^{1,2}, P. M. J. Trevelyan

^{1,3}, C. Almarcha

^{1,4}and A. De Wit

^{1}

### Abstract

In a gravitational field, a horizontal interface between two miscible fluids can be buoyantly unstable because of double diffusive effects or because of a Rayleigh-Taylor instability arising when a denser fluid lies on top of a less dense one. We show here both experimentally and theoretically that, besides such classical buoyancy-driven instabilities, a new mixed mode dynamics exists when these two instabilities act cooperatively. This is the case when the upper denser solution contains a solute A, which diffuses sufficiently faster than a solute B initially in the lower layer to yield non-monotonic density profiles after contact of the two solutions. We derive analytically the conditions for existence of this mixed mode in the (R, δ) parameter plane, where R is the buoyancy ratio between the two solutions and δ is the ratio of diffusion coefficient of the solutes. We find an excellent agreement of these theoretical predictions with experiments performed in Hele-Shaw cells and with numerical simulations.

A.D. acknowledges Prodex, the “Actions de Recherches Concertées CONVINCE” programme and FRS-FNRS for financial support. J.C.-L. thanks MICINN for funding through research (Project No. FIS2010-21023) and the FPI grant associated to FIS2007-64698.

I. INTRODUCTION

II. BUOYANCY-DRIVEN INSTABILITIES

III. MIXED MODE

IV. NONLINEAR SIMULATIONS

V. CONCLUSIONS

### Key Topics

- Solution processes
- 27.0
- Diffusion
- 16.0
- Interface diffusion
- 9.0
- Navier Stokes equations
- 8.0
- Buoyancy driven flow instabilities
- 7.0

## Figures

Buoyancy-driven instabilities in a vertical Hele-Shaw cell at the miscible interface between two fluids: (a) Rayleigh-Taylor (R, δ) = (0.82, 0.51) at t = 20 s, (b) double-diffusion (R, δ) = (2, 1.96) at t = 600 s, (c) diffusive layer convection (R, δ) = (1.25, 0.27) at t = 300 s, (d)−(f) three consecutive snapshots of the mixed mode dynamics at (R, δ) = (0.89, 0.27) taken at 40, 50, 60 s after contact. Frames size: (a)−(c) 15 mm × 9 mm, (d)−(f) 11 mm × 9 mm.

Buoyancy-driven instabilities in a vertical Hele-Shaw cell at the miscible interface between two fluids: (a) Rayleigh-Taylor (R, δ) = (0.82, 0.51) at t = 20 s, (b) double-diffusion (R, δ) = (2, 1.96) at t = 600 s, (c) diffusive layer convection (R, δ) = (1.25, 0.27) at t = 300 s, (d)−(f) three consecutive snapshots of the mixed mode dynamics at (R, δ) = (0.89, 0.27) taken at 40, 50, 60 s after contact. Frames size: (a)−(c) 15 mm × 9 mm, (d)−(f) 11 mm × 9 mm.

Classification of the various density profiles in the (R, δ) parameter plane for δ < 1. Dynamics characteristic of RT, MM, and DLC are obtained experimentally for the points above the full line (red online) and those shown by diamonds (red online) and black squares, respectively. The filled circle (green online) and diamond (red online) are those corresponding to the images of Figs. 1(a) and 1(d)–1(f) , respectively. The sketches represent the typical density profiles in each region.

Classification of the various density profiles in the (R, δ) parameter plane for δ < 1. Dynamics characteristic of RT, MM, and DLC are obtained experimentally for the points above the full line (red online) and those shown by diamonds (red online) and black squares, respectively. The filled circle (green online) and diamond (red online) are those corresponding to the images of Figs. 1(a) and 1(d)–1(f) , respectively. The sketches represent the typical density profiles in each region.

Density profiles (a) on the full curve (red online) δ = R c of Fig. 2 for R = 0.84 and (b) in the MM zone for δ = 0.3 and R = 0.9. (c) Summarizes the zone of existence of RT, MM, and DLC modes depending on the relative value of Δρ0 and Δρ′ for δ = 0.3.

Density profiles (a) on the full curve (red online) δ = R c of Fig. 2 for R = 0.84 and (b) in the MM zone for δ = 0.3 and R = 0.9. (c) Summarizes the zone of existence of RT, MM, and DLC modes depending on the relative value of Δρ0 and Δρ′ for δ = 0.3.

(a) Sketch showing how the mixing length is measured as the distance between the two black horizontal lines passing by the furthest upmost and downmost points of the fingering zone while the amplitude of the deformation of the contact line is the distance between the two dashed (red online) horizontal lines passing by the furthest upmost and downmost points of the deformed interface. The temporal evolution of the mixing length and of the amplitude of the interface modulation are given on panels (b) and (c) for δ = 0.27 and R = 0.8 (RT, crosses), 0.85 (MM, dotted line), 0.89 (MM, dashed line), 0.96 (MM, solid line), and 1.01 (DLC, squares). The mixing length continuously increases in time for all values of R but its intensity decreases as R increases. (c) The amplitude of the contact line modulation is equal to the mixing length in the RT mode, saturates to a constant value in the MM dynamics and vanishes in the DLC regime.

(a) Sketch showing how the mixing length is measured as the distance between the two black horizontal lines passing by the furthest upmost and downmost points of the fingering zone while the amplitude of the deformation of the contact line is the distance between the two dashed (red online) horizontal lines passing by the furthest upmost and downmost points of the deformed interface. The temporal evolution of the mixing length and of the amplitude of the interface modulation are given on panels (b) and (c) for δ = 0.27 and R = 0.8 (RT, crosses), 0.85 (MM, dotted line), 0.89 (MM, dashed line), 0.96 (MM, solid line), and 1.01 (DLC, squares). The mixing length continuously increases in time for all values of R but its intensity decreases as R increases. (c) The amplitude of the contact line modulation is equal to the mixing length in the RT mode, saturates to a constant value in the MM dynamics and vanishes in the DLC regime.

Numerical concentration maps of species B for R = 0.85 and in each column: δ = 0.8, 0.6, 0.4, and 0.2 on a width W=16384, 16384, 8192, 8192 (aspect ratio constant) from left to right. Times of the first row are, respectively, from left to right: t = 2 × 105, 2 × 105, 1.4 × 105, and 5 × 104; for the second row, we have t= 1.1 × 106, 1.1 × 106, 5.8 × 105, and 4 × 105.

Numerical concentration maps of species B for R = 0.85 and in each column: δ = 0.8, 0.6, 0.4, and 0.2 on a width W=16384, 16384, 8192, 8192 (aspect ratio constant) from left to right. Times of the first row are, respectively, from left to right: t = 2 × 105, 2 × 105, 1.4 × 105, and 5 × 104; for the second row, we have t= 1.1 × 106, 1.1 × 106, 5.8 × 105, and 4 × 105.

Reconstructed numerical density maps ρ = a − Rb for R = 0.85 and δ = 0.2 in a system of width 8192 (aspect ratio preserved) on a grey scale ranging from the less dense in white and the denser in black. (top left) Initially, at t = 0, a little sinusoïdal perturbation is added on the contact line between the denser solution of A on top of the less dense solution of B; (bottom left, t = 30 000) As A diffuses faster than B, it creates a denser accumulation zone (black line) beneath the miscible interface while a less dense depletion zone (white line) appears above it. This effect is weakened at the tip of the protrusions because of curvature effects; (right, t = 70 000) Across the lateral sides of the minima, A diffuses faster out than B enters creating denser sinking sides. At the tip of the minima, the concentrated inward flux of B and diluted outward flux of A lead to a mixed zone of intermediate density, which rises between the sinking denser sides. As a result, “Y” shaped sinking antennas are observed. A symmetric argument can be developed for the rising maxima.

Reconstructed numerical density maps ρ = a − Rb for R = 0.85 and δ = 0.2 in a system of width 8192 (aspect ratio preserved) on a grey scale ranging from the less dense in white and the denser in black. (top left) Initially, at t = 0, a little sinusoïdal perturbation is added on the contact line between the denser solution of A on top of the less dense solution of B; (bottom left, t = 30 000) As A diffuses faster than B, it creates a denser accumulation zone (black line) beneath the miscible interface while a less dense depletion zone (white line) appears above it. This effect is weakened at the tip of the protrusions because of curvature effects; (right, t = 70 000) Across the lateral sides of the minima, A diffuses faster out than B enters creating denser sinking sides. At the tip of the minima, the concentrated inward flux of B and diluted outward flux of A lead to a mixed zone of intermediate density, which rises between the sinking denser sides. As a result, “Y” shaped sinking antennas are observed. A symmetric argument can be developed for the rising maxima.

## Tables

Expansion coefficients α, diffusion coefficients D of the solutes used in experiments.

Expansion coefficients α, diffusion coefficients D of the solutes used in experiments.

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