Control of Rayleigh-Taylor instability onset time and convective velocity by differential diffusion effects

Fingering instabilities of a miscible interface between two fluids in a gravitational field can develop due to adverse density gradients as in the well-known Rayleigh-Taylor (RT) and double-diffusive (DD) instabilities. In the absence of differential diffusion, the mixing rate and the onset time of the RT instability developing when a denser solution of a given solute A overlies a less dense solution of a solute B are respectively proportional and inversely proportional to the initial density difference $\mathrm{\Delta }{\rho }_0$ between the two superposed layers. We show here both experimentally and theoretically for porous media flows that when the mechanisms of RT and DD instabilities are combined, the properties of the convective growth of the fingers are controlled by the dynamic density jump $\mathrm{\Delta }{\rho }_m$ of the nonmonotonic density profile induced by the differential diffusion effects. In particular, the onset time and mixing rate can be controlled by varying the ratio of the diffusion coefficients of the solutes.

Figure 1

(a) Temporal evolution of the dimensionless base state density profiles. At $t=0$, the dimensionless density difference is $\mathrm{\Delta }{\rho }_0$ with $\rho =1$ for $y<0$ and $\rho =R$ for $y>0$ (red line). Later on, the density difference $\mathrm{\Delta }{\rho }_m$ remains equal to $\mathrm{\Delta }{\rho }_0$ only if $\delta =1$ (green curves) while a nonmonotonic profile with extrema such that $\mathrm{\Delta }{\rho }_m>\mathrm{\Delta }{\rho }_0$ develops if $\delta >1$ (black curves). (b) Contours of constant $\mathrm{\Delta }{\rho }_m$ (solid lines) and constant $\mathrm{\Delta }{\rho }_0$ (dashed lines) in the $(R,\delta )$ plane.

Figure 2

Experimental snapshots of the RT instability of a denser sucrose solution overlying a less dense KCl solution ($\delta =3.67$) with equal initial density difference $\mathrm{\Delta }{\tilde{\rho }}_0=0.6\text{kg}/{\text{m}}^3$ but $\mathrm{\Delta }{\tilde{\rho }}_m$ equal to (a) $13.8\text{kg}/{\text{m}}^3$ (experiment 16) and (b) $1.3\text{kg}/{\text{m}}^3$ (experiment 13). Temporal evolution of the mixing length for variable concentrations and (c) $\mathrm{\Delta }{\tilde{\rho }}_0=0.6\text{kg}/{\text{m}}^3$ (blue: experiment 16; green: experiment 15; red: experiment 14; black: experiment 13); the onset time, denoted as ${\tau }_D$, is the time at which the initial diffusive regime is replaced by a regime where a linear increase in time of $L$ is observed; (d) $\mathrm{\Delta }{\tilde{\rho }}_m=3.7\text{kg}/{\text{m}}^3$ (black: experiment 23; green: experiment 24; blue: experiment 25; red: experiment 26). Parameter values of the experiments are provided in Table II in the Supplemental Material (SM) [21].

Figure 3

Experimental mixing velocities in the convective regime for $a=0.1$ mm as a function of (a) $\mathrm{\Delta }{\tilde{\rho }}_0$ and (b) $\mathrm{\Delta }{\tilde{\rho }}_m$ for $\delta =1$ (downward triangles), $\delta =1.25$ (squares), $\delta =1.89$ (upper triangles), and $\delta =3.67$ (circles). Solid colors represent experiments with constant $\mathrm{\Delta }{\tilde{\rho }}_0$ but variable $\mathrm{\Delta }{\tilde{\rho }}_m$. Open circles feature different $\mathrm{\Delta }{\tilde{\rho }}_0$ experiments for fixed $\delta =3.67$. The continuous lines in (a) and (b) are the equations $\tilde{U}=gK\mathrm{\Delta }{\tilde{\rho }}_0/\mu$ and $\tilde{U}=gK\mathrm{\Delta }{\tilde{\rho }}_m/\mu$, respectively. The inset in (b) shows the mixing velocity for two other gap widths: 0.25 mm (solid line) and 0.5 mm (dotted line). See Supplemental Material [21] for details on parameter values. (c) Dependence of the onset time ${\tilde{\tau }}_D$ on $\mathrm{\Delta }{\tilde{\rho }}_m$ for variable gap widths $a=0.1$ mm (solid line), $a=0.25$ mm (dotted line), and $a=0.5$ mm (dashed line).

Figure 4

(a) Mixing velocity $U$ computed as the average value of $\mathrm{\Delta }L/\mathrm{\Delta }t$ (see Fig. 1 given in the Supplemental Material [21]) in the convective regime as a function of $\mathrm{\Delta }{\rho }_m$. The color code used for the dotted symbols is the same as in the Supplemental Material [21]. The dotted red line shows the linear fit $U=0.8\mathrm{\Delta }{\rho }_m$. (b) Onset time ${\tau }_D$ as a function of $\mathrm{\Delta }{\rho }_m$. Spatial distribution of (c) the density field $\rho$ and (d) the difference $\delta \rho$ with the initial density for $R=0.90$, $\delta =10.0$, $\mathrm{\Delta }{\rho }_0=0.10$, $\mathrm{\Delta }{\rho }_m=0.54$.