Effect of the irreversible $\text{A}+\text{B}\to \text{C}$ reaction on the onset and the growth of the buoyancy-driven instability in a porous medium: Asymptotic, linear, and nonlinear stability analyses

Taking different diffusivities into account, the effect of irreversible A+B →C reaction on the growth of a buoyancy-driven instability in a Hele-Shaw cell is analyzed theoretically. For the limiting cases of infinitely fast reaction, an asymptotic stability analysis is conducted based on base density profile. To confirm the asymptotic stability analysis, under the linear stability theory, new linear stability equations are derived and solved numerically. In addition, fully nonlinear numerical simulations are conducted using the Fourier spectral method. The present asymptotic and linear stability analyses and nonlinear numerical simulations are in good agreement, and they modify the previous general classification of stability. For some cases where a stable barrier is sandwiched by two unstable regions, we also conducted linear and nonlinear analyses. It is interesting that for a certain case, instabilities with different wavelengths are possible below and above a central stable barrier.

Figure 1

Schematic diagram of the system considered here.

Figure 2

Comparison of neutral stability curves obtained from analytic approximation without the QSSA and the present numerical shooting method with the QSSA for the limiting case of ${\delta }_{\mathrm{B}}={\delta }_{\mathrm{C}}=1$.

Figure 3

Temporal evolution of (a) total flux, (b) intensity of vorticity, and (c) vorticity field for the limiting case of ${\delta }_{\mathrm{B}}={\delta }_{\mathrm{C}}=1,r_{\mathrm{B}}=0.75,{\beta }_{\mathrm{B}}=0$, and ${\beta }_{\mathrm{C}}=1$. Here, ${\tau }_c$ is obtained from the linear stability analysis.

Figure 4

Comparison between the present QSSAζ and the conventional QSSAz (Trevelyan et al. [15]) for the nonreactive case, i.e., $\mathrm{Da}=0$. The previous numerical simulations (Trevelyan et al. [15]) are also compared.

Figure 5

Effect of ${\beta }_{\mathrm{B}}$ on the (a) density profile, (b) critical time, (c) temporal evolution of the intensity of vorticity, and (d) vorticity field, where numbers in the brackets correspond to $\left[{\omega }_{y,min},{\omega }_{y,max}\right]$, for the case of ${\zeta }_f=0$ and neutrally stable top layer.

Figure 6

Effect of ${\beta }_{\mathrm{C}}$ on the (a) density profile, (b) critical time, (c) temporal evolution of intensity of vorticity and (d) vorticity field, where numbers in the brackets correspond to $\left[{\omega }_{y,min},{\omega }_{y,max}\right]$, for the case of ${\zeta }_f=0$ and neutrally stable bottom layer.

Figure 7

Effect of ${\beta }_{\mathrm{B}}$ on the (a) density profile and (b) critical time, (c) temporal evolution of the intensity of the vorticity, and (d) vorticity field, where numbers in the brackets correspond to $\left[{\omega }_{y,min},{\omega }_{y,max}\right]$, for the case of ${\zeta }_f<0$, the reaction front moving upward.

Figure 8

Effect of ${\beta }_{\mathrm{B}}$ on the (a) density profile and (b) critical time, (c) temporal evolution of the intensity of the vorticity, and (d) vorticity field, where numbers in the brackets correspond to $\left[{\omega }_{y,min},{\omega }_{y,max}\right]$, for the case of ${\zeta }_f>0$, the reaction front moving downward.

Figure 9

Stability characteristics for the case of ${\delta }_{\mathrm{B}}=0.5,{\delta }_{\mathrm{C}}=2.0,r_{\mathrm{B}}=\sqrt{2},{\beta }_{\mathrm{B}}=1$, and ${\beta }_{\mathrm{C}}=2$. (a) Concentration and density profiles, (b) neutral stability curves, (c) normalized vertical velocity disturbance and (d) temporal evolution of vorticity field where $\left[{\omega }_{y,min},{\omega }_{y,max}\right]=\left[-5.2,5.2\right]\times {10}^{-3}$.

Figure 10

Stability characteristics for the case of ${\delta }_{\mathrm{B}}=0.5,{\delta }_{\mathrm{C}}=0.25,r_{\mathrm{B}}=2^{1/2},{\beta }_{\mathrm{B}}=1$, and ${\beta }_{\mathrm{C}}=2^{1/2}$. (a) Concentration and density profiles, (b) neutral stability curves, (c) normalized vertical velocity disturbance, and (d) temporal evolution of vorticity field where $\left[{\omega }_{y,min},{\omega }_{y,max}\right]=\left[-0.01,0.01\right]$.