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.

We demonstrate experimentally that the introduction of a rail, a small height constriction, within the cross-section of a rectangular channel could be used as a robust passive sorting device in two-phase fluid flows. Single air bubbles carried within silicone oil are generally transported on one side of the rail. However, for flow rates marginally larger than a critical value, a narrow band of bubble sizes can propagate (stably) over the rail, while bubbles of other sizes segregate to the side of the rail. The width of this band of bubble sizes increases with flow rate and the size of the most stable bubble can be tuned by varying the rail width. We present a complementary theoretical analysis based on a depth-averaged theory, which is in qualitative agreement with the experiments. The theoretical study reveals that the mechanism relies on a non-trivial interaction between capillary and viscous forces that is fully dynamic, rather than being a simple modification of capillary static solutions.

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.

We investigate the effect of highly contrasting non-Newtonian liquid properties on the formation of liquid jets with a focused shape. By using two nozzle-free ejection techniques, mechanically impact- and laser-induced, fast jets of a highly elastic (sodium polyacrylate) and weakly elastic (xanthan-gum) diluted polymer solutions are generated. A unique high-speed effect is encountered at the jet ejection onset of the highly elastic solution. Its jet-tip speed is on average 1.4 times faster in comparison to a Newtonian (glycerin/water) and the weakly elastic liquids. We explain this effect occurring as a result of the high viscoelasticity of the sodium polyacrylate solution. Additionally, a ‘bungee jumper’ jet behaviour (Morrison and Harlen in Rheol Acta 49(6):619–632, 2010) is observed in a regime of high speed ($$10<V_j<40$$ 10 < V j < 40 m/s) and high viscosity ($$\mu >20$$ μ > 20 mPa s) not previously examined. We additionally characterise the viscoelastic non-breakup jet limit using the Bazilevskii et al. (Fluid Dyn 40(3):376–392, 2005) ejection criterion. Herein, the extensional rheological parameters are measured implementing a novel DoS-CaBER technique (Dinic et al. in Lab Chip 17(3):460–473, 2017). Our findings may influence results of inkjet printing technologies and recent nozzle-free ejection systems for ejecting liquids with non-Newtonian properties. Graphical abstract

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