From left to right IAA front stably propagating from top to bottom: the transient is detected by brown PVA. IAA front in a petridish propagating circularly outwards: blue fresh product, yellow burnt one, and the transient PVA color. CT reaction front stably propagating from bottom to top: the reactant is purple, the product is transparent.
Stationary chemical front in a parabolic Poiseuille flow in a rectangular Hele-Shaw cell: on the right supportive flows () and on the left: adverse () (the flat front corresponds to no flow). Sketch of the propagation front velocity versus the mean velocity of the Poiseuille flow, ; all velocities are normalized by the chemical velocity . The solid straight lines (horizontal for and the more inclined for ) correspond to the “eikonal thin front limit” (). The other straight line of constant slope corresponds to the mixing regime (); the curve close to it is the Taylor dispersion correction Eq. (4).
Sketch of the porous medium cell. The bottom of the cell is immersed in a tank of the product. The reactant fluid is sucked or flushed through a series of injectors at the top of the cell. The cell is 30 cm high, 1 cm wide, and 1 cm thick.
Top: Typical dispersion front obtained for in the 1.5–2 mm glass beads packed porous medium when dye is injected in the porous medium. The width of the picture is 10 cm. Bottom: Measurement of the hydrodynamic dispersion coefficient D of the porous medium versus the average flow velocity . The slope gives the dispersion length .
Left: Pictures of IAA chemical front in bulk fluid (top) and inside a packed beads porous medium (bottom). The pictures are 8 cm wide. Right: Front velocity, normalized by the chemical velocity , versus the bead size, d of the packed beads in absence of flow for IAA and CT reaction.
Left: pictures of the chemical wave propagation across the corner film left by an air bubble quenches in square capillary tube of edge a. Right: Front velocity, normalized by the chemical velocity , versus the square capillary tube edge a. The sketch shown in blue corresponds the corner liquid space left by the air bubble.41
Supportive flow. Left: stationary front obtained for in the 1.5–2 mm beads pack; the picture is 10 cm wide. Right: front velocity versus the mean flow velocity in the porous medium, . The straight line of slope 1.20 through the data is the best fit of the linear dependency of the front velocity with the flow rate. The black straight line is the plug flow, Galilean translation, model Eq. (3). The inset corresponds to a close-up of the same data, with the comparison with the Taylor model Eq. (4) (curve line) using the measured dispersion coefficient Fig. 4.
Stochastic porous medium. Left: Color map of the permeability field of one realization of a porous medium, characterized by a log-normal isotropic distribution of mean −1, RMS , and correlation length in lattice units. The color map on the right corresponds to the local value of the permeability. The black line corresponds to the chemical wave front inside the porous medium for a relative velocity and a mean standard deviation of the permeability distribution . Right: Measured covariance function of the log-normal isotropic distribution of the left permeability field. The value of was obtained by an exponential fit (line) using Eq. (12).
Numerical simulations. Top left: series of fronts at a given velocity and correlation length for different size distributions from 0 to 0.9. Bottom left: series of fronts at a given for different velocity from 1 to 5. Right: Plot of versus the square of the size distribution, (top) and versus the normalized flow velocity (bottom).
Numerical simulations. Plot of versus the variable (see Eq. (15)) for different series of variation of , , or and different statistical porous medium.
Top picture (width 10 cm) of the front for , which propagates upstream (). Bottom: Normalized front velocity, , versus average flow velocity for adverse flow inside the 1.5–2 mm beads pack.
Chemical front in a flow around an obstacle; the flow is from left to right. The injection of the fresh reactant produces a flow at constant velocity in a thin Hele-Shaw cell; the solid disk obstacle is a cylinder of diameter 1 cm. The chemical front in the absence of flow would propagate from right to left. Left pictures: from top to bottom, steady fronts around thesame disk for , −3, and −5. Right: 3 pictures of timesequence of detachment from the obstacle at a larger velocity (time increases from top to bottom).
Left: Velocity field around a disk, placed in a uniform adverse flow . The flow is from left to right. In the absence of flow, the chemical front propagates from right to left. Right: Steady fronts around the disk obtained from integration of Eq. (7), with , for , −2.5, −5, and −10.
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