fingers for such laterally extended Hele-Shaw cells. Experimentally, merging, fading, shielding and tip splitting of fingers are observed (B¨ockmannet al., 2003) but no detailed quantitative characterization of this nonlinear dynamics has been provided yet.
chemical reactions can change such nonlinear properties of the fingers.
In this framework, let us analyze here to what extent the density fingering of an interface between miscible fluids can be affected in spatially extended systems by a simple front generating reaction such as the redox IAA reaction. The reaction-diffusion IAA system can produce fronts between the fresh reactants and the products that travel at constant speedvand fixed widthw. We will show here that a density difference across such a chemical front dramatically modifies the stability properties and nonlinear dynamics of the hydrodynamic fingering instability. To do so, we first introduce the recent experimental results obtained in Hele-Shaw cells by the group of S.C. M¨uller in Magdeburg on this system. We next introduce the one-variable cubic model that accounts quantitatively for the properties of this reaction in some limits and recall the derivation of the front solution. We then perform a linear stability analysis of this front by considering the reaction-diffusion system coupled through an advection term to Darcy’s law describing the flow in the Hele-Shaw cells. Eventually we describe the nonlinear dynamics of fingering of IAA fronts obtained by numerical integration of the model.
3.2 The iodate-arsenous acid (IAA) system
The iodate-arsenous acid reaction is a redox reaction which exhibits a clock reaction behavior in batch reactors and bistability in tank reactors (De Kepper et al., 1981; Papsinet al., 1981). The details of the elementary kinetics are com- plex but the overall dynamics can be represented thanks to two main processes with empirical rate laws (Hannaet al., 1982; Saul & Showalter, 1985; Showalter, 1987). The first important step is the so-called Dushman reaction consisting in the reduction of iodate by iodide to give iodine:
IO−3 + 5I−+ 6H+→3I2+ 3H2O (3.1)
