Oscillatory instability in a reaction front separating fluids of different densities

Reaction fronts described by the Kuramoto-Sivashinsky (KS) equation can exhibit complex behavior as they separate reacted from unreacted fluids. If the fluid of higher density is above a fluid of lower density, then the Rayleigh-Taylor instability can lead to fluid motion. In the reverse situation, where the lighter fluid is on top, gravitationally driven forces can stabilize a convectionless flat front inhibiting the complex front propagation described by the KS equation. In these cases, a critical density difference is required to provide stability to the flat front. A linear stability analysis shows that the transition from stable to unstable flat fronts can be oscillatory for viscous fluid motion. Once the transition takes place, the fronts exhibit oscillatory convection resulting in oscillations of the shape and speed of the front.

Figure 1

The largest value of $\text{Re}\left(\sigma \right)$ as a function of the wave number for $\mathrm{S}\mathrm{c}=0.588$. Dashed lines indicate growth rates with nonzero imaginary part. Solid lines correspond to purely real growth rates.

Figure 2

The largest value of $\text{Re}\left(\sigma \right)$ as a function of the wave number for $\mathrm{Sc}=0.4$. Dashed lines indicate growth rates with nonzero imaginary part. Solid lines correspond to purely real growth rates.

Figure 3

Marginal stability curves for the stationary instability (thin line) and the oscillatory instability (bold line). For values of $\mathrm{Sc}<0.5$ the oscillatory instability takes place at Rayleigh numbers lower than the stationary transition.

Figure 4

Marginal stability curves for the transition to convection for different values of the parameter ${\gamma }_0$. The area under each curve corresponds to values of the Rayleigh and Schmidt numbers where the convectionless flat front is stable.

Figure 5

Real part of the growth rate for different values of the Schmidt number Sc. Here we set $\mathrm{Ra}=-1$ and ${\gamma }_0=0$. Dashed lines indicate growth rates with nonzero imaginary part. Solid lines correspond to purely real growth rates.

Figure 6

Growth rates for different values of the Schmidt number $\mathrm{Sc}$. Here we set $\text{Ra}=1$ and ${\gamma }_0=0$.

Figure 7

The speed of steady fronts confined in a two-dimensional tube of width $a$. Here we have $\text{Ra}=-0.66$, ${\gamma }_0=0$, and $\mathrm{Sc}=0.588$.

Figure 8

The maximum and minimum speeds of the average front position as a function of tube width for oscillatory fronts. Here we have $\text{Ra}=-1.6$, ${\gamma }_0=0$, and $\mathrm{Sc}=0.4$.

Figure 9

Time evolution of an oscillatory front. The color map indicates the front deviation relative to the average front position. In this case, the oscillatory front has a maximum on one side of the tube, changing into a minimum as the front oscillates.

Figure 10

Time evolution of an oscillatory front. The color map indicates the front deviation relative to the average front position. In this case, the oscillatory front is symmetric relative to the center of the tube, with its center oscillating between a maximum and minimum values.