Influence of interfacial rheology on viscous fingering

Interfacial instabilities present challenges in common industrial processes involving multiphase flows. One such phenomenon—the Saffman-Taylor instability—involves the formation of fingerlike protrusions when a less viscous fluid displaces a more viscous fluid. Recent studies have shown that complex surfactant-laden interfaces with surface rheological stresses resist interfacial deformation and alter the fluid dynamics of the system. Despite current progress, the impact of surface rheological stresses on the Saffman-Taylor instability is yet unknown. In this work, we demonstrate the stabilizing effect of surface rheology in radial viscous fingering using linear stability analysis. We quantify the growth rates of perturbations to show that surface viscosity slows the growth of the instability and results in thicker fingers. Finally, we highlight the quantitative changes that are predicted to occur when a typical surface viscous surfactant is present.

Figure 1

Geometry of a less viscous fluid (phase 1) radially invading a more viscous fluid (phase 2). ${\eta }_j$, with $j=1,2$, is the fluid viscosity; $b$ is the gap width of the Hele-Shaw cell; $R\left(t\right)$ is the interfacial radius in the base state; and $a\left(t,\theta \right)$ is the perturbed displacement around $R\left(t\right)$.

Figure 2

(a) Dimensionless growth rate $\alpha$ as a function of wave number $n$ at $\mathrm{Ca}=0.1,\xi =100$. Both the maximum growth rate ${\alpha }_{\max }$ and the corresponding wave number $n_{\max }$ decrease with increasing Boussinesq number $\mathrm{Bq}$, indicating that surface viscosity stabilizes perturbations. (b) Dimensionless growth rate $\alpha$ as a function of wave number $n$ at $\mathrm{Bq}=1,\xi =100$. This reveals the classically known effect of surface tension in suppressing the instability upon decreasing the capillary number $\mathrm{Ca}$.

Figure 3

(a) The most unstable wave number $n_{\max }$ as a function of Boussinesq number $\mathrm{Bq}$ at $\mathrm{Ca}=0.1$ and $\xi =100$. $n_{\max }$ asymptotes to a constant value as $\mathrm{Bq}\to 0$ (corresponding to the effect of surface tension alone) and decreases as a power law for $\mathrm{Bq}\gtrsim 1$. The dashed line with a slope of $-0.5$ is shown as a guide. (b) Maximum growth rate ${\alpha }_{\max }$ follows a similar trend, shown here for a constant $\mathrm{Ca}=0.1$.

Figure 4

(a) Dimensionless growth rate $\alpha$ as a function of wave number $n$ at $\mathrm{Ca}=0.1,\mathrm{Bq}=1$, and various values of normalized radius $\xi$. (b,c) The maximum growth rate ${\alpha }_{\max }$ decreases and the corresponding wave number $n_{\max }$ increases with increasing $\xi$, indicating more fingers if perturbed at a larger radius. The dashed lines with slopes of 1 and $-1$ are shown as guides.

Figure 5

The surface rheology model (SRM) developed here predicts lower values of the modified wave number $A_{\max }$ at larger $\mathrm{Ca}$ when compared with the viscous potential flow (VPF) model [7]. $A_{\max }$ departs further from past predictions with increasing $\mathrm{Bq}$, especially at large $\mathrm{Ca}$. The dashed horizontal line represents the expected value of $A_{\max }$ in the classic Saffman-Taylor problem which neglects viscous normal stresses and surface viscous stresses.