Onset of fingering instability in a finite slice of adsorbed solute

The effect of a linear adsorption isotherm on the onset of fingering instability in a miscible displacement in the application of liquid chromatography, pollutant contamination in aquifers, etc., is investigated. Such fingering instability on the solute dynamics arise due to the miscible viscus fingering (VF) between the displacing fluid and sample solvent. We use a Fourier pseudo-spectral method to solve the initial value problem that appears in the linear stability analysis. The present linear stability analysis is of generic type and it captures the early-time-diffusion-dominated region which was never expressible through the quasi-steady-state analysis (QSSA). In addition, it measures the onset of instability more accurately than the QSSA methods. It is shown that the onset time depends nonmonotonically on the retention parameter of the solute adsorption. This qualitative influence of the retention parameter on the onset of instability resemblances with the results obtained from direct numerical simulations of the nonlinear equations. Moreover, the present linear stability method helps for an appropriate characterization of the linear and nonlinear regimes of miscible VF instability and also can be useful for the fluid flow problems with the unsteady base state.

Figure 1

Schematic of the flow configuration for three component model with coordinate axes.

Figure 2

A straight-line fit to the normalized amplification: (a) $ln(E_{mV}\left(t\right)/E_{mV}\left(t_0\right))$ and (b) $ln(E_{cV}\left(t\right)/E_{cV}\left(t_0\right))$ for ${\kappa }^{\prime }=0.2,R=3,w=512,k=0.1$.

Figure 3

Temporal evolution of the growth functions for two semi-infinite fluids with $R=2,k=0.1$.

Figure 4

(a) Temporal evolution of the growth function, ${\sigma }_{cV}t$, for $R=2,k=0.2$ and different width of the finite slice, $w$. Symbols correspond to growth function associated with $w=128(◯),256(\square )$, and $\infty \left(◊\right)$. (b) Magnification of (a) near the onset of instability.

Figure 5

Growth functions, ${\sigma }_{cV}t$ (---) and ${\sigma }_{mV}t$ $(◯)$, for ${\kappa }^{\prime }=0,R=2,w=512,k=0.15$. The solid dot $\left(•\right)$ represents the onset of the nonlinearity. Inset: Magnified near the onset of instability.

Figure 6

Neutral curves for $R=2,w=512$ and different values of the retention parameter ${\kappa }^{\prime }=0,3$, and 8. The critical time $t^c$ and corresponding wave number $k^c$ on the parameter space is marked by a solid dot $\left(•\right)$.

Figure 7

Dispersion curves at different times for $w=512,R=2$: ${\kappa }^{\prime }=0(◯),3(\square ),8\left(◊\right)$.

Figure 8

Dominant wave number, $k_{\max }$, of the solute concentration for $w=512,R=2$ and different values of ${\kappa }^{\prime }$.

Figure 9

The optimal growth function, ${\sigma }_{mV}^{\max }t$, of the solute concentration for $w=512,R=2$ and different values of ${\kappa }^{\prime }=0,3,8$. Inset: The nonmonotonicity of onset time and dominance of diffusion at early times.

Figure 10

Onset time, $t^c$, calculated from ${\sigma }_{mV}$ as a function of the retention parameter, ${\kappa }^{\prime }$, for $R=2,3,w=512,k=0.1$.

Figure 11

For $R=2,w=512$: (a) Contribution of fingering to the standard deviation ${\tilde{\sigma }}_{a,f}$ as a function of time. (b) Onset time, $t_f^c$, obtained from DNS as a function of the retention parameter, ${\kappa }^{\prime }$. Inset: Density plot of the solute concentration field for ${\kappa }^{\prime }=0.2$ and ${\kappa }^{\prime }=5$ at the corresponding onset time, $t_f^c=740$ and $t_f^c=640$, respectively.

Figure 12

(a) Growth functions, ${\sigma }_{mV}t$, of the solute concentration for $R=2,w=512,k=0.1$, and ${\kappa }^{\prime }=0$ (solid line), ${\kappa }^{\prime }=0.2$ (dashed line): Lines without a symbol correspond to LSA and lines with symbols represent DNS. (b) Magnification of (a) near the onset of instability marked by the solid dots $\left(•\right)$.