Sensitivity of Saffman–Taylor fingers to channel-depth perturbations

Abstract

We examine the sensitivity of Saffman-Taylor fingers to controlled variations in channel depth by investigating the effects of centred, rectangular occlusions in Hele-Shaw channels. For large occlusions, the geometry is known to support symmetric, asymmetric and oscillatory propagation states when air displaces a more viscous fluid from within the channel. A previously developed depth-averaged model is found to be in quantitative agreement with laboratory experiments once the aspect ratio (width/height) of the tube's cross-section reaches a value of 40. We find that the multiplicity of solutions at finite occlusion heights arises through interactions of the single stable and multiple unstable solutions already present in the absence of the occlusion: the classic Saffman-Taylor viscous fingering problem. The sequence of interactions that occurs with increasing occlusion height is the same for all aspect ratios investigated, but the occlusion height required for each interaction decreases with increasing aspect ratio. Thus, the system becomes more sensitive as the aspect ratio increases in the sense that multiple solutions are provoked for smaller relative depth changes. We estimate that the required depth changes become of the same order as the typical roughnesses of the experimental system (1 µm) for aspect ratios beyond 155, which we conjecture underlies the extreme sensitivity of experiments conducted in such Hele-Shaw channels.

Selection of a Hele-Shaw Bubble via Exponential Asymptotics

“…If we repeat this analysis for the contribution associated with the singularity at ζ = −i/q, we can determine the equivalent late-order terms and Stokes switching behaviour associated with χ L . In this case, we find that F (ζ) is also given by (20), and that the Stokes switching occurs across a different Stokes line, satisfying Im[χ L ] = 0 with the condition that Re[χ L ] = 0. This curve is also studied in more detail in Section 5, and it is found that the contribution is switched as the curve is crossed from the negative side to the positive side.…”

Dynamics of front propagation in a compliant channel

“…As a result, the effects of the compliance-induced taper are diminished and the flow is dominated by viscous dissipation within the wedge [see figure 4(c)]. The asymmetric fingers at low Ca arise because of the central constriction of the channel ahead of the interface, which is associated with the quartic profile of the collapsed membrane in the transverse cross-section (Ducloué et al 2017b); asymmetric fingers minimise viscous dissipation in rigid channels of similar geometry (Franco-Gómez et al 2016). The contrast between peeling modes is further illustrated in figure 4(c); here the total pressure drop within the oil is given by the difference between the bubble pressure p B and the hydrostatic pressure head p HS far ahead of the interface, which sets the initial collapse.…”

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

“…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.…”

“…Figure 2(a) shows the bubble shape and speed for the first three solution branches in channels with a rectangular cross-section. Introducing a depth-perturbation to the system (see sketch in figure 1) allows the solution branches to interact, which results in the diverse range of observed stable steady states and time-dependent behaviour [4][5][6][7][8][9][10][11].…”