Interfacial elastic fingering in Hele-Shaw cells: A weakly nonlinear study

We study a variant of the classic viscous fingering instability in Hele-Shaw cells where the interface separating the fluids is elastic, and presents a curvature-dependent bending rigidity. By employing a second-order mode-coupling approach we investigate how the elastic nature of the interface influences the morphology of emerging interfacial patterns. This is done by focusing our attention on a conventionally stable situation in which the fluids involved have the same viscosity. In this framework, we show that the inclusion of nonlinear effects plays a crucial role in inducing sizable interfacial instabilities, as well as in determining the ultimate shape of the pattern-forming structures. Particularly, we have found that the emergence of either narrow or wide fingers can be regulated by tuning a rigidity fraction parameter. Our weakly nonlinear findings reinforce the importance of the so-called curvature weakening effect, which favors the development of fingers in regions of lower rigidity.

Figure 1

Representative sketch of a radial Hele-Shaw cell with an elastic interface separating fluids 1 and 2.

Figure 2

Snapshots of the evolving interface for the interaction of two cosine modes $n=4$ and $2n=8$. Purely linear evolution for (a) $R_0=1.2$ cm, and (b) $R_0=1.145$ cm. Weakly nonlinear evolution for (c) $R_0=1.2$ cm, and (d) $R_0=1.145$ cm. Recall that here $C=0.99$.

Figure 3

Weakly nonlinear time evolution of the cosine perturbation amplitude $a_{2n}$ for different values of $C$ when (a) $R_0=1.2$ cm and (b) $R_0=1.145$ cm.

Figure 4

Time evolution of the bending rigidity field $\nu$ [Eq. (6)] in units of $\mathrm{g}{\mathrm{cm}}^2$/${\mathrm{s}}^2$, for the evolving interfaces in Fig. 2. Purely linear evolution for (a) $R_0=1.2$ cm, and (b) $R_0=1.145$ cm, and weakly nonlinear evolution for (c) $R_0=1.2$ cm and (d) $R_0=1.145$ cm. Note that smallest $\nu$ observed is $0.0013$.