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All figures (17)
FIGURE 2. (Colour online) Sketch of the experimental set-up. See Enríquez et al. (2013) for a detailed description.
FIGURE 10. Contour lines of the tangent-sphere η, ξ coordinates, plotted in the y = 0 (φ = 0) Cartesian plane. η = 0 lies on the z-axis, η → ∞ at the contact point. The horizontal wall lies on the ξ = 0 isosurface, while the bubble surface is always mapped by ξ = 1. The separation of the plotted contours is uniform (1η=1ξ = 0.1).
FIGURE 8. (Colour online) Bubble snapshots at both extremes of the bubble size range measured during our experiments. The largest radius is (a) R = 358 µm, corresponding to the maximum radius attained during experiment 1 (see later figure 4), whereas (b) R = 92 µm is the smallest radius, obtained during the dissolution experiment 4 (see figure 7). The radius is computed by means of the light-blue circumference fitted to the bubble contour. The horizontal red line marks the height of the bubble–substrate contact line, below which there is the reflection of the bubble on the substrate surface.
FIGURE 9. Sketch of a spherical CO2 bubble adhered to a flat plate. The relevant parameters and functions used in the formulation of the mass transfer problem are also indicated.
FIGURE 14. (Colour online) Dimensionless mass flux across the bubble interface as a function of the angle θ for different times during the first growth stage of experiment 2. The step-like markers indicate the angle θ∗ delimiting the effective bubble area available for mass transfer, where cos(θ∗)=−a/(a+ √ πτ) according to Enríquez et al. (2014).
FIGURE 15. (Colour online) Simulation snapshots for solution (ii) of experiment 1, in which natural convection has been neglected. The snapshots are taken at τ = 3.5, corresponding to the first growth stage (see figure 11a). These show (a) the velocity field (arrows) and streamlines, (b) the vorticity field contours and (c) the concentration field contours. The darker vorticity and concentration contours (from pink to green to dark blue) indicate positively increasing values.
TABLE 1. Values of the parameters used in the simulations, corresponding to the experiments discussed in § 2. For completeness, the Schmidt number is Sc= ν/Dm = 523.
FIGURE 3. Experimental procedure during which the bubble is exposed to two identical supersaturation–undersaturation cycles. The lower plot shows the pressure that the bubble is subjected to, whilst the upper one illustrates its radius time evolution. In brief, once a bubble is stabilised at a given radius Ri, it is forced to a growth cycle for a prescribed time T and then to a dissolution cycle down to a size slightly smaller than Ri (t = t3a), such that a short time Ts after the pressure returns to the initial level P0 (t = t3b), the combination of previous gas expansion (during t3a–t3b) and history lead the bubble size to the initial radius (t = t4). An identical pressure cycle is immediately imposed, which results in a different time evolution of the bubble radius due to the history effect.
FIGURE 11. (Colour online) Time evolution of the dimensionless bubble radius, a(τ ). Simulation (black curves) is compared to experimental data (markers) for experiments (a) 1, (b) 2, (c) 3 and (d) 4. The time evolution of the dimensionless ambient pressure, p(τ ), is also included (grey curve). The different simulation curves correspond to (i), (iv) the full solution, (ii) solution where density-induced convection is neglected, (iii) solution for pure diffusion (both density-induced convection and boundary-induced advection are neglected). Moreover, case (iv) is only used to model experiment 4 as seen in (d), corresponding to the full solution coupled with the CO2 stratification model, with Υwall = 1.25.
FIGURE 7. (Colour online) Results for experiment 4 (see caption of figure 4). This time, the direction of the pressure jumps is inverted thereby replacing the two growth stages observed in experiments 1–3 with two dissolution stages.
FIGURE 4. (Colour online) Results for experiment 1 showing the time-histories of (a) the measured bubble radius R in response to (b) the imposed pressure P∞(t). In (c), the rate of growth of the ambient radius R0, defined in (1.1), is plotted for the two growth cycles. The time axis is initialised at t1 or t4 accordingly.
FIGURE 1. Dissolution of a CO2 spherical-cap bubble tangent to a flat chip immersed in a CO2-water solution under pressurised conditions (see later figure 9). The bubble is subjected to (b) a pressure jump P∞(t), from P∞(0) = P0 = 7.4 bar to 6.5 bar. Both pressures are above the saturation pressure, Psat = 6.1 bar (according to simulation). Panel (a) shows the evolution in time of the measured bubble radius R(t) (white markers) and ambient radius R0(t) (dark markers). The former is compared to simulation, which in addition was employed to depict (c) the concentration profile along the z-axis above the bubble at three different instants in time. The employed experimental and numerical techniques are detailed in the main text.
FIGURE 16. (Colour online) Simulation snapshots for solution (i) of experiment 1, which takes into account natural convection. The snapshots of (a) the velocity field and streamlines, (b) the vorticity field contours and (c) the concentration field contours are taken at τ = 3.5, corresponding to the first growth stage (see figure 11a). Snapshots (d–f ) show the same fields as (a–c) above, but are taken at τ = 9.5, corresponding to the first shrinkage stage. The darker vorticity and concentration contours (from pink to green to dark blue) indicate positively increasing values. The thick black contour lines in (b) and (e) mark the zero-vorticity contour. Despite the significant changes that natural convection induces in the velocity field, its influence on the concentration field in the vicinity of the bubble is minute, as is revealed by the comparison of the isoconcentration lines with and without convection (panel c versus figure 15c).
FIGURE 12. Sketch of the ‘effective’ far-field concentration as a function of the instantaneous bubble radius as a means to model the effect of stratification on the mass transfer rate across the bubble interface.
FIGURE 13. (Colour online) Dimensionless concentration field contours for experiment 2 according to simulation at the instant of time when the bubble radius is a= 1.05 during the (a) first growth stage and (b) second growth stage. The darker concentration contours (from pink to green to dark blue) indicate positively increasing values.
FIGURE 5. (Colour online) Results for experiment 2 (see caption of figure 4). The range of pressures is slightly different to the ones exposed in figure 4. However, the history effect is repeatable.
FIGURE 6. (Colour online) Results for experiment 3 (see caption of figure 4).
Journal Article
•
DOI
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The history effect on bubble growth and dissolution. Part 2. Experiments and simulations of a spherical bubble attached to a horizontal flat plate
[...]
Pablo Peñas-López
1
,
Álvaro Moreno Soto
2
,
Miguel A. Parrales
3
,
Devaraj van der Meer
2
,
Detlef Lohse
2
,
Javier Rodríguez-Rodríguez
1
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+2 more
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Institutions (3)
Charles III University of Madrid
1
,
University of Twente
2
,
Technical University of Madrid
3
10 Jun 2017
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Journal of Fluid Mechanics