Control of viscous fingering by nanoparticles

A substantial viscosity increase by the addition of a low dose of nanoparticles to the base fluids can well influence the dynamics of viscous fingering. There is a lack of detailed theoretical studies that address the effect of the presence of nanoparticles on unstable miscible displacements. In this study, the impact of nonreactive nanoparticle presence on the stability and subsequent mixing of an originally unstable binary system is examined using linear stability analysis (LSA) and pseudospectral-based direct numerical simulations (DNS). We have parametrized the role of both nondepositing and depositing nanoparticles on the stability of miscible displacements using the developed static and dynamic parametric analyses. Our results show that nanoparticles have the potential to weaken the instabilities of an originally unstable system. Our LSA and DNS results also reveal that nondepositing nanoparticles can be used to fully stabilize an originally unstable front while depositing particles may act as temporary stabilizers whose influence diminishes in the course of time. In addition, we explain the existing inconsistencies concerning the effect of the nanoparticle diffusion coefficient on the dynamics of the system. This study provides a basis for further research on the application of nanoparticles for control of viscosity-driven instabilities.

Figure 1

Schematic of viscosity profiles: (a) nondepositing and (b) depositing cases. Arrow shows the direction for increase of time.

Figure 2

Schematic of the problem. A mixture consisting of species $a$ and nanoparticles $n_p$ displaces the fluid containing component b.

Figure 3

Parameter maps for the case with nondepositing nanoparticles in the flow (a) ${\alpha }_b=1$, (b) ${\alpha }_{np}=1$.

Figure 4

Parameter map for the case with depositing nanoparticles in the flow (a) ${\alpha }_b=1$, (b) ${\alpha }_{np}=1$.

Figure 5

Plot of maximum growth rate with time for $R_{np}=6$ and different ${\alpha }_{np}$ in the absence of deposition ($\mathrm{D}{\mathrm{a}}_{\mathrm{dep}}=0$) at ${\alpha }_b=1$, for regions (a) III, ${\mathrm{IV}}_{\mathrm{a}}$, and ${\mathrm{IV}}_{\mathrm{b}}$; (b) ${\mathrm{IV}}_{\mathrm{c}}$ and V. The region numbers are based on the parameter map in Fig. 3.

Figure 6

Schematic plot of the maximum growth rate with time in the presence of deposition.

Figure 7

Effect of nanoparticle mobility ratio on the maximum growth of instabilities at ${\alpha }_b=1$ and $\mathrm{D}{\mathrm{a}}_{\mathrm{dep}}=0$ for (a) ${\alpha }_{np}=0.5$ and (b) ${\alpha }_{np}=2$.

Figure 8

Effect of different nanoparticle mobility ratios on the maximum growth rate of instabilities at ${\alpha }_b=1$ and $\mathrm{D}{\mathrm{a}}_{\mathrm{dep}}=0.01$, for (a) ${\alpha }_{np}=0.5$ and (b) ${\alpha }_{np}=2$.

Figure 9

Effect of deposition rate on the maximum growth of instabilities at ${\alpha }_b=1$ and $R_{np}=4$ for (a) ${\alpha }_{np}=0.5$ and (b) ${\alpha }_{np}=2$.

Figure 10

Effect of nanoparticle diffusion coefficient on the maximum growth rate of instabilities at $\mathrm{D}{\mathrm{a}}_{\mathrm{dep}}=0.005$ and ${\alpha }_b=1$ for (a) $R_{np}=6$ and (b) $R_{np}=2$.

Figure 11

Effect of nanoparticle diffusion coefficient (larger values) on the maximum growth rate of instabilities at $\mathrm{D}{\mathrm{a}}_{\mathrm{dep}}=0.005$ and ${\alpha }_b=1$ for (a) $R_{np}=2$ and (b) $R_{np}=6$.

Figure 12

The plot of maximum growth rate of instabilities versus time at different ${\alpha }_{np}$ for ${\alpha }_b=0.5$. (a) $R_{np}=2, \mathrm{D}{\mathrm{a}}_{\mathrm{dep}}=0$; (b) $R_{np}=6, \mathrm{D}{\mathrm{a}}_{\mathrm{dep}}=0$; (c) $R_{np}=2, \mathrm{D}{\mathrm{a}}_{\mathrm{dep}}=0.005$; (d) $R_{np}=6, \mathrm{D}{\mathrm{a}}_{\mathrm{dep}}=0.005$.

Figure 13

Comparison of numerical simulations between this study and Singh [45] for a binary mixture with $R=3, \mathrm{Pe}=1000, A=1$ (contours between 0.1 and 0.6 are depicted with increments of 0.1).

Figure 14

Contours of $a$ at three different deposition rates for $t=200$, 400, and 600 with $R_{np}=1, {\alpha }_b=1$, and ${\alpha }_{np}=1$.

Figure 15

Transverse average concentration of nanoparticles versus location for (a) $\mathrm{D}{\mathrm{a}}_{\mathrm{dep}}=0.001$ and (b) $\mathrm{D}{\mathrm{a}}_{\mathrm{dep}}=0.01$. Other parameters are set to $R_{np}=1,{\alpha }_b=1$, and ${\alpha }_{np}=1$.

Figure 16

Contours of $a$ at two different nanoparticle log mobility ratios with $\mathrm{D}{\mathrm{a}}_{\mathrm{dep}}=0.001, {\alpha }_b=0.5$, and ${\alpha }_{np}=1$.

Figure 17

Comparison of mixing length between two nanoparticle log mobility ratios ($\mathrm{D}{\mathrm{a}}_{\mathrm{dep}}=0.001,{\alpha }_b=0.5,{\alpha }_{np}=1$).

Figure 18

Contours of $a$ at two different ${\alpha }_b$ with $\mathrm{D}{\mathrm{a}}_{\mathrm{dep}}=0.001, R_{np}=1$, and ${\alpha }_{np}=1$.

Figure 19

Contours of $a$ at two different ${\alpha }_{np}$ with $\mathrm{D}{\mathrm{a}}_{\mathrm{dep}}=0.001, R_{np}=1$, and ${\alpha }_b=1$.

Figure 20

Contours of $a$ (left) and $n_p$ (right) for $\mathrm{D}{\mathrm{a}}_{\mathrm{dep}}=0, R_{np}=2, {\alpha }_b=1$, and ${\alpha }_{np}=0.1$.

Figure 21

Contours of $a$ (left) and $n_p$ (right) for $\mathrm{D}{\mathrm{a}}_{\mathrm{dep}}=0.001, R_{np}=2, {\alpha }_b=1$, and ${\alpha }_{np}=0.1$.

Figure 22

Comparison of the base-state viscosity profile for (a) $\mathrm{D}{\mathrm{a}}_{\mathrm{dep}}=0$ and (b) $\mathrm{D}{\mathrm{a}}_{\mathrm{dep}}=0.001$ at different times ($R_{np}=2$ and ${\alpha }_b=1, {\alpha }_{np}=0.1$).

Figure 23

Comparison of nanoparticle base state concentration between Eqs. (A13) (open circles) and (B11) (solid lines) for $\mathrm{Pe}=1000, {\alpha }_{np}=1, \mathrm{D}{\mathrm{a}}_{\mathrm{dep}}=0.001$, and $n_p^0=1$.