(a) Plan view of the Hele–Shaw cell and (b) side view of the complete experimental setup (not to scale).
The red (▲), green ( ▼), and blue (•) channel concentration–intensity relationships, together with the curve fits used.
Horizontal, pure diffusion experiment. The concentration profile along a z-transect for times , , , and in (a) dimensional and (b) similarity form. (c) Dissolved mass per unit interfacial-area. In (a)–(c), the thin solid curves are experimental results and the bold dotted curves are the analytic pure diffusion solutions Eqs. (3) and (4) . (d) Amplitude of perturbation away from pure diffusion (solid curve) together with an estimate of the measurement-error amplitude (dashed curve).
“Jan11” convection experiment. Snapshots of the concentration field at times t ( , t H ) after inception (a) (12, 0.016), (b) (160, 0.20), (c) (320, 0.40), (d) (710, 0.90), (e) (960, 1.2), (f) (3900, 4.9), (g) (12 000, 15), and (h) (21 000, 27). Solid curves in (a)–(f) are the 0.05c sat contour. Insets show in (b) a vertical concentration slice at various times (thin solid curves) together with the analytic solution Eq. (3) (bold dotted curve), in (e) an early merger (at intervals of approximately , , 0.15 H ), and in (g) a merger of a reinitiated finger (at intervals of , , 0.60 H ). The solid curves are the 0.05c sat contour. White spots are air bubbles (except along the left boundary where the camera was obscured).
“Jan11” convection experiment. Temporal behavior of (a) the concentration profile along the horizontal slice below the KMnO4-water interface, (b) the amplitude (the lower curve is a lower bound, subtracting the estimated measurement-error amplitude), (c) dissolution flux and dissolved mass per unit interfacial-area (the bold dotted curves are the analytic, pure diffusion solution Eqs. (4) ), and (d) horizontally averaged concentration profile as a function of z. Contours in (d) are at intervals of 0.1c sat from 0.05c sat. Left panels show early times; right panels show the full time history. For the flux, circles are raw time-differentiated mass data and the curve is a smooth interpolation.
Amplitude against time for all convection experiments without bubbles at the interface in (a) dimensional and (b) advection–diffusion forms. In (b), the bold dotted line at t = 47.9 is the theoretical lower bound on onset time. The second dotted line is proportional to , an excellent fit to both the maximum amplification 25 across all horizontal wavenumbers and the growth of “white noise” initial conditions 11 for horizontal wavenumber and for and .
Flux against time across all convection experiments in (a) dimensional, (b) advection–diffusion, and (c) layer-thickness formulations. In (a), the bold dotted curve is the analytic, pure diffusion solution Eq. (4b) . In (c), the bold dotted curve is the late-time phenomenologically derived scaling Eq. (9c) . Fluxes have been smoothed to improve readability.
Horizontally averaged concentration profile against vertical coordinate for the “Jan11” experiment at times t = 13.5, 26.0, 51.1, 63.6, and in the direction of the arrow (t H = 8.0, 15, 30, 38, and 45). Fingers first impact the base of the cell at (t H ≈ 6). The bold dotted profiles are the bulk concentration estimates from the phenomenological relation Eq. (9a) .
Dissolved mass per unit interfacial-area against time across all convection experiments in (a) dimensional and (b) layer-thickness forms. The thick, dotted curve in (a) is the analytic, pure diffusion solution Eq. (4a) and in (b) is the late-time phenomenologically derived scaling Eq. (9d) , omitting the term.
Landmark times. (a) Amplitude onset (+), flux onset (▼), flux maximum (△), and nonlinear amplitude saturation (×) in advection–diffusion scalings. (b) Fingers half-way down the cell (▲), fingers impacting bottom (▽), and 50% average saturation (■) in layer-thickness scalings. Finger location measures are taken when the horizontally averaged concentration reaches 0.05c sat at that location.
Wavelength in time in the advection–diffusion framework across all convection experiments. The bold dotted curve is the stability theory prediction 10 . Experimental wavelengths are defined as the average distance between peaks on a horizontal slice beneath the interface.
Vertical fingertip locations against (a) time and (b) horizontal coordinate in the advection–diffusion framework for the “Jan 11” experiment. In (a), the bold dotted line is ; the inset shows the same plot logarithmically and includes the curve .
Summary of the six distinct regimes as a function of Rayleigh number and advection–diffusion time.
Experiment names, symbols, and parameters. Symbols are (sd) for short-dashed, (ld) for long-dashed, and (s) for solid. Errors in θ are ±0.02° and in H are . To evaluate , , T H , and , we take , , and as described in Secs. III and IV , together with and . Errors in and are approximately 20%, 10%, and 7% for angles 0.11°, 0.21°, and 0.33°, respectively (principally due to the error in θ). Errors in T H are twice this percentage. Asterisks (*) denote experiments with a small bubble at the interface.
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