Buoyancy-Driven Chemohydrodynamic Patterns in A + B → Oscillator Two-Layer Stratifications

Abstract

Chemohydrodynamic patterns due to the interplay of buoyancy-driven instabilities and reaction-diffusion patterns are studied experimentally in a vertical quasi-two-dimensional reactor in which two solutions A and B containing separate reactants of the oscillating Belousov-Zhabotinsky system are placed in contact along a horizontal contact line where excitable or oscillating dynamics can develop. Different types of buoyancy-driven instabilities are selectively induced in the reactive zone depending on the initial density jump between the two layers, controlled here by the bromate salt concentration. Starting from a less dense solution above a denser one, two possible differential diffusion instabilities are triggered depending on whether the fast diffusing sulfuric acid is in the upper or lower solution. Specifically, when the solution containing malonic acid and sulfuric acid is stratified above the one containing the slow-diffusing bromate salt, a diffusive layer convection (DLC) instability is observed with localized convective rolls around the interface. In that case, the reaction-diffusion wave patterns remain localized above the initial contact line, scarcely affected by the flow. If, on the contrary, sulfuric acid diffuses upward because it is initially dissolved in the lower layer, then a double-diffusion (DD) convective mode develops. This triggers fingers across the interface that mix the reactants such that oscillatory dynamics and rippled waves develop throughout the whole reactor. If the denser solution is put on top of the other one, then a fast developing Rayleigh-Taylor (RT) instability induces fast mixing of all reactants such that classical reaction-diffusion waves develop later on in the convectively mixed solutions.

Figure 1

Sketch of the double-layer systems α + β → BZ where the acidic solution β is respectively on the top (upper panels, (a)) or on the bottom (lower panels, (b)). Panels (a) and (b) illustrate for both configurations the concentration profiles of the BZ reactants and intermediates forming in the mixing zone. Density profiles, shown on the right, can be reconstructed via eq 1 as a weighted combination of the species concentrations.

Figure 2

Sketch of the experimental setup. (a) Schematic of the Hele-Shaw (HS) cell and related exploded view showing the two glass plates (middle), a spacer (left), and a metal frame (right). (b) Basic features of a Schlieren optic setup: (from the left) point light source, collimator lens, the Hele-Shaw cell, collimator lens, cutoff, and collecting camera.

Figure 3

Classification of the main chemohydrodynamic scenarios in the parameter space spanned by the density ratio between the bottom and the top solution, R = ρ^bottom/ρ^top, and the diffusivity ratio, δ = D^bottom/D^top, defined using the sulfuric acid and bromate diffusivities. δ_α/β and δ_β/α identify the two diffusivity ratios characterizing the two reactant distributions considered. Different scenarios are accompanied by representative density profiles along the gravitational axis which can explain the observed dynamics. Scenarios obtained under excitable conditions across ζ₀ are identified by gray symbols, and those corresponding to oscillatory regimes are in red.

Figure 4

(a) Chemohydrodynamic scenario I develops when solution β3 ([MA] = 2.2 × 10^–3 M) overlies the denser solution α4 ([BrO₃^–] = 0.3 M). Fingering in the bottom layer is due to the local accumulation of fast-diffusing sulfuric acid below the initial contact line and to the autocatalytic transformation of ferroin into denser ferriin. The contribution of the reaction is shown by comparing the density profile resulting from simple species diffusion (in black) and in the presence of the reaction (in red). (b) Scenario II: combination of ascending and sinking convective patterns obtained when solution β3 is put on top of solution α2 ([BrO₃^–] = 0.15 M). Diffusive layer convection sets in due to differential diffusion between the counter diffusing bromate salt and sulfuric acid. In both scenarios, chemical waves forming in the top layer remain localized. In each line, the first panel shows the reconstructed density profiles. The next panels present snapshots of the cell at successive times. Real size = 2 cm × 3 cm. The horizontal lines in the first snapshots of the dynamics indicate the position of the initial contact line while the vertical dashed line in the second snapshot indicates where space–time plots are built.

Figure 5

Space–time maps of the dynamics of the BZ reaction in the double-layer configuration in the (a) absence and (b) presence of buoyancy-driven flows when the system is locally excitable in the solution contact area (real dimensions = 4.2 cm × 180 min). (b) Space–time plot describes the experiment shown in Figure 4a along the vertical dashed line shown in the second snapshot.

Figure 6

Scenario III: chemohydrodynamic patterns controlled by a double-diffusive (DD) instability quickly developing when the top solution α1 (with [NaBrO₃] = 0.1 M) is put on top of a denser solution β3 ([MA] = 2.2 × 10^–3 M). The first snapshots are made binary to obtain a better contrast. Real size = 2 cm × 3 cm. The horizontal line in the first snapshot of the dynamics indicates the position of the initial contact line.

Figure 7

Typical space–time maps (dimensions 5 cm × 220 min) of the dynamics of the BZ reaction in the double-layer system in a gel, without buoyancy-driven flows when the system is locally oscillatory at the initial contact line between the two solutions (see ref (36)). Decreasing the concentration of malonic acid increases the period of the waves developing in the contact area.

Figure 8

(a) Scenario IV: transient periodic fingering induced by the oscillatory kinetics in the reactive zone. Solution β7 is put on top of solution α4. (b) Scenario V showing three different convective instabilities concurrently at play and interacting with chemical waves in the isopycnal case. The dynamics are obtained with solution β8 ([MA] = 0.6 M) layered on top of α4. Waves no longer remain localized along the initial contact zone. Real size = 2 cm × 3 cm. The horizontal lines in the first snapshots of the dynamics indicate the position of the initial contact line.

Figure 9

(a) Scenario VI: dynamics observed for the isopycnal case when solution α4 ([BrO₃^–] = 0.3 M) is on top of solution β8 ([MA] = 0.6 M). A double-diffusion instability is responsible for the mixing of the reactants in the whole spatial domain. Waves propagate preferentially upward and are broken in segments which are correlated with the wavelength of the convective patterns after these have merged. (b) Scenario VII: deformation of ascending fronts when [MA] = 0.8 M (α4 on top of β9). Note that the stratification is initially statically buoyantly stable. As can be observed from the comparison between the density profile of the isopycnal case (in gray), the stabilization is due to a larger density gap between the top and the bottom solutions. Real size = 2 cm × 3 cm. The horizontal lines in the first snapshots of the dynamics indicate the position of the initial contact line.

Figure 10

Interplay of localized oscillatory kinetics with an initially unstable stratification. (a) Scenario VIII: propagation of ascending periodic fronts in an oscillatory medium mixed by a Rayleigh–Taylor instability (solution β9, [MA] = 0.8 M, on top of solution α4). Broken waves curl into the beginning of a spiral and further collide with and annihilate the surrounding waves. (b) Scenario IX: propagation of descending fronts after a Rayleigh–Taylor instability causes vigorous stirring through the whole spatial domain (solution β7, [MA] = 0.4 M, underlying solution α4). Real size = 2 cm × 3 cm. The horizontal lines in the first snapshots of the dynamics indicate the position of the initial contact line.