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Fingering Instabilities in Compliant Channels
What links a baby’s first breath to adhesive debonding, enhanced oil recovery, or even drop-on-demand devices ? All these processes involve moving or expanding bubbles displacing fluid in a confined space, bounded by either rigid or elastic walls. In this talk, we show how spatial confinement may either induce or suppress interfacial instabilities and pattern formation in such flows. Viscous fingering in a radial Hele-Shaw cell is a classical and widely studied fluid-mechanical instability : when air is injected into the narrow, liquid-filled gap between parallel rigid plates, the axisymmetrically expanding air-liquid interface tends to be unstable to non-axisymmetric disturbances. We show how the introduction of wall elasticity (via the replacement of the upper bounding plate by an elastic membrane) can weaken or even suppress the fingering instability by allowing changes in cell confinement through the flow-induced deflection of the boundary. For sufficiently large driving flow rate, a fingering instability arises that is reminiscent of a printer’s instability. The amplitude of the interfacial deformation saturates as the interface expands radially by contrast with the rigid Hele-Shaw cell, where fingers grow unbounded, and successive tip-splitting instabilities results in a highly branched pattern. The presence of a deformable boundary also makes the system prone to additional solid-mechanical instabilities, so that in elastic-walled Hele-Shaw cells that are bounded by sufficiently thin and elastic sheets, the (fluid-based) viscous fingering instability can arise concurrently with a (solid-based) wrinkling instability. We then consider a rectangular Hele-Shaw geometry with a flexible top boundary in the context of pulmonary airway reopening. We find that the front propagation - a steadily advancing air finger - is significantly altered by the compliant boundary, which broadens the reopening modes. Broad fingers exhibit a flat tip region, which in turn promotes a compliant fingering instability in the form of stubby fingers superposed onto the broad tip of the reopening mode. Hence, compliance in the rectangular channel is found to promote fingering instability.