Oscillatory bubbles induced by geometrical constraint

Abstract

We show that a simple change in pore geometry can radically alter the behavior 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 centered finger to a state that exhibits spatial oscillations formed by periodic sideways motion of the interface at a fixed distance behind the moving finger tip. We characterize the dynamics of the oscillations, which suggest that they arise from a global homoclinic connection between the stable and unstable manifolds of a steady, symmetry-broken solution.

Sensitivity of Saffman–Taylor fingers to channel-depth perturbations

“…B. Thompson, A. L. Hazel and A. Juel of the channel height (de Lózar et al 2009;Pailha et al 2012;Hazel et al 2013). Thompson et al (2014) found that all finger propagation modes in partially occluded Hele-Shaw channels were qualitatively reproduced in an adapted version of the depthaveraged model used by McLean & Saffman (1981) to show that finite surface tension selects the half-width finger in unoccluded Hele-Shaw channels.…”

“…The subcritical symmetry-breaking bifurcation is also associated with oscillatory solutions, identified by Pailha et al (2012), who proposed a surface tension-based mechanism to explain their genesis. A fast local decrease in the cross-sectional curvature occurs when the interface of a finger passes over the edge of the occlusion to the less occluded region.…”

“…In summary, the qualitative stages in evolution of the system with increasing occlusion height are: steady symmetric propagation; supercritical symmetry breaking; For low values of α h , we observe a supercritical symmetry-breaking bifurcation at a critical capillary number Ca c1 ; as the value of α h increases, the symmetry breaking becomes subcritical and is associated with a critical capillary number Ca c2 , and a limit point at Ca LP ; larger values of α h lead to a subcritical symmetry breaking with Hopf bifurcations on the asymmetric branch at Ca H1 and Ca H2 ; the largest values of α h promote an asymmetric solution without oscillations, which is disconnected from the symmetric branch, and continues to a localised state as Ca is decreased (see Pailha et al (2012) for a similar bifurcation structure in a small aspect ratio occluded tube). subcritical symmetry breaking; subcritical symmetry breaking with oscillations; appearance of localised asymmetric solutions; disappearance of oscillations.…”

“…More recently, motivated by the physiological problem of airway reopening, de Lózar et al (2009), Pailha et al (2012) and Hazel et al (2013) investigated two-phase flow through rectangular tubes with aspect ratio α 13 that were partially occluded by axially uniform rectangular blocks occupying 50 % and 33.3 % of the tube height. Multiple stable modes of propagation were found in these geometries including asymmetric and oscillatory modes as well as the expected symmetric mode.…”

Viscous fingering and dendritic growth under an elastic membrane

“…These patterns are shed from the propagating front, that undergoes oscillations in width driven by the change in cross-sectional curvature of the interface as it crosses the edge of the block. This mechanism has been described in rigid constricted tubes [11,18,19] and is also the dominant mechanism in Fig. 1(b), where fingers are generated in the direction of the strongest depth gradient.…”