1 research outputs found
The influence of invariant solutions on the transient behaviour of an air bubble in a Hele-Shaw channel
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 two or more 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; a consequence of the model's intricate
solution structure. For the bubble volumes studied, a stable asymmetric bubble
exists for all flow rates of interest, whilst 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
emanate from the Hopf bifurcation at the lower flow rate and, moreover, that
these orbits are edge states that influence the transient behaviour of the
system