Thermal and compositional driven convection in thin reaction fronts

Chemical reaction fronts separate regions of reacted and unreacted substances as they propagate in liquids. These fronts may induce density gradients due to different chemical compositions and temperatures across the front. In this paper, we investigate buoyancy-induced convection driven by both types of gradients. We consider a thin front approximation where the normal front velocity depends only on the front curvature. This model applies for small curvature fronts independent of the specific type of chemical reaction. For density changes due only to heat variations near the front, we find that convection can take place for either upward or downward propagating fronts if density gradients are above a threshold. Convection can set in even if the fluid with lower density is above the higher density fluid. Our model consists of Navier-Stokes equations coupled to the front propagation equation. We carry out a linear stability analysis to determine the parameters for the onset of convection. We study the nonlinear front propagation for liquids confined in narrow two-dimensional domains. Convection leads to fronts of steady shape, propagating with constant velocities.

Figure 1

The largest value of $\text{Re}\left(\sigma \right)$ as a function of the wave number for $\mathrm{Ra}=0.5,-0.5,-1.5$. The front is stable for $-0.5$, but it becomes unstable for positive and negative thermal Ra.

Figure 2

The largest value of $\text{Re}\left(\sigma \right)$ as a function of the wave number for $\mathrm{Ra}=0.5$. The front is unstable for ${\mathrm{Ra}}_c=0.0$ since wave numbers below a critical value have positive growth rate. This critical value increases for a positive compositional Rayleigh number (${\mathrm{Ra}}_c=0.2$), thus the front is unstable over a larger range of wave numbers. The opposite is true for ${\mathrm{Ra}}_c$ negative (${\mathrm{Ra}}_c=-0.2$).

Figure 3

The largest value of $\text{Re}\left(\sigma \right)$ as a function of the wave number for $\mathrm{Ra}=-4.$ The front is unstable for ${\mathrm{Ra}}_c=0.0$ since wave numbers below a critical value have positive growth rate. This critical value increases for a positive compositional Rayleigh number (${\mathrm{Ra}}_c=0.25$), thus the front is unstable over a larger range of wave numbers. The opposite is true for ${\mathrm{Ra}}_c$ negative (${\mathrm{Ra}}_c=-0.5$).

Figure 4

Stability of flat fronts as a function of thermal (Ra) and compositional (${\mathrm{Ra}}_c$) Rayleigh numbers. For a fixed value of ${\mathrm{Ra}}_c$, we observe a region where the front can be unstable for both positive and negative thermal Rayleigh numbers.

Figure 5

Diagram showing a harmonic perturbation to a flat front on an unstable density configuration due to compositional density changes (${\mathrm{Ra}}_c>0, \mathrm{Ra}=0$). The color map indicates a denser fluid above a less dense fluid. The horizontal arrows indicate the direction of density increments in the horizontal direction, leading to a baroclinic torque. The curved arrows indicate the direction of flow induced by the change in vorticity. In this configuration, the flow raises the peaks of the front and lowers the troughs. This mechanism leads to the Rayleigh-Taylor instability.

Figure 6

Diagram showing an unstable configuration due to thermal density changes generated by a harmonic perturbation to a flat front. The color map represents the fluid temperature near the front. The front propagates downwards, releasing heat. The fluid has a positive thermal expansion coefficient, resulting in a negative thermal Rayleigh number ($\mathrm{Ra}<0, {\mathrm{Ra}}_c=0$). The horizontal arrows indicate the direction of density increments in the horizontal direction, leading to a baroclinic torque. The curved arrows indicate the direction of flow induced by the change in vorticity. In this configuration, the flow raises the peaks and troughs of the front. Comparing the regions far away from the front, the fluid below the front has the higher density.

Figure 7

Increase of front speed as a function of tube width. In this case, $\mathrm{Ra}=0.5, {\mathrm{Ra}}_c=0.1,0,-0.1$.

Figure 8

Increase of front speed as a function of tube width. In this case, $\mathrm{Ra}=-4, {\mathrm{Ra}}_c=0.25,0,-0.25$.

Figure 9

Nonaxisymmetric front propagating upward with $\mathrm{Ra}=0.5, {\mathrm{Ra}}_c=0$. The velocity field consists of two counter-rotating convective rolls. The reaction front remains closer to the upper roll.

Figure 10

Nonaxisymmetric front propagating upward with $\mathrm{Ra}=-4, {\mathrm{Ra}}_c=0$. The velocity field consists of two counter-rotating convective rolls. The reaction front remains closer to the lower roll.

Figure 11

The largest value of $\text{Re}\left(\sigma \right)$ as a function of the wave number for ${\mathrm{Ra}}_c=0.29$ and $\mathrm{Ra}=-2.2$. The figure shows two positive relative maxima separated by a range of wave numbers with $\text{Re}\left(\sigma \right)<0$. Growing perturbations occur in two separate wave number intervals where $\text{Re}\left(\sigma \right)>0$.

Figure 12

Increase of speed as a function of tube width. Convection takes place only for widths above 2.32, except in the interval [3.61,4.62] where the flat front is stable. In this case, $\mathrm{Ra}=-2.3$ and ${\mathrm{Ra}}_c=0.29$.

Figure 13

Front of steady shape propagating upward coupled to convective cells. In this case, $\mathrm{Ra}=-0.75$ and ${\mathrm{Ra}}_c=0.16$.

Figure 14

Temperature deviation from the average for a front of steady shape propagating upward coupled to convective cells. In this case, $\mathrm{Ra}=-0.75$ and ${\mathrm{Ra}}_c=0.16$. The color map corresponds to ${10}^{-2}$ dimensionless temperature units.

Figure 15

Value of stream function for a front of steady shape propagating upward coupled to convective cells. In this case, $\mathrm{Ra}=-0.75$ and ${\mathrm{Ra}}_c=0.16$. The color map corresponds to ${10}^{-3}$ dimensionless units described in the text.