arXiv subjects

Jack Lawless

Publications and source records attributed to Jack Lawless.

3 recordsLinked to original sources

arXiv2025

On the oscillatory dynamics of a Saffman--Taylor finger with a bubble at its tip

The complex behaviour of air-liquid interfaces driven into Hele-Shaw channels at high speeds could arise from oscillatory dynamics; yet, both the physical and dynamical mechanisms that lead to interfacial oscillations remain unclear. We extend the experiments by Couder \textit{et. al.} (\textit{Phys. Rev. A}, vol. 34, 1986, p. 5175) to present a systematic investigation of the dynamics that result when a small air bubble is placed at the tip of a steadily propagating air finger in a Hele-Shaw channel. The system can exhibit steady and oscillatory behaviour, and we show that these different behaviours each occur in well-defined regions of the phase space defined by flow rate and bubble size. For sufficiently large flow rates, periodic finger oscillations give way to disordered dynamics characterised by an irregular meandering of the finger's tip. We demonstrate that at a fixed flow rate, the oscillations commence when the bubble size is increased sufficiently so that the decreased in-plane curvature of the bubble tip matches the in-plane curvature of the finger tip. The equality between the two in-plane curvatures causes the axial pressure gradient across the bubble, which drives the finger, to vanish, thus rendering the finger susceptible to lateral perturbations. Differing timescales for finger and bubble restoral under perturbation allow sustained oscillations to develop in the finger-bubble system. The oscillations cease when the bubble is sufficiently large that it can act as the tip of a compound finger. The disordered dynamics at high flow rates are consistent with the transient exploration of unstable periodic states, which suggests that similar dynamics may underlie the observed disordered dynamics in viscous fingering.

physics.flu-dyn↗

arXiv2024

Periodic dynamics in viscous fingering

The displacement of a viscous liquid by air in the narrow gap between two parallel plates - a Hele-Shaw channel - is an exemplar of complex pattern formation. Typically, bubbles or fingers of air propagate steadily at low values of the driving parameter. However, as the driving parameter increases, they can exhibit disordered pattern-forming dynamics. In this paper, we demonstrate experimentally that a remote perturbation of the bubble's tip can drive time-periodic bubble propagation: a fundamental building block of complex unsteady dynamics. We exploit the propensity of a group of bubbles to self-organise into a fixed spatial arrangement in a Hele-Shaw channel with a centralised depth-reduction in order to apply a sustained perturbation to a bubble's shape as it propagates. We find that the bubble with a perturbed shape begins to oscillate after the system undergoes a supercritical Hopf bifurcation upon variation of the tip perturbation and dimensionless flow rate. The oscillation cycle features the splitting of the bubble's tip and advection of the resulting finger-like protrusion along the bubble's length until it is absorbed by the bubble's advancing rear. The restoral of the bubble's tip follows naturally because the system is driven by a fixed flow rate and the perturbed bubble is attracted to the weakly unstable, steadily propagating state that is set by the ratio of imposed viscous and capillary forces. Our results suggest a generic mechanism for time-periodic dynamics of propagating curved fronts subject to a steady shape perturbation.

physics.flu-dyn↗

arXiv2023

Stable bubble formations in a Hele-Shaw channel

Deformable bubbles propagated by the flow of a viscous liquid in a planar Hele-Shaw channel of uniform depth tend to travel steadily along the channel's streamwise axis and pairs of neighbouring bubbles will either separate or coalesce because an individual bubble's propagation speed increases monotonically with its size. Thus, any group of bubbles will eventually rearrange itself in order of decreasing size and all of the bubbles will separate. We show that, by introducing a small geometric perturbation to the channel in the form of an axially-uniform depth reduction along its centreline, the system supports a multitude of stable bubble formations and this can disrupt the usual reordering by bubble size. The constituent bubbles of a stable formation lie in alternation on opposite sides of the depth-perturbation, retain fixed shapes and propagate steadily at the same speed. A stable formation is always led by the smallest of its constituent bubbles, which would be the slowest bubble in isolation. The leading bubble propagates as if it were isolated and the trailing bubbles reduce their speeds by adjusting their shapes and overlaps of the depth-perturbation in the perturbation fields of their preceding nearest-neighbours in order to match that of the leading bubble. The trailing bubbles can be arranged in any order and, hence, the number of stable formations increases factorially as the number of bubbles is increased.

physics.flu-dyn↗