Electrically controlled self-similar evolution of viscous fingering patterns

We design a controlling protocol for the traditional viscous fingering instability by taking advantage of an electro-osmotic flow generated by an external electric field. Under the coupled action of time-varying electric currents and injection rates, fully nonlinear simulations show that, besides setting the number of fingers on the interface, our strategy can either delay or promote the self-similar regime by several orders of magnitude, or even suppress it. In addition, we can produce self-similar shapes ranging from almost circular, mildly perturbed boundaries to fully developed fingered structures with large perturbation amplitudes. All these features are controlled without altering the material properties of the fluids and cell's geometry.

Figure 1

Representative sketch of the electrohydrodynamic Hele-Shaw flow.

Figure 2

Left panel: Behavior of the shape factor $\mathrm{\Delta }\left(t\right)/R\left(t\right)$ with respect to variations in the ratio $R\left(t\right)/R_0$, for flows performed utilizing the controlling protocol with $n_{\max }=7$ and various injection rates $Q\left(t\right)=\alpha Q_{\mathrm{crit}}\left(t\right)$. Right panel: Plot of the electric current Eq. (5) as a function of $R\left(t\right)/R_0$ for the injection rates utilized in the left panel.

Figure 3

Gallery of representative fully nonlinear sevenfold patterns (see Supplemental Material) [44] corresponding to the cases with (a) $\alpha =0.49$, (b) $\alpha =0.45$, (c) $\alpha =0.3$, (d) $\alpha =0.23$, (e) $\alpha =0.21$, and (f) $\alpha =0.1$, shown in Fig. 2. The interfaces (a)–(d) are plotted at the onset of the self-similar regime, i.e., for (a) $R/R_0={10}^{36}$, (b) $R/R_0=1.75\times {10}^9$, (c) $R/R_0=1.25\times {10}^5$, and (d) $R/R_0=2.29\times {10}^6$. The patterns depicted in panels (e) and (f) do not reach the self-similar regime, and are plotted for (e) $R/R_0=114$ and (f) $R/R_0=1.97$.

Figure 4

Left panel: Behavior of the shape factor $\mathrm{\Delta }\left(t\right)/R\left(t\right)$ with respect to variations in the ratio $R\left(t\right)/R_0$, for flows performed utilizing the controlling protocol with $n_{\max }=3$ and various injection rates $Q\left(t\right)=\alpha Q_{\mathrm{crit}}\left(t\right)$. Right panel: Plot of the electric current Eq. (5) as a function of $R\left(t\right)/R_0$ for the injection rates utilized in the left panel.

Figure 5

Gallery of representative fully nonlinear threefold patterns corresponding to the cases (a) $\alpha =0.45$, (b) $\alpha =0.37$, and (c) $\alpha =0.35$, shown in Fig. 4. The interfaces (a) and (b) are plotted at the onset of the self-similar regime, i.e., for (a) $R/R_0=8.2\times {10}^{10}$ and (b) $R/R_0=3.2\times {10}^{11}$. The pattern depicted in (c) does not reach the self-similar regime, and is plotted for $R/R_0=1.06\times {10}^3$.

Figure 6

Behavior of the shape factor $\mathrm{\Delta }\left(t\right)/R\left(t\right)$ with respect to variations in the ratio $R\left(t\right)/R_0$, for flows performed utilizing the controlling protocol with $n_{\max }=3,4,5,6,$ and 7. Here $Q\left(t\right)=0.37\times Q_{\mathrm{crit}}\left(t\right)$, where $Q_{\mathrm{crit}}\left(t\right)$ is evaluated for each mode $n_{\max }$ utilizing Eq. (6). The resulting self-similar shapes are depicted as insets.

Figure 7

Weakly nonlinear time evolution of the expanding interfacial patterns generated by utilizing the controlling protocol for $n_{\max }=7$, and three injection rates $Q\left(t\right)=\alpha Q_{\mathrm{crit}}\left(t\right)$: (a) $\alpha =0.25$, (b) $\alpha =0.23$, and (c) $\alpha =0.2$. Here we include modes $2\le n\le 20$ and a random initial phase. The corresponding time evolution of the rescaled perturbation mode amplitudes $|{\delta }_n\left(t\right)|/R\left(t\right)$ are shown in the bottom panels. The final times used are (a) $t_f=480$ s, and (b) $t_f=600$ s, and the remaining physical parameters are listed in Table 1.