Dynamics of $A+B \to C$ reaction fronts under radial advection in three dimensions

The dynamics of $A+B\to C$ reaction fronts is studied both analytically and numerically in three-dimensional systems when $A$ is injected radially into $B$ at a constant flow rate. The front dynamics is characterized in terms of the temporal evolution of the reaction front position, $r_f$, of its width, $w$, of the maximum local production rate, $R^{\text{max}}$, and of the total amount of product generated by the reaction, $n_C$. We show that $r_f, w$, and $R^{\text{max}}$ exhibit the same temporal scalings as observed in rectilinear and two-dimensional radial geometries both in the early-time limit controlled by diffusion, and in the longer time reaction-diffusion-advection regime. However, unlike the two-dimensional cases, the three-dimensional problem admits an asymptotic stationary solution for the reactant concentration profiles where $n_C$ grows linearly in time. The timescales at which the transition between the regimes arise, as well as the properties of each regime, are determined in terms of the injection flow rate and reactant initial concentration ratio.

Figure 1

(a) Numerical radial profile $u(r,t)$ for $Q=100, \gamma =1$ at different times and analytical stationary solution (6). (b) Numerical stationary concentration profiles $a_s$ and $b_s$, production rate, $R_s=a_s{b}_s$, and front width, $w_s$, for $Q=100$ and $\gamma =1$. The evolution of concentration profile of the product $C$ is also shown. The vertical dashed line indicates $r_{\mathrm{fs}}$. Numerical solutions are considered stationary for $t>{10}^3{t}_{\mathrm{TS}}$, where $t_{\mathrm{TS}}$ is defined as in Eq. (34).

Figure 2

Stationary maximum production rate, $R_s^{\text{max}}$, and reaction front width, $w_s$, as a function of $K_s$ obtained through numerical computations for different values of $Q\in [4,{10}^3]$ and $\gamma \in [1/40,20]$. The dashed and solid lines represent the scalings (17) and (20), respectively.

Figure 3

Temporal evolution of the front position. Solid lines refer to numerical solutions of Eqs. (3) with $Q=10$ and different values of $\gamma$. Horizontal dashed lines are the stationary front positions, $r_{\mathrm{fs}}$, given by Eq. (7). Vertical dashed lines show the transition times, $t^{*}$, defined by Eq. (23). The inset shows a zoom in the evolution at small time delimited by the dashed rectangle in the main graph.

Figure 4

Space-time maps of the evolution of the production rate $R(r,t)$ for the polar 2D (a) and the spherical 3D systems (b). The dashed line is $r_f\left(t\right)$.

Figure 5

Temporal evolution of the maximum production rate (a) and of the reaction front width (b) obtained by solving numerically Eqs. (3) with $\gamma =1/4$. The power laws characterizing the early-time and transient regimes are shown. The vertical dashed-dotted lines indicate the timescale, $t_{\mathrm{ET}}$, at which the transition between the early-time and the transient regimes occurs; see Eq. (33). The vertical dashed lines indicate the timescale, $t_{\mathrm{TS}}$, at which the transition between the transient and the stationary regimes occurs; see Eq. (34).

Figure 6

Early-time behavior of the maximum production rate (a) and the reaction front width (b). The horizontal dotted lines in panel (a) represent the analytical early-time behavior (26).

Figure 7

Evolution of the coefficients $c_r$ (a), $c_R$ (b), and $c_w$ (c) defined by Eqs. (31a), (31b), and (31c) as a function of $\gamma$. The following values of $Q$ were used. For the first numerical dataset (blue circles), $Q=5$ for $\gamma <0.1, Q=100$ for $\gamma \in [0.1,1], Q=200$ for $\gamma =5$, and $Q=300$ for $\gamma =10$. For the second numerical dataset (red squares), $Q=10$ for $\gamma =0.01, Q=300$ for $\gamma =0.05, Q=200$ for $\gamma \in [0.1,1], Q=300$ for $\gamma =5$, and $Q=400$ for $\gamma =10$.

Figure 8

Temporal evolution of $n_C$ computed using Eq. (37) and $R$ obtained by solving numerically Eqs. (3) with $Q=5$ and $\gamma =1/20$. The scalings (41) valid at early times, (42) for the transient regime, and (43) for the stationary regime are also shown.

Figure 9

Summary of the temporal scalings of $r_f, w$, and $R^{\max }$ for $Q=100$ and $\gamma =1$.

Figure 10

Early-time concentration profiles $a$ and $b$ for two values of $\gamma$ and $Q=100$ at $t={10}^{-4}$. The solid and dashed lines represent the asymptotic solutions (A8), and the symbols represent the solutions obtained by solving numerically Eqs. (3).