Bubble propagation in Hele-Shaw channels with centred constrictions

Abstract

We study the propagation of finite bubbles in a Hele-Shaw channel, where a centred occlusion (termed a rail) is introduced to provide a small axially-uniform depth constriction. For bubbles wide enough to span the channel, the system's behaviour is similar to that of semi-infinite fingers and a symmetric static solution is stable. Here, we focus on smaller bubbles, in which case the symmetric static solution is unstable and the static bubble is displaced towards one of the deeper regions of the channel on either side of the rail. Using a combination of experiments and numerical simulations of a depth-averaged model, we show that a bubble propagating axially due to a small imposed flow rate can be stabilised in a steady symmetric mode centred on the rail through a subtle interaction between stabilising viscous forces and destabilising surface tension forces. However, for sufficiently large capillary numbers Ca, the ratio of viscous to surface tension forces, viscous forces in turn become destabilising thus returning the bubble to an off-centred propagation regime. With decreasing bubble size, the range of Ca for which steady centred propagation is stable decreases, and eventually vanishes through the coalescence of two supercritical pitchfork bifurcations. The depth-averaged model is found to accurately predict all the steady modes of propagation observed experimentally, and provides a comprehensive picture of the underlying steady bifurcation structure. However, for sufficiently large imposed flow rates, we find that initially centred bubbles do not converge onto a steady mode of propagation. Instead they transiently explore weakly unstable steady modes, an evolution which results in their break-up and eventual settling into a steady propagating state of changed topology.

The influence of invariant solutions on the transient behaviour of an air bubble in a Hele-Shaw channel

“…Additionally we define the aspect ratio of the channel α = W * /H * . Details of the derivation of the depth-averaged equations from the Navier-Stokes equations can be found in [8,10]. The critical assumptions are that the channel aspect ratio α is large; the reduced Reynolds number U * 0 W * /(ρµα 2 ) and Bond number Bo = ρgL * 2 /γ are small; the bubble occupies the full height of the channel; and the component of curvature as z varies corresponds to a semi-circle filling the channel height with the fluid perfectly wetting the upper and lower walls.…”

“…The critical assumptions are that the channel aspect ratio α is large; the reduced Reynolds number U * 0 W * /(ρµα 2 ) and Bond number Bo = ρgL * 2 /γ are small; the bubble occupies the full height of the channel; and the component of curvature as z varies corresponds to a semi-circle filling the channel height with the fluid perfectly wetting the upper and lower walls. We neglect the thin film corrections proposed by Homsy [22] and Reinelt [23], because they do not change the qualitative comparison with the experiment, and it is not obvious how they should be modified due to the presence of the depth-perturbation and multiple solutions [8][9][10]. The nondimensional domain is shown in figure 3.…”

“…A typical example of this smoothed profile is shown in figure 3 [24]. Details of the implementation can be found in [8] for the case of a propagating air-finger and [10] for a finite air bubble. It is convenient to keep h, w and α constant and analyse the system mainly through variations in Q and V , which are easily manipulated in a physical, experimental set-up.…”

“…A summary of the qualitative behaviour is shown in figure 5, with regions of parameter space classified according to the characteristic shape first adopted by the bubble. As Q and V are varied, Figure 10 of [10], classifying the fate of circular bubbles with static diameter, 2r and non dimensional flow-rate Q = µU * 0 /γ according to whether they develop one, two, three or four tips before the bubble has either broken up (hollow symbols) or remains in a steady mode of propagation (filled symbols, only at the lowest flow rate). This data is from experiments with α = 40, h = 0.024 and w = 0.…”

“…We study the behaviour theoretically by finding invariant solutions, namely steady states and periodic orbits, of the system. We pursue the idea, hypothesised by Franco-Gómez et al [10], that when the flux is large enough the complex time-dependent behaviour of a single bubble can be interpreted as a transient exploration of the stable manifolds of weakly unstable edge states. We consider only the dynamics before the changes in topology when the bubble breaks up into two or more separate bubbles: systems with their own dynamics that we do not pursue here.…”