Stationary patterns in centrifugally driven interfacial elastic fingering

A vortex sheet formalism is used to search for equilibrium shapes in the centrifugally driven interfacial elastic fingering problem. We study the development of interfacial instabilities when a viscous fluid surrounded by another of smaller density flows in the confined environment of a rotating Hele-Shaw cell. The peculiarity of the situation is associated to the fact that, due to a chemical reaction, the two-fluid boundary becomes an elastic layer. The interplay between centrifugal and elastic forces leads to the formation of a rich variety of stationary shapes. Visually striking equilibrium morphologies are obtained from the numerical solution of a nonlinear differential equation for the interface curvature (the shape equation), determined by a zero vorticity condition. Classification of the various families of shapes is made via two dimensionless parameters: an effective bending rigidity (ratio of elastic to centrifugal effects) and a geometrical radius of gyration.

Figure 1

Typical stationary shapes that occur during the conventional rotating Hele-Shaw problem [21, 22] (i.e., in the absence of an elastic interface). In this case, capillary and centrifugal forces equally balance at the two-fluid interface. Increasingly larger centrifugal driving results in patterns containing more fingers, that eventually tend to pinch off. The same class of patterns appear in various other contexts [26, 27, 28, 29, 30, 31, 32, 33, 34].

Figure 2

Schematic representation of a rotating Hele-Shaw cell, where the fluid-fluid boundary is an elastic layer. We consider that ${\rho }_2>{\rho }_1$.

Figure 3

$B-R_g$ parametric map showing the disposition of the families of equilibrium patterns I–V.

Figure 4

Stationary shapes—Family I. (a) $B=0.0333$, $R_g=1.4768$; (b) $B=0.0347$, $R_g=1.7004$; (c) $B=0.0483$, $R_g=2.3253$; (d) $B=0.0500$, $R_g=2.5826$. Observe the serpentinelike interface arising in each of the $n$-folded structures.

Figure 5

Stationary shapes—Family II. (a) $B=0.0070$, $R_g=1.4582$; (b) $B=0.0058$, $R_g=1.4032$; (c) $B=0.0052$, $R_g=1.3051$; (d) $B=0.0038$, $R_g=1.2547$. Note the swirling nature of all patterns, and the droplet entrapment phenomenon of inward pointing fingers of the outer fluid in (d).

Figure 6

Stationary shapes—Family III. (a) $B=0.0087$, $R_g=1.2640$; (b) $B=0.0065$, $R_g=1.2876$; (c) $B=0.0050$, $R_g=1.3296$; (d) $B=0.0018$, $R_g=1.3633$. As in Fig. 5 the patterns also swirl, but now we observe tendency toward pinch off of outward pointing fingers of the inner in (d).

Figure 7

Stationary shapes—Family IV. (a) $B=0.0034$, $R_g=1.6714$; (b) $B=0.0054$, $R_g=1.8260$; (c) $B=0.0074$, $R_g=1.9525$; (d) $B=0.0096$, $R_g=2.0633$. From (a) to (d) one notices increasing interface buckling and finger sidebranching formation.

Figure 8

Stationary shapes—Family V. (a) $B=0.0060$, $R_g=0.9539$; (b) $B=0.0045$, $R_g=0.9851$; (c) $B=0.0032$, $R_g=1.0408$; (d) $B=0.0029$, $R_g=1.0938$. Sidebranching and buckling occurs, but the fingering protrusions are considerably larger than the ones shown in Fig. 7.