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Interfacial instabilities on the pore scale

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.

We demonstrate that a simple change in the bounding geometry can radically alter the behaviour of a fluid-displacing air finger both in rigid and elastic vessels. A rich array of propagation modes, including symmetric, asymmetric and localised fingers, is uncovered when air displaces oil from axially uniform tubes that have local variations in flow resistance within their cross-sections. An unexpected and novel propagation mode exhibits spatial oscillations formed by periodic sideways motion of the interface at a fixed relative distance behind the moving finger-tip. The presence of multiple steady and unsteady modes is in contrast to the single, symmetric mode observed in tubes of regular cross-section, e.g. circular, elliptical, rectangular and polygonal. Moreover, we show that the experimentally observed states are all captured by a two-dimensional depth-averaged model for bubble propagation through wide channels with a smooth occlusion, which is similar to a model describing viscous fingering, but with a spatially varying channel height.

Viscous fingering in Hele-Shaw cells 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. The presence of a deformable boundary also makes the system 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 study the interaction between these distinct instabilities, using a theoretical model that couples the depth-averaged lubrication equations for the fluid flow to the Föppl-von Kármán equations, which describe the deformation of the thin elastic sheet.