Fingering dynamics driven by a precipitation reaction: Nonlinear simulations

A fingering instability can develop at the interface between two fluids when the more mobile fluid is injected into the less-mobile one. For example, viscous fingering appears when a less viscous (i.e., more mobile) fluid displaces a more viscous (and hence less mobile) one in a porous medium. Fingering can also be due to a local change in mobility arising when a precipitation reaction locally decreases the permeability. We numerically analyze the properties of the related precipitation fingering patterns occurring when an $A+B\to C$ chemical reaction takes place, where $A$ and $B$ are reactants in solution and $C$ is a solid product. We show that, similarly to reactive viscous fingering patterns, the precipitation fingering structures differ depending on whether $A$ invades $B$ or vice versa. This asymmetry can be related to underlying asymmetric concentration profiles developing when diffusion coefficients or initial concentrations of the reactants differ. In contrast to reactive viscous fingering, however, precipitation fingering patterns appear at shorter time scales than viscous fingers because the solid product $C$ has a diffusivity tending to zero which destabilizes the displacement. Moreover, contrary to reactive viscous fingering, the system is more unstable with regard to precipitation fingering when the high-concentrated solution is injected into the low-concentrated one or when the faster diffusing reactant displaces the slower diffusing one.

Figure 1

Schematic diagram of a two-dimensional porous medium of dimensions $L_x\times L_y$ with permeability ${\kappa }_0$ in which a solution of reactant $B$ sandwiched between the solutions of reactant $A$ is displaced from left to right at a constant speed $U$. $x_l$ and $x_r$ are the initial positions of the left and the right miscible interfaces, respectively.

Figure 2

Concentration of the product $C$ (first row) and reaction rate $\mathcal{R}$ (second row) for ${\delta }_c=0.01,{\delta }_b=1,\varphi =1,R=2$, and $D_a=1$, at $t=20$, 40, 60, 80, and 100 (from left to right).

Figure 3

Reaction-diffusion profiles for (a) concentrations and (b) permeability for three values of ${\delta }_c$: 0.01 (thickest blue line), 0.1 (thicker green line), and 0.2 (thick red line). Here the dashed, dash-dotted, and solid lines represent $A,B$, and $C$, respectively. The initial location at $x=x_l=x_r=x_0$ corresponds here to the black dotted line at origin. The other parameters are the same as those of Fig. 2.

Figure 4

Comparison of the concentrations of the product for decreasing values of ${\delta }_c$ at $t=80$, 90, 100, and 110. The other parameters are the same as in Fig. 2.

Figure 5

Transverse averaged (a) concentration profiles of $A$ (dashed line), $B$ (dash-dotted line), and $C$ (solid line), and (b) reaction rate profiles for the simulation of Fig. 2 at $t=0$, 20, 40, 60, 80, 100. Here $x_0$ is the initial location of the left interface $x_l$.

Figure 6

Temporal variation of the mixing length $L$ for various values of ${\delta }_c=0.01$ (solid line), 0.05 (dashed line), 0.1 (dash-dotted), 0.15 (dotted), and 0.2 (solid line with plus symbol). The inset shows the variation of $L$ with ${\delta }_c$ for two values of $t=10$ and 20. Other parameters are the same as in Fig. 2.

Figure 7

Temporal variation of the center of mass for the (a) product $m_c$ and (b) reaction rate $m_R$ for various values of ${\delta }_c$. The insets of panel (a) and panel (b) show the corresponding variation of total concentration of the product, $C_{\mathrm{tot}}$, and reaction rate, ${\mathcal{R}}_{\mathrm{tot}}$, respectively. Parameters and line conventions are the same as those of Fig. 6.

Figure 8

Asymmetric fingering: Comparison of the product concentration for various values of $\varphi$: (a) 1.0, (b) 1.5, (c) 2.0, and (d) 2.5 at time $t=40,50,60$, and 70 (from top to bottom). Other parameters are the same as Fig. 2.

Figure 9

Asymptotic RD profiles for (a) concentrations and (b) permeability for $\varphi$: 1.5 (thickest blue line), 2.0 (thicker green line), and 2.5 (thick red line). The dashed, dash-dotted, and solid lines represent the concentrations of $A,B$, and $C$, respectively. The initial location at $x=x_l=x_r=x_0$ corresponds here to the black dotted line at origin. The other parameters are the same as those of Fig. 8.

Figure 10

Temporal variation of the mixing length for various values of $\varphi$: 1.0 (solid line), 1.5 (square), 2.0 (star), and 2.5 (circle). All solid and dashed lines represent the left and right reactive zones, respectively. The other parameters are the same as those of Fig. 2.

Figure 11

Temporal variation of (a) $m_c$ (main panel) and $C_{\mathrm{tot}}$ (inset) and (b) $m_R$ (main panel) and ${\mathcal{R}}_{\mathrm{tot}}$ (inset). Other parameters along with convention of lines and symbols are the same as Fig. 10.

Figure 12

Asymmetric fingering: Concentration of the product at $t=40,60,80$, and 100 (from top to bottom) for ${\delta }_b$: (a) 0.25, (b) 0.5, (c) 0.8, and (d) 1.0. Other parameters are the same as in Fig. 2.

Figure 13

Asymptotic transverse averaged RD profiles for (a) concentrations and (b) permeability for ${\delta }_b$: 0.25 (thickest blue line), 0.5 (thicker green line), and 0.8 (thick red line). The dashed, dash-dotted, and solid lines represent the concentrations of $A,B$, and $C$, respectively. The initial location at $x=x_l=x_r=x_0$ corresponds here to the black dotted line at origin. The other parameters are the same as those of Fig. 12.

Figure 14

Variation of the mixing length with time for ${\delta }_b$: 0.25 (square), 0.5 (star), and 0.8 (circle). The solid line without symbol represents ${\delta }_b=1$. All solid and dashed lines represent left and right interfaces, respectively. Other parameters are the same as Fig. 2.

Figure 15

Temporal variation of (a) $m_c$ (main panel) and $C_{\mathrm{tot}}$ (inset) and (b) $m_R$ (main panel) and ${\mathcal{R}}_{\mathrm{tot}}$ (inset). Other parameters along with convention of lines and symbols are the same as those of Fig. 14.