Picture of the stationary IAA autocatalytic reaction front in a rectangular cell of height H = 15 mm and thickness b = 1 mm. is the velocity of the front, propagating from left to right. is the local unit vector normal to the interface, z = h(x), and is the angle between and the horizontal cell axis, x. is the gravity acceleration. L is the extension length of the front.
Pictures of the stationary IAA autocatalytic reaction fronts in six rectangular cells and in a cylindrical tube of diameter 6.93 mm (bottom). From top to bottom, first row: H = 4 mm, b = 0.2 mm (left), and b = 0.4 mm (right). Second row: H = 6 mm, b = 0.3 mm (left), and b = 0.6 mm (right). Third row: H = 8 mm, b = 0.4 mm (left), and b = 0.8 mm (right).
Log–log plot of the nondimensional front velocity, (left), and of the nondimensional front extension, L/H (right), versus the buoyancy characteristic velocity, , for different aspect ratios: Γ = 1/3 (⋄) and 0.4 (⧫), 1 (■), 3 (▲), 6 (△), 10 (×), 15 (□), and 20 (+). The open circles (○) correspond to cylindrical tubes where the tube diameter, d, replaces the cell thickness, b, in the expressions of ε and Γ. The bullets (•) correspond to the lattice BGK numerical simulations of Jarrige et al. (Ref. 1) performed at Γ = 10. The overall accuracies in ε and are of the order of 50% and 20%, respectively.
Nondimensional front velocity, , versus the nondimensional front extension, L/H. The symbols are the same as in Fig. 3. The dashed straight line of slope 0.74 corresponds to the 2D simulations of Rongy et al. (Ref. 2) The solid line through the data corresponds to the relationship found from the eikonal regime [Eq. (8)] in Jarrige et al. (Ref. 1) which, for large extensions, leads to .
Normalized lock-exchange diffusion coefficient for rectangular cells of aspect ratio Γ = H/b [from Martin et al. (Ref. 9)]. The coefficient is normalized by leading to the function G(Γ) [Eq. (9)]. With such a normalization, the dashed horizontal line of value 1/12 corresponds to a Hele–Shaw cell of infinite aspect ratio, i.e., a homogeneous porous medium.
Product of the front velocity by the length of the front, , versus the lock-exchange diffusion coefficient, , for various cell aspect ratios, Γ. Same symbols as in Fig. 3. The full line corresponds to Eq. (10).
Log–log plot of (left) and (right) versus the scaling variable Λ, where for rectangular cells and for cylindrical tubes. The symbol (□) corresponds to the experiments plotted in the previous figures with different symbols. The symbol (▲) corresponds to lattice BGK simulations of Jarrige et al. (Ref. 1) for small aspect ratios whereas the dashed line represents the 2D numerical simulations of Rongy et al. (Ref. 2) between two parallel boundaries separated by a height, H. The full line on the graphs corresponds to our empirical law [Eqs. (11) and (12)]. Note that the dashed line on the right figure is hardly distinguishable from the full line for Λ > 10. The circles (•) correspond to cylindrical tube experiments: the two large Λ values (Λ > 10) correspond to our experiments and the three data points at low values (Λ < 10) on the left figure correspond to the data of Pojman et al. (Ref. 3).
Top: pseudointerface, z = h(x), between the reactant and the product obtained by integration of Eq. (20), using the front velocity, , measured in the experiment. Bottom: a tentative superimposition of the calculated front and the experimental front.
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