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Miscible density-unstable displacement flows in inclined tube
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Image of FIG. 1.
FIG. 1.

(a) Schematic view of experimental set-up (interface shape is illustrative only). (b) Change in iso-viscous displacement mixing with β for = 0.0035 and kinematic viscosity . Mean flow speed is (see Fig. 3 ). The field of view is 1330 × 19 mm located 1740  mm downstream of the gate valve. The color bar at the top left of the figure shows the corresponding concentration value, , with 0 referring to the pure displaced fluid and 1 to the pure displacing fluid.

Image of FIG. 2.
FIG. 2.

Comparison of the experimental front velocity values for the exchange flow (solid data) with the data reported by Seon (open data). The solid line shows . Squares correspond to = 0.001 and circles to = 0.01. Inclination angle, β varies between 20° and 70° in the experiments shown. The dashed lines show the limit of above which the exchange front velocity values deviate from , for corresponding ( for = 0.001, and = 0.01, respectively).

Image of FIG. 3.
FIG. 3.

Spatio-temporal diagrams of depth-averaged concentration field, for the same experiments as shown in Figure 1(b) : (a) β = 0°, ; (b) β = 30°, ; (c) β = 45°, ; (d) β = 60°, ; (e) β = 70°, ; (f) β = 85°, . The dashed line in (f) indicates the position of the displacing front and its slope is . The spatiotemporal diagram gives a front speed .

Image of FIG. 4.
FIG. 4.

Ultrasound Doppler Velocimetry (UDV) measurements obtained for = 0.004, and (a) β = 0°, , (b) β = 60°, , and (c) β = 85°, .

Image of FIG. 5.
FIG. 5.

(a) Evolution of the depth-averaged concentration field, with time, s, and streamwise location, , measured form the gate valve for the same experiment as in Figures 1(b) (β = 30°) and 3(b) . The dashed line shows = 0.1 which is used for measuring the displacing front velocity, , consistently. (b) Evolution of the front velocity value, , with time for the same experiment. The inset shows the constant value of the front velocity when the flow is fully developed ( in this case with 2% standard deviation).

Image of FIG. 6.
FIG. 6.

Change in displacing front velocity, with tilt angle, β, and imposed velocity, for and (a) = 0.001, (b) = 0.0035, and (c) = 0.01. Different markers represent (▷), 10 (◀), 20 (▲), 40 (*), 60 (•), 80 (▼), and 100  mm/s (■). The dashed lines are guide to the eye plotted at

Image of FIG. 7.
FIG. 7.

(a) Presentation of our results for the full range of experiments: normalized front velocity , plotted against and  cos β/. The rectangular area (0 <  cos β/ < 120, 0 < < 6) indicated by dashed lines, locates the range of nearly horizontal experiments and is studied in full details in Ref. . (b) Instantaneous displacement flows marked by the superposed circles plotted against and  cos β/. The heavy line represents the prediction of the lubrication model for the stationary interface. The dashed lines  cos β/ = 650 and = 2 roughly indicate the boundary between instantaneous and non-instantaneous displacement. Normalized front velocity scale is limited to 2 in Figure 7(b) to better show the variations of the front velocity in the plane of and  cos β/.

Image of FIG. 8.
FIG. 8.

Collapse of depth-averaged concentration profiles with for and (a) β = 45°, , (b) β = 20°, , (c) β = 45°, , and (d) β = 85°, . In (a) = 0.08 and in (b)–(d) = 0.0035. The solid lines in (a) and (b) show the error-function fits with , respectively. The upper right insets show the qualitative flow pattern in each case. In (d) the lower left inset shows the collapse of the concentration profiles when is used instead of .

Image of FIG. 9.
FIG. 9.

Variation of the dimensionless diffusion coefficient versus tilt angle. The circles are based on current measurements and the crosses are added from the experiments of Seon for comparison. The curve fit shown is also suggested by Seon which is Eq. (3) .

Image of FIG. 10.
FIG. 10.

Variation of the diffusion coefficient, versus mean imposed flow velocity, for and (a) = 0.0035 and (b) = 0.01.

Image of FIG. 11.
FIG. 11.

Contours of the normalized diffusion coefficient (a) and (b) versus and  cos β/.

Image of FIG. 12.
FIG. 12.

Reynolds number, , plotted against and  cos β/. The illustrative curves correspond to different inclination angles, β, and are show = = 2100. Flows marked by the superposed triangles and/or squares satisfy the criterion > 2100. The symbols used are for β = 70° (△), β = 60° (▽), β = 45° (▷), β = 30° (◁), and β = 0° (□).

Image of FIG. 13.
FIG. 13.

Classification of our results for the full range of experiments, presented in the (,  cos β/)-plane: (i) instantaneous displacement flows are colored in blue and non-instantaneous flows in red; (ii) fully diffusive flows have no superposed symbol; (iii) non-diffusive flows are marked as viscous (superposed circles) or inertial (superposed squares). The heavy line represents the prediction of viscous backflows, from the lubrication model in Ref. , (χ = χ = 116.32). The thick broken line represents  cos β/ = −50 + 500. The point of intersection of the two lines is ≈ 4.62 and  cos β/ ≈ 270.

Image of FIG. 14.
FIG. 14.

(a) Comparison of the front velocity values obtained from experiments classified as viscous with the predictions (χ) from the lubrication model in Refs. and (thick solid line). Color values indicate  cos β/ for each experiment. The data with the solid boundary are for near-horizontal β, taken from Ref. . The thick white circle indicates the theoretical transition (χ = χ ≈ 116.32). The thin solid line shows below which leading front velocities are not possible due to conservation of mass (b) comparison of the experimental and predicted values of , for intermittent flows with the predictions following (6) . The dashed line in Figure 14(b) indicates . The contour values in both figures show the corresponding  cos β/ to each experiment.

Image of FIG. 15.
FIG. 15.

(a) Comparison of the experimental macroscopic diffusion diffusion coefficient, , against the prediction of (7) . The broken line shows . Meanings of the symbols are the same as in Figure 10 . (b) Front velocity values, , plotted against the mean flow velocity, for all fully diffusive experiments shown in Figure 13 . The color values show the corresponding  cos β/. The solid line is the linear fit . The broken line represents .


Generic image for table
Table I.

Parameter range of our experimental study. Note that most of the experiments were run for = 0.001, 0.0035, and 0.01.


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Miscible density-unstable displacement flows in inclined tube