Two-Phase Fluid Displacement and Interfacial Instabilities Under Elastic Membranes

We study the elastic version of the Saffman-Taylor problem: the Hele-Shaw displacement of a viscous liquid by a gas underneath an elastic membrane. We derive the dynamics of the propagating gas-liquid interface and of the deforming membrane. Even though the displacement of a viscous liquid by a gas is susceptible to viscous fingering, the presence of the elastic boundary can lead to the suppression of the instability. We demonstrate how the mechanism of suppression is provided by surface tension at the gas-liquid interface owing to the tapered flow geometry underneath the deflected membrane. We also determine the critical conditions for the onset of the fingering instability in the presence of the elastic boundary.