Abstract
We hypothesize that dynamical systems concepts used to study the transition to turbulence in shear flows are applicable to other transition phenomena in fluid mechanics. In this paper, we consider a finite air bubble that propagates within a Hele-Shaw channel containing a depth-perturbation. Recent experiments revealed that the bubble shape becomes more complex, quantified by an increasing number of transient bubble tips, with increasing flow rate. Eventually, the bubble changes topology, breaking into multiple distinct entities with non-trivial dynamics. We demonstrate that qualitatively similar behaviour to the experiments is exhibited by a previously established, depth-averaged mathematical model and arises from the model’s intricate solution structure. For the bubble volumes studied, a stable asymmetric bubble exists for all flow rates of interest, while a second stable solution branch develops above a critical flow rate and transitions between symmetric and asymmetric shapes. The region of bistability is bounded by two Hopf bifurcations on the second branch. By developing a method for a numerical weakly nonlinear stability analysis we show that unstable periodic orbits (UPOs) emanate from the first Hopf bifurcation. Moreover, as has been found in shear flows, the UPOs are edge states that influence the transient behaviour of the system.
Highlights
A Hele-Shaw channel consists of two parallel glass plates, separated by a distance much smaller than the width of the channel
For higher flow rates in large aspect ratio channels, propagating air fingers develop complex patterns via multiple tip splitting events as well as side-branching [2,3]. The onset of this complex interfacial dynamics appears to be a subcritical transition, a feature that it shares with other systems including the buckling of elastica and the transition to turbulence in shear flows
The bubble will eventually reach a stable asymmetric state on one side of the depth perturbation; but if the volume flux exceeds a critical threshold the bubble shape becomes increasingly deformed before eventually breaking up into two or more distinct parts, as sketched in figure 2b, despite theoretical predictions that a stable steady state exists for all flow rates, a feature preserved from the unperturbed Hele-Shaw channel
Summary
A Hele-Shaw channel consists of two parallel glass plates, separated by a distance much smaller than the width of the channel. The bubble will eventually reach a stable asymmetric state on one side (or the other) of the depth perturbation; but if the volume flux exceeds a critical threshold the bubble shape becomes increasingly deformed before eventually breaking up into two or more distinct parts, as sketched, despite theoretical predictions that a stable steady state exists for all flow rates, a feature preserved from the unperturbed Hele-Shaw channel This model system allows us to explore whether the dynamical systems concepts applied in the study of transition to turbulence in shear flows apply to the transient behaviour of other canonical problems in fluid mechanics.
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