Chemically Driven Hydrodynamic Instabilities

In the gravity field, density changes triggered by a kinetic scheme as simple as $A+B\to C$ can induce or affect buoyancy-driven instabilities at a horizontal interface between two solutions containing initially the scalars $A$ and $B$. On the basis of a general reaction-diffusion-convection model, we analyze to what extent the reaction can destabilize otherwise buoyantly stable density stratifications. We furthermore show that, even if the underlying nonreactive system is buoyantly unstable, the reaction breaks the symmetry of the developing patterns. This is demonstrated both numerically and experimentally on the specific example of a simple acid-base neutralization reaction.

Figure 1

2D density fields obtained by numerically solving (1) using $\mathrm{Br}=0$, $R_b={\delta }_b=\mathcal{D}=\varphi =1$ with (a) ${\delta }_c=1$, $R_c=2.5$, $t=3.3\times {10}^5$ and domain width $1.9\times {10}^4$; (b) ${\delta }_c=0.5$, $R_c=2$, $t={10}^5$ and domain width $8.2\times {10}^3$. The black line corresponds to the initial contact line.

Figure 2

Total dimensionless density profile $\rho (x)={\rho }_a+{\rho }_b+{\rho }_c$ at $t=3\times {10}^5$ where ${\rho }_{\xi }$ corresponds to the density contribution from species $\xi$. (a) Nonreactive case ($\mathcal{D}=0$, ${\delta }_b=0.3$, $R_b=2.22$). (b) Reactive case with experimental values. Shaded zones show the sources of instability, i.e., with unstable density stratifications.

Figure 3

(a) Interferometry picture of experiments using a 1 M HCl solution on top of a 1 M NaOH solution in a vertical Hele-Shaw cell of gap width 0.5 mm. The initial contact line position is indicated by the black segments at the left and right. The width of the field of view is 13 mm. (b) Corresponding interferometry figure reconstructed at the same times from numerical integration of system (1) for $\varphi =\mathcal{D}=1$, ${\delta }_b=0.61$, ${\delta }_c=0.5$, $R_b=2.22$, $R_c=2.17$ and $\mathrm{Br}=3.49\times {10}^4$.

Figure 4

Superposition of the interferometry figure and velocity map at time 85 s for the fingers of Fig. 3. The black arrow denotes the line of initial contact between solutions. The maximum velocity measured by PIV is $132\mu \mathrm{m}/\mathrm{s}$.

Figure 5

Experimental nondimensional wavelength (▵) and inverse growth rate (□) of the perturbations as a function of Br, and corresponding numerical results shown by the solid and dashed lines, respectively. The porous media limit (Darcy limit) is recovered for $\mathrm{Br}\to 0$ (dotted line).