Nonlinear Saffman-Taylor Instability

We show, both theoretically and experimentally, that the interface between two viscous fluids in a Hele-Shaw cell can be nonlinearly unstable before the Saffman-Taylor linear instability point is reached. We identify the family of exact elastica solutions [Nye et al., Eur. J. Phys. 5, 73 (1984)] as the unstable branch of the corresponding subcritical bifurcation which ends up at a topological singularity defined by interface pinchoff. We devise an experimental procedure to prepare arbitrary initial conditions in a Hele-Shaw cell. This is used to test the proposed bifurcation scenario and quantitatively asses its practical relevance.

Figure 1

Sketch of a rectangular Hele-Shaw cell of width $L$ and gap thickness $b$. The gap step used to prepare arbitrary initial conditions (see text) is also sketched.

Figure 2

Bifurcation diagram showing the stationary solutions of Eq. (1). The horizontal axis corresponds to the planar interface and the parabola to the subcritical branch. Linearly stable (unstable) solutions are represented by solid (dashed) lines.

Figure 3

Exact stationary solutions of the ST problem with nonzero surface tension, in channel geometry (left) and circular geometry with rotation (right).

Figure 4

In the top panel, the distance $H$ between the maximum and the minimum height of the interface is plotted vs $B$, in scaled variables $\overline{H}=H(2\pi /\lambda )$, $B^{*}=B(2\pi /\lambda {)}^2$, up to $B^{*}=3$. The branch of elastica solutions (subcritical) is represented by a light solid line. The branch ends at $B^{*}=23.9$, where interface pinchoff occurs. The dashed line is the weakly nonlinear approximation obtained from Eq. (1). The bottom panel shows a magnification of the upper unstable branch, together with the set of initial conditions used in our experiments, represented by different symbols.

Figure 5

Snapshots of two experiments for two different initial conditions. Dark and light gray regions, above and below the interface, correspond, respectively, to silicone oil and air. A dashed line represents the initial condition, designed as sketched on the bottom left panel (see text for details). The shaded strips on the sides define the regions disregarded in the quantitative analysis.

Figure 6

Time evolution of the measured maximum-to-minimum interface height for different experiments. The experimental parameters are indicated in Fig. 4 with corresponding symbols.