Control of chemically driven convective dissolution by differential diffusion effects

Chemically driven convective dissolution can occur upon reaction of a dissolving species in a host phase when the chemical reaction destabilizes an otherwise stable density stratification. An $\mathrm{A}+\mathrm{B}\to \mathrm{C}$ reaction is known to trigger such convection when, upon dissolution into the host solution, A reacts with B present in the solution to produce a sufficiently denser product C. We study numerically the effect of differential diffusion on such a chemically driven convective dynamics. We show that below the reaction front either double-diffusive or diffusive-layer convection can arise, modifying the local Rayleigh-Taylor instability. When B diffuses faster than C, the density profile contains a local maximum at the reaction front, followed by a local minimum below it. A double-diffusive instability can develop below the reaction front, accelerating the convective dynamics and thereby enhancing the dissolution rate of A into the host phase. Conversely, when B diffuses slower than C, the density profile exhibits a local maximum below the reaction front and diffusive-layer convection modes stabilize the dynamics compared to the equal diffusivity case. When B and C diffuse at equal rates but faster than A, the convective dynamics is accelerated with respect to the equal diffusivity case.

Figure 1

Classification of the RD density profiles $\rho \left(z\right)$ in the ($R_B/R_C,{\delta }_B/{\delta }_C$) space for $R_A=-1$ and $R_C>0$ with horizontal dashed black lines on the density profiles representing the location of the reaction front: regimes NM (nonmonotonic with maximum at the reaction front), M (monotonically increasing), RT-DD indicated by the red region (maximum at the reaction front followed by a minimum below), and RT-DLC indicated by the green region (maximum below the reaction front). Adapted from Ref. [6]. The equal diffusivity cases [9] with C denser than B (indicated by the blue solid line) and C less dense than B (indicated by the pink solid line) are similar to regimes NM and M, respectively. Note that the profiles are sketched for $\beta =1$ but a similar classification can be made for different $\beta$ as well.

Figure 2

Density $\rho$ fields at times $t=5000,10000,20000,\text{and}30000$ from left to right in the RT-DD regime indicated in red in Fig. 1 with ${\delta }_C=1,{\delta }_B=3,R_B=1$ for $R_C=1$ (top), $R_C=1.5$ (middle), and $R_C=2$ (bottom). Typical 1D RD density profiles are enclosed in the first panel. Density scales between its minimum (blue) and maximum (red) values in each line.

Figure 3

Concentration fields of A, B, and C and the distribution of the reaction rate AB corresponding to the density field shown in the top line of Fig. 2 at $t=30000$. Concentration and reaction rate scale between their minimum (blue) and maximum (red) values.

Figure 4

Temporal evolution of the dissolution flux $J$ for different ${\delta }_B,{\delta }_C$ indicated in the inset for the destabilizing RT-DD cases with ${\delta }_B/{\delta }_C=3$, $R_B=1$, and (a) $R_C=1.5$; (b) $R_C=2$. Solid black curves correspond to the cases with equal diffusion coefficients.

Figure 5

Space-time plots of the density computed at $z=128$ below the interface for $R_B=1$ with $R_C=1.5$ (left) and $R_C=2$ (right). Destabilizing RT-DD cases with ${\delta }_B/{\delta }_C=3$ (top) and stabilizing RT-DLC cases with ${\delta }_B/{\delta }_C=1/3$ (bottom) are compared to the equal diffusivity ${\delta }_B={\delta }_C=1$ cases (middle).

Figure 6

Density $\rho$ fields at times $t=5000,10000,20000,\text{and}30000$ from left to right in the RT-DLC regime indicated in green in Fig. 1 with ${\delta }_C=1,{\delta }_B=1/3,R_B=1$ for $R_C=1.5$ (top) and $R_C=2$ (bottom). Typical 1D RD density profiles are enclosed in the first panel. Density scales between its minimum (blue) and maximum (red) values in each line.

Figure 7

Temporal evolution of the dissolution flux $J$ for different diffusivities indicated in the inset for the stabilizing RT-DLC cases with ${\delta }_B/{\delta }_C=1/3$, $R_B=1$, and (a) $R_C=1.5$; (b) $R_C=2$. Solid black curves correspond to the cases with equal diffusion coefficients.

Figure 8

RD density profiles for different diffusion coefficients (${\delta }_B$, ${\delta }_C$) indicated in the inset for the stabilizing RT-DLC cases ${\delta }_B/{\delta }_C=1/3$ with $R_B=1$ and (left) $R_C=1.5$; (right) $R_C=2$.

Figure 9

RD density profiles for different diffusion coefficients (${\delta }_B$, ${\delta }_C$) indicated in the inset for ${\delta }_B/{\delta }_C=1$ with $R_B=1$ for $R_C=1.5$ (solid curves) and $R_C=2$ (dashed curves).

Figure 10

Temporal evolution of the dissolution flux $J$ when B and C diffuse at equal rates ${\delta }_B/{\delta }_C=1$ for different values of the diffusivities indicated in the inset with $R_B=1$ and (a) $R_C=1.5$; (b) $R_C=2$.

Figure 11

Onset time $t_0$ for different diffusivity ratios ${\delta }_B/{\delta }_C$ and $R_B=1$ when (a) $R_C=1.5$ and (b) $R_C=2$. Dashed lines represent the equal diffusivity cases. For a given ratio ${\delta }_B/{\delta }_C$, various combinations of absolute values of diffusion coefficients (${\delta }_B$, ${\delta }_C$) are used. The letters S and D refer to stabilizing RT-DLC and destabilizing RT-DD cases, respectively. NM stands for nonmonotonic as in Fig. 1.

Figure 12

Asymptotic flux $J^{*}$ for different diffusivity ratios ${\delta }_B/{\delta }_C$ and $R_B=1$ when (a) $R_C=1.5$ and (b) $R_C=2$. Same conventions are used as in Fig. 11.