Stabilization of periodically forced Hele-Shaw flows by means of a nonmonotonic viscosity profile

The onset of viscous fingering in the presence of a viscosity profile is investigated theoretically for two immiscible fluids undergoing a time-dependent injection. Here, we show that the presence of a positive viscosity gradient at the interface between both fluids stabilizes the interface facilitating the spread of the perturbation. This effect is much more pronounced in the case of sinusoidal injection flows. The influence of the viscosity gradient on the dispersion relation is analyzed. Numerical simulations of the Navier-Stokes equation confirm the linear stability analysis.

Figure 1

Sketch of the Hele-Shaw cell (a) and the piecewise viscosity profile $\mu \left(x_1\right)$ (b) used here. $L$ stands for the region where a linear viscosity gradient is considered between both fluids with constant viscosities ${\mu }_a$ and ${\mu }_b$.

Figure 2

Maximum growth rate ${\omega }_{\text{max}}$ as a function of $\alpha$, ${\mu }_a^{\prime }=(\alpha {\mu }_b-{\mu }_a)/L$, for the unstable case (${\mu }_b>{\mu }_a$). Dashed line corresponds to the Saffman-Taylor case. Set of parameters: $h={10}^{-3}$ m, $U=0.01$ m/s, water ${\mu }_a=0.896\times {10}^{-3}$ Pa s, glycerine ${\mu }_b=0.95$ Pa s, $\gamma =29.06\times {10}^{-3}$ N/m, $L=0.01$, and ${\mu }_b^{\prime }=0$.

Figure 3

Interface fronts between fluids for three forcing frequencies $\mathrm{\Omega }$, with (blue and green dashed lines) and without periodic forcing (red dot-dashed line). Green dashed interface corresponds to an intermediate time. $\alpha =0.4$, $G=-1$ Pa/m, $P^{\prime }=1$ Pa/m, and the rest of parameters as in Fig. 2. From left to right, $\mathrm{\Omega }=0.004,0.012,0.026$, respectively.

Figure 4

Time evolution of the surface ratio $S_f$ for two values of the ratio $\alpha =\mu \left(0\right)/{\mu }_b$. $\mathrm{\Omega }=0.012$.

Figure 5

Surface ratio $S_f$ covered by the fluid interfaces, with and without periodic forcing, as a function of the forcing frequency $\mathrm{\Omega }$ for two values of the ratio $\alpha =\mu \left(0\right)/{\mu }_b$. Parameters as in Fig. 2.

Figure 6

Resonant period $T_{\text{res}}$ as a function of the viscosity (a) and the surface tension $\gamma$ (b). In all cases, the behavior of $T_{\text{res}}$ coincides with the characteristic time of the flow $t_{\text{ch}}\propto \gamma /({\mu }_b-{\mu }_a)$. $\gamma =29.06\times {10}^{-3}$ N/m (a) and ${\mu }_b=0.2$ Pa s (b). $\alpha =0.4$ and ${\mu }_a=0.89\times {10}^{-3}$ Pa s in all cases.

Figure 7

Maximum surface ratio $S_f$ as a function of the ratio $\alpha =\mu \left(0\right)/{\mu }_b$ (a) and the length $L$ of the middle layer (b). In panel (b), lines correspond to $\alpha =0.4$ (circles and blue solid line), $\alpha =0.5$ (squares and dot-dashed red line), and $\alpha =0.8$ (triangles and dashed green line). Parameters are as in Fig. 2.