2 Overview

The research undertaken by Ducloué et al. (Reference Ducloué, Hazel, Pihler-Puzović and Juel2017a ) was preceded by some key influential studies. The idea of controlling viscous fingering in a Hele-Shaw flow by manipulating time-dependent experimental inputs (such as injection rate or gap thickness) has been explored in some detail (Li et al. Reference Li, Lowengrub, Fontana and Palffy-Muhoray2009). Further control strategies were developed by altering the physical apparatus itself; for example, Al-Housseiny et al. (Reference Al-Housseiny, Tsai and Stone2012) constructed a tapered Hele-Shaw cell and showed that a negative gradient in depth could inhibit the normal fingering instability for sufficiently low capillary numbers. Similar results were obtained by replacing the Hele-Shaw plates with elastic membranes, as locally the flow is analogous to that in a tapered cell (Pihler-Puzović et al. Reference Pihler-Puzović, Illien, Heil and Juel2012; Al-Housseiny et al. Reference Al-Housseiny, Christov and Stone2013), and so the geometric mechanism for suppressing the instability is the same.

The experiments of Pihler-Puzović et al. (Reference Pihler-Puzović, Illien, Heil and Juel2012) in a radial geometry motivated Ducloué et al. (Reference Ducloué, Hazel, Pihler-Puzović and Juel2017a ) to study a steadily moving front in a converging Hele-Shaw channel, since such a set-up allows for a more controlled study of the unique fingering patterns at the decelerating interface observed by Pihler-Puzović et al. (Reference Pihler-Puzović, Illien, Heil and Juel2012). The first stage was achieved by the authors in their very recent paper (Ducloué et al. Reference Ducloué, Hazel, Thompson and Juel2017b ), where they investigated flow through an initially collapsed channel via a Hele-Shaw cell with an elastic upper boundary. This type of flow geometry is interesting as it has depth gradients in the axial direction and exhibits similar features to that observed when peeling off a strip of viscous adhesive. However, in order to achieve the ideal conditions for studying short, flat-tipped, viscous fingers, one final modification was required, namely, the inclusion of a solid uniform rectangular block that runs along the length of the Hele-Shaw channel, including transverse depth gradients. Ducloué et al. (Reference Ducloué, Hazel, Pihler-Puzović and Juel2017a ) refer to this block as an occlusion.

The experimental set-up used by Ducloué et al. (Reference Ducloué, Hazel, Pihler-Puzović and Juel2017a ) is illustrated in figure 1(a), with the cross-section of the occluded channel shown in figure 1(b). The width of the shallow central layer of the flow is referred to as the peeling region; its dimensions are controlled by the width of the block. The deep side channels are required in order to maintain a steadily moving almost-flat front.

With these ingredients, Ducloué et al. (Reference Ducloué, Hazel, Pihler-Puzović and Juel2017a ) were able to exploit the induced axial and transverse depth gradients to generate a range of fingering patterns along the almost-flat peeling front, examples of which are included in figure 1(c–k). These almost-steady fingers have a constant length $l$ and become narrower and shorter as the capillary number increases. Despite the complexity of the underlying flow, the authors determined that the finger length $l$ is inversely proportional to the tangent of the elastic wall’s inclination angle; this remarkably simple relationship suggests that, for a given fluid layer thickness, the fingers extend to a constant reopening height. Interestingly, these properties appear analogous to those observed in the viscous adhesive problem as well as the so-called printer’s instability (Rabaud et al. Reference Rabaud, Michalland and Couder1990). Such short, flat-tipped, viscous fingers are very different to those normally observed in a Hele-Shaw flow, and in fact look more like molar teeth than fingers!

Other striking fingering patterns are also observed. For example, the flow regimes illustrated in figure 2 do not give rise to a flat peeling front, but instead a curve front emerges. Tip-splitting instabilities appear to drive periodic fingering along the sides of the single long finger (figure 2 a,b). For a very thin rectangular block, dendritic patterns develop (figure 2 c), with very small fingers at the very tip of the long finger. The possibilities for further families of fingering patterns appear endless.

3 Future

By extending the work of Ducloué et al. (Reference Ducloué, Hazel, Pihler-Puzović and Juel2017a ), there is a plethora of exciting opportunities for studying the connections between various types of viscous fingering patterns. One immediate direction is to explore further the transition from the printer’s instability to Saffman–Taylor fingering. Another is to expand the experimental set-up to allow for a broader range of fluid–structure interactions with flexible plates that have more complex constitutive properties than elastic membranes. More generally, there is a need to connect all previously observed viscous fingering patterns, for example via a phase diagram that demonstrates a dependence on geometry, membrane flexibility and flow rate.

Finally, from a modelling perspective, studying viscous fingering in a Hele-Shaw cell provides a number of mathematical challenges. For flow configurations of the type explored in Ducloué et al. (Reference Ducloué, Hazel, Pihler-Puzović and Juel2017a ) with elastic membranes and/or occlusions, analytical insight may be sought via depth-averaged models based on the lubrication approximation (Al-Housseiny et al. Reference Al-Housseiny, Christov and Stone2013; Peng et al. Reference Peng, Pihler-Puzović, Juel, Heil and Lister2015).

Figure 2.

The wide variety of interfacial patterns generated by Ducloué et al. (Reference Ducloué, Hazel, Pihler-Puzović and Juel2017a ) include spatially periodic sideways oscillations (a,b) and dendritic-like patterns (c).