Schematic diagram of the experimental setup. The end view shows the cross-section of the tube, which is rectangular with a centred step-change in depth. The dimensions of the cross-section of the flow tube are: H = 3.07 ± 0.01 mm, 3 ⩽ W ⩽ 45 mm, w = 4.49 ± 0.01 mm, h = 1.09 ± 0.01 mm or h = 1.50 ± 0.02 mm. The side view illustrates how liquid is withdrawn at constant flow rate Q from one end of the initially liquid-filled tube, while the other end of the tube is open to the atmosphere.
Dependence of the finger tip offset from the tube centreline on Ca for α w = 1/4, α h = 0.49, and α = 6.0. The value of the offset δ was measured 50 mm behind the finger tip, as shown schematically. Inset top-views are shown of the main finger types: symmetric (solid black symbols), asymmetric non-localised (white, (*) and (+) symbols), and localised state (solid light grey symbols). Both localised and symmetric fingers are stable for very low values of Ca. The symbols distinguish the perturbations applied to the propagating finger in order to reach the different states: (○) experiments start from rest, (△) experiments start at a high flow rate that is then reduced. A localised finger was formed as a static initial condition (□) by driving a finger at high flow rate before interrupting the flow, and then restarting the experiment at the desired value of Ca. The other perturbations were applied by blocking parts of the cross-section over a short distance at the inlet of the tube: (▷) one side-channel and region over obstacle blocked, (+) only region over obstacle blocked, or (*) only one side-channel blocked.
Wet fraction m as a function of the capillary number Ca for α h = 0.49 and different values of α w . Black markers: experiments start from rest; grey markers: experiments start at a high flow rate that is then reduced. Markers with a white cross correspond to an oscillatory state: (■) α w = 2/5, (✦) α w = 1/3, (●) α w = 2/7, (▼) α w = 1/4, (★) α w = 1/5, (▲) α w = 1/8. The arrow indicates the direction of increasing α w and illustrative finger shapes are shown as insets. The data for α = 2/7 are explicitly marked because it divides regions of different qualitative behaviour: for α < 2/7, the system does not return to the symmetric state when Ca is reduced from an initially high flow rate. The data for α w = 2/5 and α w = 1/3 were previously presented in Ref. 7 .
Oscillatory pattern “shed” behind a finger propagating under constant flow rate from right to left for α h = 0.49: (a) α w = 1/8, Ca = 3.69 × 10−2, (b) α w = 1/8, Ca ≃ 6.8 × 10−2, (c) α w = 1/3, Ca = 5.74 × 10−3, (d) α w = 1/3, Ca = 7.14 × 10−3. The field of view is 400 mm long. Despite the occurrence of oscillations at very different values of Ca, the mechanism underlying their formation is similar in narrow and wide tubes. In particular, the spatial wavelength of the pattern decreases in both cases with increasing Ca, as discussed by Pailha et al. 7 in the case of narrow tubes. Figures 4(c) and 4(d) are reprinted with permission from Phys. Fluids24, 021702 (Year: 2012)10.1063/1.3682772. .
Schematic of a three-dimensional air finger propagating from right to left in an axially uniform tube with a partially occluded rectangular cross-section. Behind the finger tip, the finger becomes axially uniform and an illustrative cross-section is shown in white. The remaining thin liquid films are essentially at rest and the majority of the viscous stress on the interface occurs near the tip region.
Sketches of symmetric static equilibria in tubes of rectangular cross-section in the absence of gravity: (a) unoccluded, (b) low occlusion, and (c) high occlusion.
Sketches of asymmetric static equilibria in tubes of rectangular cross-section in the absence of gravity: (a) high occlusion, (b) medium occlusion, and (c) low occlusion.
Static prediction of the existence of asymmetric localised fingers in terms of the relative obstacle size α h and α w . The experimental tube had H = 3.07 ± 0.01 mm and Bo = 1.058 ± 0.008. The solid line shows SE computations for H = 3.06 ± 0.01 mm and Bo = 1.050, while the almost indistinguishable dashed line is for H = 3.08 ± 0.01 mm and Bo = 1.065. Circles indicate tube geometries where localised fingers are stable when the flow is suddenly stopped (Ca = 0), while squares correspond to tubes where localised fingers are unstable at Ca = 0. The dotted-dashed line is the analytic prediction in the absence of gravity (Bo = 0) corresponding to Eq. (A6) .
Comparison between two-dimensional SE calculations of cross-sectional finger shapes and experiments for α h = 0.35 and α w = 1/3. Experimental results were obtained with silicone oils of different dynamic viscosities: circles μ = 4.81 × 10−3 kg m−1s−1 and squares μ = 9.7 × 10−2 kg m−1s−1. The two-dimensional static solutions, shown as the solid line (green), are in excellent agreement with the experiments, suggesting that the loss of stability of the symmetric finger can be predicted from existence of the non-localised asymmetric state that forms behind the finger tip in the SE calculations. The value of δ was measured at a distance of 45 mm behind the finger tip.
Comparison between two-dimensional SE calculations of cross-sectional finger shapes and experiments for α h = 0.49 and α w = 1/4. Experimental results (circles) were obtained with silicone oil of dynamic viscosity μ = 4.81 × 10−3 kg m−1s−1. The localised solution occurs at the point m ≈ 0.6, δ ≈ 0.3 and is predicted accurately by the static calculations (solid line and dots, green). The broadening of the finger is qualitatively captured by the static calculations, with good agreement once the finger is detached from both side walls, but the exact details of the evolution are different in the simulations and experiments. The value of δ was measured at a distance of 45 mm behind the finger tip.
The geometry of an asymmetric static bubble with maximum curvature in a tube with partially occluded cross-section. The half-height of the channel is chosen to be the reference length-scale and is set to 1.
Wet fractions at Ca = 0 computed using SE corresponding to the experimental data for α h = 0.49 presented in Fig. 3 . The experimental values correspond to straight line extrapolation of the two data points with smallest Ca to the value Ca = 0.
Wet fractions computed using SE for the proposed limiting steady asymmetric fingers for different tube widths corresponding to the experimental data presented in Fig. 3 . The experimental values correspond to the steady (non-oscillatory) asymmetric solution with minimum wet fraction. Note that when α w = 2/7 the data point with m = 0.33 is not chosen because it is believed to correspond to an oscillatory solution.
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