The displacement of a liquid by an air finger is a generic two-phase flow that
underpins applications as diverse as microfluidics, thin-film coating, enhanced
oil recovery, and biomechanics of the lungs. I will present two intriguing
examples of such flows where, firstly, oscillations in the shape of propagating
bubbles are induced by a simple change in tube geometry, and secondly, flexible
vessel boundaries suppress viscous fingering instability.
1) A simple change in pore geometry can radically alter the behaviour of a
fluid displacing air finger, indicating that models based on idealized pore
geometries fail to capture key features of complex practical flows. In
particular, partial occlusion of a rectangular cross-section can force a
transition from a steadily-propagating centred finger to a state that exhibits
spatial oscillations via periodic sideways motion of the interface at a fixed
location behind the finger tip. We characterize the dynamics of the
oscillations and show that they arise from a global homoclinic connection
between the stable and unstable manifolds of a steady, symmetry-broken
solution.
2) 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
analyse the dependence of the instability on the parameters of the system and
present scaling arguments to explain the experimentally observed behaviour.