Reaction-subdiffusion in moving fluids

To capture the dynamic behaviors of reaction-subdiffusion in linear flow fields, in the present paper we analyze a simple monomolecular conversion $A\to B$. We derive the corresponding master equations for the distribution of $A$ and $B$ particles in continuous time random walk schemes. The results are then used to obtain the generalizations of the advection-diffusion reaction equation in which the diffusion and advection operators both depend on the reaction rate.

Figure 1

The time evolution of the PDF for the $A$ particle in the reaction-subdiffusion process in flows with reaction rate $\alpha =0.15$ and anomalous diffusion exponent $\beta =0.5$. The solution is shown for the dimensionless times $t=0.2$ (solid line), 1 (dashed line), and 2 (dotted line). It is asymmetric with respect to the $Y$ axis, and the plume stretches more and more into the direction of the convection velocity. Moreover, the decay with time is faster than that in nonreactive process in Fig. 2 because of the negative exponential decay accounting for the effect of the $A\to B$ reaction.

Figure 2

The evolution of the PDF for the particle in the nonreactive Galilei variant subdiffusive model with reaction rate $\alpha =0$ and mean flight time $\beta =0.5$. The solution is shown for the dimensionless times $t=0.2$ (solid line), 1 (dashed line), and 2 (dotted line). The propagator is asymmetric with respect to the maximum which stays fixed at the origin since the anomalous particle is dragged along with the drift velocity [4, 22].

Figure 3

The evolution of the PDF for the $B$ particle in the reaction-subdiffusion process in flows with reaction rate $\alpha =0.15$ and anomalous exponent $\beta =0.5$. The solution is shown for the dimensionless times $t=0.2$ (solid line), 1 (dashed line), and 2 (dotted line). The moving flows also result in the asymmetry of the solution for created particles with respect to the $Y$ axis. The growth over time compares with the decay with time for $A(x,t)$ in Fig. 1.