Schematic of the experimental set-up (the shape of the interface is illustrative only).
Density-unstable and density-stable displacements at β = 70° (see Figure 1 for definition) and . (a) At = 0.0035 (dark fluid is less dense than the lighter coloured fluid) and mean velocity mm/s; the sequence starts 16.25 s after opening the gate valve and the time interval between images is 2.5 s. (b) At = −0.0035 (dark fluid is more dense than the lighter coloured fluid) and mm/s; the sequence starts 21.5 s after opening the gate valve and the time interval between images is 3.25 s. The field of view is 430 × 19 mm2, taken 800 mm below the gate valve. The last image at the bottom of (a) shows the gray scale for the concentration values, which can also be used for (b). The arrows indicate the flow direction and the left end of the tube is higher than the right (the fluid flows downhill).
(a,b) Spatiotemporal diagrams obtained for the same experiments as Figures 2(a) and 2(b). (c) The spatiotemporal diagram of an iso-density experiment (At = 0) for mm/s. The dashed line is an eye guide representing . Note that is the streamwise distance measured from the gate valve and is the time spent after the start of the experiment. The colour bar in (a) refers to all figures and indicates depth-averaged concentration values with 0 indicating pure displaced fluid and 1 indicating pure displacing liquid.
(a) Experimental profiles of normalized for s for the same experiments as Figures 2(b) and 3(b). The dotted lines show and 0.15. The horizontal solid line indicates and the inclined bold dashed line (eye guide) shows the slope of the collapsed curves which stays the same for different profiles. (b) Collapse of the normalized profiles with for the same experiment as part (a). (c) The evolution of stretch length value, , with time for the same experiment as part (a). The fully developed stretch length is attained for approximately s. The inset in (c) shows the dependency of the scaled stretch length for different choices of lower and upper bounds of (0.15–0.85, 0.1–0.9, and 0.2–0.8).
The dependency of the dimensionless time taken to reach fully developed flow, t FD , on (a) χ and (b) Fr. The data in parts (a) and (b) correspond to: (■); (▶); (*); (▲); (▼); (•).
Front velocity values, , plotted against the mean flow velocity, for different experiments as in Figure 5. The dashed-dotted line is the linear fit . In this case an average displacement efficiency for all density stable experiments is 1/1.04 ∼ 96%. The solid line represents the ideal displacement case with .
Experimental profiles of at different times but at around the same position in the pipe for β = 60° and ; (a) At = −0.0035 different ; (b) similar and different At < 0. All experiments are density stable.
(a) Variations in dimensional stretch length , for different β and . The solid symbols are for At = −0.0035 and the open symbols for At = −0.01. Note that . (b) Dimensional stretch length versus V 0 for two different dynamic viscosities at fixed At = −0.0035 and β = 20°.
Snapshots of the vertical displacement flow experiment, β = 0°, carried for , At = −0.0035 and (a) mm/s, (b) mm/s, and (c) mm/s. The sequence for part a starts 30.75 s after opening the gate valve and the time interval between images is 5 s. For parts (b) and (c) the sequences start 80.75 s and 19.5 s after opening the gate valve and the time intervals between images are 2 s and 1.25 s, respectively. The arrow indicates the flow direction and the left end of the tube is higher than the right (the fluid flows vertically downhill). The field of view is 700 × 19 mm2, taken 1550 mm below the gate valve. The last image at the bottom of (a) is the colourbar of the concentration values and can be used for (b) and (c) too. The white dashed lines in (a)-(c) are eye guide and indicate the boundaries of the mixed region moving upfront.
(a) Experimental dimensionless stretch length values L, plotted against χ. The solid line is L − tan β = −3680/χ; (b) the same data plotted in log-log scale. The data in parts (a) and (b) correspond to: (■); (▶); (*); (▲); (▼); (•).
Parameter range of our experimental study.
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