Suppression of Complex Fingerlike Patterns at the Interface between Air and a Viscous Fluid by Elastic Membranes

Growth of complex dendritic fingers at the interface of air and a viscous fluid in the narrow gap between two parallel plates is an archetypical problem of pattern formation. We find a surprisingly effective means of suppressing this instability by replacing one of the plates with an elastic membrane. The resulting fluid-structure interaction fundamentally alters the interfacial patterns that develop and considerably delays the onset of fingering. We analyze the dependence of the instability on the parameters of the system and present scaling arguments to explain the experimentally observed behavior.

Figure 1

(a)–(c) Top view of the front propagation in a radial Hele-Shaw cell in which a growing air bubble displaces viscous fluid that occupies the narrow gap between two parallel plates. Fluids are injected through a small nozzle with an internal diameter of 2.28 mm embedded in a brass fitting, which is mounted flush with the lower bounding plate of the cell and appears as the central black circle. Four successive positions of the interface are shown in each image, with the smallest interface recorded $t=0.4\mathrm{s}$ after air injection started: (a) elastic cell with membrane a; see 17, separator b; see 18, $Q=145{\mathrm{cm}}^3{\min }^{-1}$, $\Delta t=4.28\mathrm{s}$; (b) same elastic cell as in (b) with $Q=1250{\mathrm{cm}}^3{\min }^{-1}$, $\Delta t=0.71\mathrm{s}$; (c) rigid cell, $Q=145{\mathrm{cm}}^3{\min }^{-1}$, $\Delta t=0.57\mathrm{s}$. (d) Schematic diagram of the experimental apparatus.

Figure 2

(a) Dependence of scaled $⟨r⟩$ (in units of mm ${\mathrm{s}}^{-3/8}$) on $t$ (in s), both on a logarithmic scale, for different experimental runs ($h$ and $b_0$ are given in 17, 18, respectively; units of $Q$ are ${\mathrm{cm}}^3{\min }^{-1}$). (b) Displacement of membrane c for separator $b$ and $Q=50{\mathrm{cm}}^3{\min }^{-1}$ at $t=69\mathrm{s}$ from the start of the experiment.

Figure 3

(a) Typical image analysis procedure—the corresponding experimental run is presented in Fig. 4j. (b) The excess perimeter ${\delta }_{\ell }=\tilde{\ell }-\ell$ as a function of the average interfacial radius $⟨r⟩$ for membrane a, see 17, separator $b$, see 18, and several experimental runs with injection rates $Q=200{\mathrm{cm}}^3{\min }^{-1}$ (“stable”) and $Q=700{\mathrm{cm}}^3{\min }^{-1}$ (“unstable”). (c) Close-up of fingers 7 and 8 from (a) with the schematics for defining the depth $d$.

Figure 4

Superimposed snapshots of the interface when $⟨r⟩=4\mathrm{cm}$ and $⟨r⟩=5\mathrm{cm}$. First row: the variation in the typical top view for membrane $h=0.69\pm 0.02\mathrm{mm}$, $b_0=0.56\pm 0.02\mathrm{mm}$, and injection flow rates $75{\mathrm{cm}}^3{\min }^{-1}$ (a), $200{\mathrm{cm}}^3{\min }^{-1}$ (b), $500{\mathrm{cm}}^3{\min }^{-1}$ (c), $700{\mathrm{cm}}^3{\min }^{-1}$ (d), and $900{\mathrm{cm}}^3{\min }^{-1}$ (e); second row: the variation in the typical top view for $Q=400{\mathrm{cm}}^3{\min }^{-1}$, $b_0=0.56\pm 0.02\mathrm{mm}$, for sheet thicknesses given in 17 increasing from left to right.

Figure 5

Dependence of $\Delta$ on injection flow rates $Q$ for the different membranes given in 17.

Figure 6

The critical injection flow rate at the onset of the instability for (a) different latex sheets and (b) different separators at $⟨r⟩=5\mathrm{cm}$.