Hydrodynamic instabilities of autocatalytic reaction fronts:

1. Hydrodynamic instabilities of autocatalytic reaction fronts: A. De Wit Unité de Chimie Physique non Linéaire Université Libre de Bruxelles, Belgium

2. Scientific questions • Can chemical reactions be at the originof hydrodynamic instabilities (and notmerely be passively advected by the flow) ? • What are the properties of the new patterns that can then arise ? • Influence on transport and yield of reaction ?

3. Outline I. Vertical set-ups • Convective deformation of chemical fronts • Experiments in Hele-Shaw cells • Model of hydrodynamic instabilities of fronts • Rayleigh-Bénard, Rayleigh-Taylor and double-diffusive instabilities • Reactive vs non-reactive system II. Horizontal set-ups • Marangoni vs buoyancy-driven flows

4. Buoyancy-driven instability of a chemical front in a vertical set-up Heavy A Light B Hydrodynamic Rayleigh-Taylor instability of autocatalytic IAA fronts ascending in the gravity field in a capillary tube because reactant A is heavier than the product B Heavy A: stable descending front Light B: unstable ascending front Bazsa and Epstein, 1985; Nagypal, Bazsa and Epstein, 1986 Pojman, Epstein, McManus and Showalter, 1991 (Courtesy D. Salin)

5. Model system: fingering of chemical fronts in Hele-Shaw cells Density of reactants different than density of products: ∆c=prod-react≠0 Fresh reactants Products Buoyantly unstablefronts Carey, Morris and Kolodner, PRE (1996)Böckmann and Müller, PRL (2000)Horvath et al. (2002)

6. Böckmann and Müller, PRL (2000) Horvàth et al., JCP (2002) Chlorite-tetrathionate (CT) reaction Isothermal system Density of products (top) larger than density of reactants (bottom) Drc>0 Iodate-arsenous acid (IAA) reaction Buoyantly unstableDESCENDING fronts Reactants heavier than productsDrc<0: Buoyantly unstableASCENDING fronts

7. J.A. Pojman and I.R. Epstein, “Convective effects on chemical waves: 1. Mechanisms and stability criteria”, J. Phys. Chem., 94, 4966 (1990) r= ro[1-aT(T-To)+S aci(Ci-Cio)] Across a chemical front: Dr=rproducts-rreactants=Drc+DrT Drc<0:reactants are heavier than products (IAA)Drc>0:products are heavier than reactants (CT) DrT<0:exothermic reaction, products are hotterthan reactants

8. Antagonist solutal and thermal effects Cooperative solutal andthermal effects

9. Questions • Which kind of hydrodynamic instabilities can the competition • or cooperative effects between solutal and thermal density effects generate across a chemical front ? • Are there new instabilities possible with regard to the • non reactive case ?

10. -r(C,T)g Theoretical model with Le=DT/D Rayleigh numbers Lewis number

11. Linear stability analysis of pure hydrodynamic instabilities F(C) = 0 Base state :linear concentration and temperature gradients with a corresponding density gradient

12. Hydrodynamic Rayleigh-Bénard instability Fluid heated from below RT>0 COLD HOT

13. Hydrodynamic Rayleigh-Taylor instability Rc>0 Heavy fluid on top of a light one Fernandez et al., J. Fluid Mech. (2002)

14. Double diffusive instability Salt fingers:Hot salty water lies over cold fresh water of a higher density. The stratification is kept gravitationally stable. The key to the instability is the fact that heat diffuses much more rapidly than salt (hence the term double-diffusion). A downward moving finger of warm saline water cools off via quick diffusion of heat, and therefore becomes more dense. This provides the downward buoyancy force that drives the finger. Similarly, an upward-moving finger gains heat from the surrounding fingers, becomes lighter, and rises. RT<0, Rc>0 Salt fingers: Instability even if light solution on top of a heavy solution (statistically stable density gradient) !

15. Pure double diffusion (without chemistry): Le=20 Heated from below UNSTABLE Thermal Rayleigh-Bénard OSCILLATING Solutal Rayleigh-Taylor Heavy at bottom Light at the bottom Salt fingers STABLE Cooled from below Turner

16. Chemical fronts F( C ) = - C (C-1) (C+d)

17. Base state for the linear stability analysis: reaction-diffusion fronts for both concentration and temperature, connecting the reactants where (C,T)=(0,0) to the products for which (C,T)=(1,1)and traveling at a speed v T profile is function of Le Hot products v F( C ) = - C (C-1) (C+d) Reactants at room temperature g

18. Convection with chemistry : Le = 1 Ascending Exothermic reaction IAA 1: light but cold on top of heavy but hot:unstable if sufficiently exothermic Thermal plumes2: heavy and cold on top of light and hot:always unstable 3: heavy but hot on top of light but cold: stable descending fronts if sufficiently exothermic CT Lighter reactants Heavier reactants CT IAA Descending

19. UNSTABLE Stable UNSTABLE

20. Instability due to thermal diffusion and chemistry Le>1 Descending exothermic front F(C1) < F(C2) T1>T2 C1,T1 Light and hot products g C1,T2 C2,T2 The little protrusion reaches rapid thermal equilibriumbut still reacts at a rate F(C1) smaller than the rate F(C2)of its surroundings. It gains thus less heat (the reactionbeing exothermic) and it thus continues to sink. Heavy and cold reactants

21. Properties of this instability • Because the region with F’(c) >0 is followed by a regionwith F’(c) <0 , the local instability is constrained by the region • of local stability. • This unusual instability magnifies with a larger negative RT • and larger Le since r(x) = -RT T -Rc C Light and hot g Heavy and cold

22. Rayleigh-Taylor (heavy over light) Stable New instability of light over heavy

23. Antagonist solutal and thermal effects: double diffusiveinstabilities Cooperative solutal andthermal effects: candidates for the new instability for descendingfronts

24. Conclusions and perspectives • Classification of the various hydrodynamic instabilities ofexothermic reaction-diffusion fronts in the (RT,Rc) plane • Double-diffusive instabilities of chemical fronts have some • differences with pure hydrodynamic DD instabilities: • Different base state • stability boundaries depend on the kinetics and on Le • different nonlinear dynamics: frozen fingers • Uncovering of a new instability due to the coupling betweenthermal diffusion and spatial variations in reaction rate: should be observed in families of exothermic reactions for which Drc and DrT are both negative

25. Take home message When chemical reactions are at the core of density gradients, the possible resulting hydrodynamic instabilities in the corresponding reaction-diffusion-convection system is not always the simple addition of the usual buoyancy related instabilities on a chemical pattern. New chemically-driven instabilities can arise ! References: J. D'Hernoncourt, A. Zebib and A. De Wit, Phys. Rev. Lett. 96, 154501 (2006). J. D'Hernoncourt, A. De Wit and A. Zebib, J. Fluid Mech. 576, 445-456 (2007). J. D'Hernoncourt, A. Zebib and A. De Wit, Chaos, 17, 013109 (2007).

26. Front in horizontal set-ups 1 0 ≠ 1

27. Equations where

28. 3. Boundary conditions and Marangoni boundary condition at the free surface : (4) g M > 0 : C with g M < 0 : C

29. Open surface with no buoyancy effects (Ra=0) Marangoni effects M = 0 : reaction-diffusion front M = 500 M = - 500

30. Asymptotic dynamics : focus on the deformed front surrounded by a stationary asymmetric convection roll M>0 M<0

31. Closed surface: no Marangoni effects (Ma=0) Buoyancy effects Ra = 0 : reaction-diffusion front Ra=100: rp  rr : products lighter go on top Ra= - 100: rp > rr : products heavier sink

32. Ra = 100 Ra = -100

33. Experiments in capillary tubes with the Bromate-Sulfite reaction : products heavier than reactants => Ra < 0 products reactants A. Keresztessy et al., Travelling Waves in the Iodate-Sulfite and Bromate-Sulfite Systems , J. Phys. Chem. 99, 5379-5384, 1995. Qualitative agreement between experiments and theoretical model Buoyancy effects: Comparison with experiments

34. Experiments in capillary tubes with the Iodate-Arsenous Acid reaction : dr/d[I-] = -1,7.10-2g/cm3M Numerical front velocities (10-3 cm/s) Vnum = 3.24 Vnum = 4.84 Vnum = 6.44 J. Pojman et al., Convective Effects on Chemical Waves, J. Phys. Chem. 95, 1299-1306, 1991. Quantitative agreement between experiments and theoretical model

35. Asymptotic mixing length Symmetric buoyancy effects Asymmetric Marangoni effects

36. Constant propagation speed Asymmetric Marangoni effects Symmetric buoyancy effects

37. Conclusions Marangoni effects: Asymmetry between the results for M > 0 and M < 0 Increase of the front deformation, the propagation speed and the convective motions with M and Lz Buoyancy effects : Symmetry between the results for Ra > 0 and Ra < 0 Increase of the front deformation, the propagation speed and the convective motions with Ra and Lz

38. References • Marangoni effects: • L. Rongy and A. De Wit, ``Steady Marangoni flow traveling with a • chemical front", J. Chem. Phys. 124, 164705 (2006). • - L. Rongy and A. De Wit, ``Marangoni flow around chemical fronts traveling • in thin solution layers: influence of the liquid depth", J. Eng. Math. 59, • 221-227 (2007). • Buoyancy effects: • L. Rongy, N. Goyal, E. Meiburg and A. De Wit, ``Buoyancy-driven convection • around chemical fronts traveling in covered horizontal solution layers", • J. Chem. Phys. 127, 114710 (2007).

39. 2008 Gordon conference“Oscillations and dynamic instabilities in chemical systems” July 13-18, 2008 Colby College, Waterville, USA Chair: Vice Chair: Anne De Wit Oliver Steinbock http://www.grc.org/programs.aspx?year=2008&program=oscillat