^{1}, N. Jarrige

^{1}, J. Martin

^{1}, N. Rakotomalala

^{1}, L. Talon

^{1}and D. Salin

^{1,2,a)}

### Abstract

A viscous lock-exchange gravity current corresponds to the reciprocal exchange of two fluids of different densities in a horizontal channel. The resulting front between the two fluids spreads as the square root of time, with a diffusion coefficient reflecting the buoyancy, viscosity, and geometrical configuration of the current. On the other hand, an autocatalytic reaction front between a reactant and a product may propagate as a solitary wave, namely, at a constant velocity and with a stationary concentration profile, resulting from the balance between molecular diffusion and chemical reaction. In most systems, the fluid left behind the front has a different density leading to a lock-exchange configuration. We revisit, with a chemical reaction, the classical situation of lock-exchange. We present an experimental analysis of buoyancy effects on the shape and the velocity of the iodate arsenous acid autocatalytic reaction fronts, propagating in horizontal rectangular channels and for a wide range of aspect ratios (1/3 to 20) and cylindrical tubes. We do observe stationary-shaped fronts, spanning the height of the cell and propagating along the cell axis. Our data support the contention that the front velocity and its extension are linked to each other and that their variations scale with a single variable involving the diffusion coefficient of the lock-exchange in the absence of chemical reaction. This analysis is supported by results obtained with lattice Bathnagar-Gross-Krook (BGK) simulations Jarrige *et al.* [Phys. Rev. E **81**, 06631 (2010)], in other geometries (like in 2D simulations by Rongy *et al.* [J. Chem. Phys. **127**, 114710 (2007)] and experiments in cylindrical tubes by Pojman *et al.* [J. Phys. Chem. **95**, 1299 (1991)]), and for another chemical reaction Schuszter *et al.* [Phys. Rev. E **79**, 016216 (2009)].

It is a pleasure to thank Patrick De Kepper for giving us the PVA recipe. This work was partly supported by CNES (No. 793/CNES/00/8368), ESA (No. AO-99-083), Réseaux de Thématiques de Recherches Avancées “Triangle de la physique,” and the Initial Training Network (ITN) “Multiflow.” One of us (I.B.M.) was supported by a postdoctoral grant from the CNRS, whereas N.J. was supported by a grant from the French Ministry of Research (MESRT). All these sources of support are gratefully acknowledged.

I. INTRODUCTION

II. EXPERIMENTS

III. RESULTS: FRONT VELOCITY AND EXTENSION

IV. RELATION BETWEEN CHEMICAL REACTION AND PURE LOCK-EXCHANGE

V. FRONT REACTION PROFILE

VI. COMPARISON AND ANALYSIS OF RELATED PAPERS

VII. CONCLUSION

### Key Topics

- Chemical reactions
- 22.0
- Viscosity
- 19.0
- Diffusion
- 15.0
- Hydrogen reactions
- 6.0
- Velocity measurement
- 6.0

## Figures

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.

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).

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.

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 .

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.

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).

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).

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.

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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