Poiseuille Advection of Chemical Reaction Fronts

Poiseuille flow between parallel plates alters the shapes and velocities of chemical reaction fronts. In the narrow-gap limit, the cubic reaction-diffusion-advection equation predicts a front-velocity correction equal to the gap-averaged fluid velocity $ϵ$. In the singular wide-gap limit, the correction equals the midgap fluid velocity $3ϵ/2$ when the flow is in the direction of propagation of the reaction front, and equals zero for adverse flow of any amplitude for which the front has a midgap cusp. Stationary fronts are possible only for adverse flow and finite gaps. Experiments are suggested.

Figure 1

Dimensionless reaction front velocity $u$ vs the dimensionless average velocity $ϵ$ of the Poiseuille flow between parallel plates, for various ratios $\eta$ of the gap half-width to the front thickness. The front velocity increases monotonically with increasing $\eta$ at fixed $ϵ$, and is bounded from below by the $\eta \to 0$ result $u=1+ϵ$ (solid trace) and from above by the singular $\eta \to \infty$ results $u=1$ for $ϵ<0$ and $u=1+3ϵ/2$ for $ϵ>0$ (shortest dashes). Each trace for finite $\eta$ has continuous slope, and matches the value and slope of the $ϵ\to 0$ result $u=1+ϵ$ near $ϵ=0$. Stationary fronts and reversed-direction fronts with $u=0$ and $u<0$ are possible only for adverse flow satisfying $ϵ\le -1$. Results for $\eta \to 0$, $\eta \to \infty$, and $ϵ\to 0$ were obtained analytically, and all other results were obtained numerically.

Figure 2

Front distortion $\Gamma$ vs $ϵ$, for various values of $\eta$. Results for $\eta \to \infty$ (shortest dashes) are obtained by integrating Eq. (6). Numerical results for finite $\eta$ (longer dashes) evidently match the analytical result $\Gamma =ϵ\eta /16$ for small $\eta$ and finite $ϵ$, and for small $ϵ$ and finite $\eta$.

Figure 3

Front and flow profiles for supportive Poiseuille flow with $ϵ=2$, for which the flow is in the same direction as the chemical reaction, with parallel plates located at $x=\pm 1$. For each value of $\eta =1$, 4, 16, and $\infty$, a pair of constant-concentration traces with catalyst concentrations $c=2/5$ (dotted trace, reaction 40% completed) and $c=3/5$ (solid trace, reaction 60% completed) is shown. The separation between these traces is comparable to the front thickness, which vanishes for $\eta \to \infty$. Arrows with solid heads represent the front velocities $u=3.04$, 3.42, 3.85, and 4.00 for $\eta =1$, 4, 16, and $\infty$. Arrows with hollow heads represent the quadratic Poiseuille flow. The $\eta \to \infty$ trace was obtained by integrating Eq. (6).

Figure 4

Front and flow profiles for adverse Poiseuille flow with $ϵ=-2$, similar to Fig. 3. The front velocities $u=-0.98$, $-0.49$, 0.37, and 1.00 for $\eta =1$, 4, 16, and $\infty$ indicate that the flow succeeds in reversing the direction of propagation of the reaction front only for narrow gaps. In the wide-gap limit $\eta \to \infty$, the singularity of Eq. (5) demands a trailing midgap cusp in the front, which propagates at its advection-free velocity $u=1$. The $\eta \to \infty$ trace was obtained by integrating Eq. (6).