Figure 1

(a) Single-phase spreading. Schematic of a system in which a fluid of viscosity $\mu$ is injected at a constant flow rate $Q$ underneath a thin elastic membrane with zero initial height. (b) Two-phase displacement. Schematic of a radial Hele-Shaw cell where the bottom boundary is a rigid horizontal plate and the top boundary is a thin elastic sheet. A gas is injected at the center of the cell at a constant flow rate $Q$, displacing a viscous liquid of viscosity $\mu$ and deflecting the elastic membrane. The height of the membrane $h(r,t)$ is initially $b$ and the gas-liquid interface is located at $r=r_0(t)$.Reuse & Permissions

Figure 2

Propagation of the gas-liquid interface. Comparison between our prediction (solid line) in Eq. (14a) and the experimental results in Ref. 6, which were conducted using a liquid of viscosity $\mu =1.04\mathrm{kg}{\mathrm{m}}^{-1}{\mathrm{s}}^{-1}$ and a membrane with bending modulus $B$. The initial membrane height is $b$, and then nitrogen gas is injected at a constant flow rate $Q$. (Filled circles) $Q=55\mathrm{mL}{\min }^{-1}$, $B=1.46\times {10}^{-1}\mathrm{N}\mathrm{mm}$, and $b=0.56\mathrm{mm}$. (Filled squares) $Q=300\mathrm{mL}{\min }^{-1}$, $B=2.81\times {10}^{-2}\mathrm{N}\mathrm{mm}$, and $b=0.79\mathrm{mm}$. The dashed line is Eq. (14a) with the prefactor adjusted to be $1.25\times C_r$. The units of the vertical axis are $s^{5/14}$.Reuse & Permissions

Figure 3

The influence of the bending modulus on the onset of viscous fingering. Comparison between our scaling for the critical flow rate (19) and the experimental results (open circles) in Ref. 6. To vary the bending modulus $B$ in the experiments, membranes of different thicknesses were used. The critical flow rate $Q_c$ is the minimum gas flow rate required to initiate an interfacial instability at $R=5\mathrm{cm}$. The initial height of the elastic membrane is $b=0.56\mathrm{mm}$. The viscosity and the surface tension of the displaced wetting liquid are $\mu =1.04\mathrm{kg}{\mathrm{m}}^{-1}{\mathrm{s}}^{-1}$ and $\gamma =22\mathrm{mN}{\mathrm{m}}^{-1}$, respectively. The solid line represents Eq. (19) with the prefactor adjusted to be $1.7\times C_Q$. This offset in the prefactor is expected due to the cumulative discrepancies resulting from the asymptotic approximation (15) and from neglecting the effect of the wetting film [12, 26] on the stability threshold (17).Reuse & Permissions