Reactive convective dissolution with differential diffusivities: Nonlinear simulations of onset times and asymptotic fluxes

Convection can develop upon dissolution of a given species A in a host phase when dissolution leads to a buoyantly unstable density stratification. If A reacts with a solute B present in the host solution according to a bimolecular $\mathrm{A}+\mathrm{B}\to \mathrm{C}$ reaction, convective dissolution can be enhanced or slowed down depending on the relative contribution to density of each chemical species. We study numerically the influence of differential diffusion on such reactive convective dissolution in the nonlinear regime. In particular we compute the temporal evolution of the dissolution flux, its asymptotic value and the onset time of convection as a function of the ratio of the diffusion coefficients. We find that, when B diffuses faster than C, the density profiles can exhibit a local minimum below the reaction front where a double-diffusive instability develops. This has a destabilizing effect and leads to enhanced mixing, earlier onset of convection, and increased asymptotic fluxes. On the other hand, when B diffuses slower than C, the density profiles can contain a local minimum at the reaction front followed by a local maximum below, which gives rise to two convection zones with a diffusive-layer convection instability occurring below the reaction front. The overall dynamics is stabilizing with delayed onset of convection and with smaller asymptotic fluxes. When B and C diffuse at an equal rate but differently from A, differential diffusion can accelerate or slow down convection.

Figure 1

Reaction-diffusion (RD) concentration profiles for species A, B, and C developing upon dissolution of A from above in a host phase containing B, followed by an $\mathrm{A}+\mathrm{B}\to \mathrm{C}$ reaction. The dashed line represents the location of the reaction front where C is produced by the reaction. The zone between the interface and the reaction front contains mainly A and C while only B is present far away in the bulk. Here $\beta =1$. The right part of the figure depicts schematically the different zones of composition below the interface with the possible instabilities.

Figure 2

Classification of the RD density profiles $\rho \left(z\right)$ in the ($R_B/R_C, {\delta }_B/{\delta }_C$) space for $R_A,R_C>0$. The dashed black lines on the density sketches represent the location of the reaction front. Regimes are R1 (monotonic decreasing), R2 (minimum at the reaction front), R3a, R3b (minimum below the reaction front), R4a, R4b (minimum at the reaction front followed by a maximum below). The equal diffusivity cases from Ref. [26] are indicated by the horizontal line with ${\delta }_B={\delta }_C=1$; those when C is denser or less dense than B, with density profiles similar to regimes R1 and R2, are indicated by the pink and blue lines, respectively. Adapted from Refs. [16, 24].

Figure 3

Density $\rho$ field at different times $t$ for $R_B=1,R_C=2,{\delta }_C=1$ with (top) ${\delta }_B=1$: in regime R1; (middle) ${\delta }_B=3$: RT-DD case in regime R3a; (bottom) ${\delta }_B=0.33$: RT-DLC case in regime R4a. Typical RD density profiles are shown in the left panels. Density scales between its minimum (blue) and maximum (red) values in each line. Note that the effect of differential diffusivities is stronger for $\beta =1$ [24].

Figure 4

Density $\rho$ field at different times $t$ for $R_B=2,R_C=1,{\delta }_C={\delta }_B=1$. A typical RD density profile is shown in the left panel. Density scales between $-1.5$ (blue) and 0 (red).

Figure 5

Density $\rho$ field at different times $t$ for $R_B=2,R_C=1,{\delta }_C=1$ with (top) ${\delta }_B=3$: RT-DD case in regime R3b; (middle) ${\delta }_B=3/2$: in regime R2; (bottom) ${\delta }_B=1/3$: RT-DLC case in regime R4b. Typical RD density profiles are shown in the left panels. Density scales between its minimum (blue) and maximum (red) values in each line.

Figure 6

Temporal evolution of the dissolution flux $J$ for C denser than B ($R_B/R_C=0.5$) and different diffusivities $({\delta }_B,{\delta }_C)$ indicated in the inset: (top) ${\delta }_B/{\delta }_C=3$: RT-DD case in regime R3a; and (bottom) ${\delta }_B/{\delta }_C=1/3$: RT-DLC case in regime R4a. Solid black curves correspond to the equal diffusivity case with ${\delta }_B={\delta }_C=1$ [26].

Figure 7

Temporal evolution of the dissolution flux $J$ for C less dense than B ($R_B/R_C=2$) and different diffusivities $({\delta }_B,{\delta }_C)$ indicated in the inset: (top) ${\delta }_B/{\delta }_C=3$: RT-DD case in regime R3b; and (bottom) ${\delta }_B/{\delta }_C=1/3$: RT-DLC case in regime R4b. Solid black curves correspond to the equal diffusivity case with ${\delta }_B={\delta }_C=1$ [26].

Figure 8

Temporal evolution of flux $J$ for different values of ${\delta }_B={\delta }_C$ indicated in the inset for (top) C denser than B with $R_B/R_C=0.5$ and (bottom) C less dense than B with $R_B/R_C=2$.

Figure 9

Onset time $t_0$ for different diffusivity ratios ${\delta }_B/{\delta }_C$ when (a) C is denser than B ($R_B/R_C=0.5$) and (b) C is less dense than B ($R_B/R_C=2$). Dashed lines represent the equal diffusion cases while the dotted lines correspond to the nonreactive (NR) case. For a given ratio of ${\delta }_B/{\delta }_C$, various combination of absolute values of diffusivities (${\delta }_B,{\delta }_C$) are used.

Figure 10

Asymptotic flux $J^{*}$ for different diffusivity ratios ${\delta }_B/{\delta }_C$ when (a) C is denser than B ($R_B/R_C=0.5$) and (b) C is less dense than B ($R_B/R_C=2$). Dashed lines represent the equal diffusion cases and dotted lines represent the nonreactive (NR) case. For a given ratio of ${\delta }_B/{\delta }_C$, various combination of absolute values of diffusivities (${\delta }_B,{\delta }_C$) are used.