A Theory for Reaction-Diffusion-Advection Fluid System in Terms of One-sided Barriers

Classical fluid mechanics describes a standard reaction-diffusion-advection fluid system in terms of a set of differential equations, which must heavily rely on numerical computations to simulate the fluid flow. This research analyzes one-sided invariant barriers of chemical reaction fronts found in the fluid flow, which we call those barriers Burning Invariant Manifolds (BIMs). Experimentally, the flows were magneto-hydrodynamically-generated, quasi-two-dimensional, and vortex-dominated, on centimeter length scales. BIM theory analyzes the system in terms of only these BIM barriers, which can provide a more visually intuitive approach. By looking at the BIMs, one can visually approximate and predict the reaction front propagation behavior. Furthermore, these BIMs are intrinsic to the fluid flow; they are independent of the initial chemical stimulation fronts in either time-independent or periodically driven systems. This theoretical research improves the understanding of the BIM theory by numerically solving for and analyzing the fixed points in the reaction front dynamical system, where BIM theory simplified this front dynamical system into 3D ODE. Our study also includes the rules of bifurcation among the fixed points in the system using tools such as local linearization and eigenvalues. We also did numerical simulation to examine and confirm the front propagation, which can result in pinning phenomenon, where pinning is the situation that the advection cancels the reaction forming a stationary reaction front. Finally, we determine the basins of attraction of different types of pinning. In conclusion, this theoretical research provided a broader understanding of this new BIM theory and its application.