We show that the dynamics of a spatially closed Friedmann -Robertson -Walker Universe conformally coupled to a real, free, massive scalar field, is chaotic, for large enough field amplitudes. We do so by proving that this system is integrable under the adiabatic approximation, but that the corresponding KAM tori break up when non adiabatic terms are considered. This finding is confirmed by numerical evaluation of the Lyapunov exponents associated with the system, among other criteria. Chaos sets strong limitations to our ability to predict the value of the field at the Big Crunch, from its given value at the Big Bang.PACS 04.20.Cv, 98.80.Bp, 03.20.+i

' INTRODUCTIONIn reactive systems, forced convection is an efficient way to mix reactants and hence increase the reaction rate. This is of particular importance in chemical engineering processes. Conversely, one can address the question: how can chemical reactions influence natural convection or even be at the very source of hydrodynamic motion? These issues are at the heart of numerous applications in combustion, 1,2 polymer processing, 3,4 extraction techniques, 5,6 microfluidic devices, 7À9 bioconvection, 10 traveling fronts, 11À13 and CO 2 sequestration, 14,15 to name a few.To answer such questions, experimental studies have for instance investigated chemically driven convective mixing and enhanced extraction from one phase to another, induced by reactions between reactants initially contained separately in immiscible solvents. 5,16À18 In that case, it has been shown that the flow around the interface and within the bulk solutions result from (i) the coupling between transfer of chemical species at the interface, (ii) changes by the reaction of the density of the solutions which can trigger buoyancy-driven convective motions, and (iii) reaction-induced Marangoni effects, that is, fluid motion generated by surface tension changes at the immiscible interface. The situation is therefore quite complex, and even if theoretical studies 19À21 provide some help in understanding the influence of the various parameters, there is a need to gain insight also into simpler situations where some of the various effects are isolated. In this regard, the use of miscible solvents removes the influence of both transfer rate and Marangoni effects and allows one to separately analyze the influence of purely buoyancy-driven convection.For such miscible solvents, it has been shown experimentally that putting in contact aqueous solutions of an acid and of a base in the gravity field allows one to observe a wealth of beautiful convective patterns and instabilities. 22À25 More specifically, ascending plumes can develop above the reaction front when a solution of hydrochloric acid is put on top of a denser miscible equimolar aqueous solution of sodium hydroxide. 23 The patterns are different in presence of a color indicator, 22 indicating that this species is not neutral to the convective dynamics. 24 In this context, it is of interest to analyze such miscible systems in which a simple acidÀbase reaction takes place to understand the various possible buoyancy-driven instabilities induced by the presence, in aqueous solutions, of the neutralization reaction H + + OH À f H 2 O. To do so, we study experimentally chemically driven convective motions arising when putting in contact an aqueous solution of a strong acid on top of a denser aqueous solution of a strong base in the gravity field. We explain the influence on the dynamics of changing the type of reactants used and their concentrations. In a first part, we vary the type of counterion in the basic solution at fixed concentrations. We next vary the ratio in concentrations between th...

We report the hydrodynamic instabilities found in a simple exothermic neutralization reaction. Although the heavier aqueous NaOH solution was put below the lighter layer of aqueous HCl solution, fingering at the interface in a Hele-Shaw cell was observed. The reaction front, which propagates downward, becomes buoyantly unstable in the gravity field. The mixing zone length and wave number depend on the reactant concentrations. The mixing zone length increases and the wave number decreases when the reactant concentrations decrease.

Buoyancy-driven hydrodynamic instabilities of acid-base fronts are studied both experimentally and theoretically in the case where an aqueous solution of a strong acid is put above a denser aqueous solution of a color indicator in the gravity field. The neutralization reaction between the acid and the color indicator as well as their differential diffusion modifies the initially stable density profile in the system and can trigger convective motions both above and below the initial contact line. The type of patterns observed as well as their wavelength and the speed of the reaction front are shown to depend on the value of the initial concentrations of the acid and of the color indicator and on their ratio. A reaction-diffusion model based on charge balances and ion pair mobility explains how the instability scenarios change when the concentration of the reactants are varied.

Density driven instabilities produced by CO2 (gas) dissolution in water containing a color indicator were studied in a Hele Shaw cell. The images were analyzed and instability patterns were characterized by mixing zone temporal evolution, dispersion curves, and the growth rate for different CO2 pressures and different color indicator concentrations. The results obtained from an exhaustive analysis of experimental data show that this system has a different behaviour in the linear regime of the instabilities (when the growth rate has a linear dependence with time), from the nonlinear regime at longer times. At short times using a color indicator to see the evolution of the pattern, the images show that the effects of both the color indicator and CO2 pressure are of the same order of magnitude: The growth rates are similar and the wave numbers are in the same range (0-30 cm(-1)) when the system is unstable. Although in the linear regime the dynamics is affected similarly by the presence of the indicator and CO2 pressure, in the nonlinear regime, the influence of the latter is clearly more pronounced than the effects of the color indicator.