Elastic fingering in rotating Hele-Shaw flows

The centrifugally driven viscous fingering problem arises when two immiscible fluids of different densities flow in a rotating Hele-Shaw cell. In this conventional setting an interplay between capillary and centrifugal forces makes the fluid-fluid interface unstable, leading to the formation of fingered structures that compete dynamically and reach different lengths. In this context, it is known that finger competition is very sensitive to changes in the viscosity contrast between the fluids. We study a variant of such a rotating flow problem where the fluids react and produce a gellike phase at their separating boundary. This interface is assumed to be elastic, presenting a curvature-dependent bending rigidity. A perturbative weakly nonlinear approach is used to investigate how the elastic nature of the interface affects finger competition events. Our results unveil a very different dynamic scenario, in which finger length variability is not regulated by the viscosity contrast, but rather determined by two controlling quantities: a characteristic radius and a rigidity fraction parameter. By properly tuning these quantities one can describe a whole range of finger competition behaviors even if the viscosity contrast is kept unchanged.

Figure 1

Perspective view of a rotating Hele-Shaw cell with an elastic interface separating fluids 1 and 2.

Figure 2

Linear growth rate $\Lambda \left(n\right)$ as a function of mode $n$, for three values of $C$ and two values of $\lambda$. Here $B=2.5\times {10}^{-3}$.

Figure 3

Finger competition function $\mathcal{C}\left(n\right)$ plotted in terms of $\lambda$, for $B=2.5\times {10}^{-3}$, $C=0.5$, and three values of $A=1,0$, and $-1$.

Figure 4

Snapshots of the evolving interface for the interaction of the fundamental mode $n=6$ and its subharmonic mode $n=3$ for the situations corresponding to points $P_1$ (a) $\lambda =0.8$, $P_2$ (b) $\lambda =0.926$, and $P_3$ (c) $\lambda =1.05$ that have been indicated in Fig. 3 for $A=0$. The interfaces are plotted in intervals of $t_f/10$, where $t_f=1.25$ is the final time. The darker interfaces correspond to $t=t_f$.

Figure 5

Dimensionless radial coordinate $\mathcal{R}$ of the finger tips for each inward- and outward-pointing fingers, as a function of the polar angle $\theta$, when (a) $\lambda =0.8$, (b) $\lambda =0.9286$, and (c) $\lambda =1.05$. These data are taken from the corresponding patterns plotted in Figs. 4–(c) at $t=t_f$, and $A=0$.

Figure 6

Finger competition function $\mathcal{C}\left(n\right)$ plotted in terms of $\lambda$, for $B=2.5\times {10}^{-3}$, $A=0$, and five values of $C$.

Figure 7

Finger competition function $\mathcal{C}\left(n\right)$ as a function of the viscosity contrast $A$, for $C=0.5$, and three different values of $\lambda$. These are the values of $\lambda$ utilized to get the interfacial evolutions presented in Figs. 4 and 5.