Stream lines of the velocity field (see the Appendix), one obstacle (central circle), and the motion (arrows) of a particle (small circle) close to the obstacle. The dashed circle indicates the area of the obstacle's influence due to the finite size of the particle. The radius of the obstacle a and the potential d are indicated.
Periodic pattern (PP) with tilted obstacles. The black circles are the obstacles and the red area corresponds to the range of the interaction potential. The dashed lines show the positions where incoming and final flows, respectively, are numerically measured. The blue and red bars on the y axis denote the inlets through which the two reacting species are introduced into the system (see Fig. 3 ).
Distribution of the two species, black and white, at the final time τ = 60, associated with the periodic pattern (PP) of Fig. 2 (top) and with a random pattern (RP) of obstacles (bottom). In both cases .
Left column: concentration profiles c a (black) and c b (red), at x = 0.5 (dashed lines) and x = 40 (solid lines) for τ = 60, when the species do not react ( ). The upper left panel shows the profiles for the periodic pattern (PP), the middle left panel for the random pattern (RP), and the lower left panel when there are no obstacles (NP) at all on the surface. Right column: the same three cases (PP, RP, and NP)) as in the left column but now incorporating the reaction with rate coefficient .
Reaction efficiency versus reaction rate constant at the final position x F = 40 (τ = 60) for the periodic pattern (PP, diamonds), random pattern (RP, circles), and without obstacles (NP, triangles). The dashed line corresponds to the reaction efficiency for perfect mixing, cf. Eq. (11) . The full lines are the fittings of Eq. (12) .
Left panel: plot of the flux for the PP configuration and three values of the diffusion coefficient: (solid line), 0.05 (dashed line) and 0.1 (dotted line). Inset: Efficiency versus diffusion coefficient. Right panel: fluxes for the three configurations considered earlier ( ), from top to bottom: NP, RP, and PP (same colors as in Fig. 6 ). In both panels and τ = 60.
Comparison of fluxes obtained by using patterns with different radii of obstacles a and of potential range d. Solid lines: d = 0.20; dashed lines: d = 0.25. For each d value, different values of obstacle radius are represented: from top to bottom a = 0.15 (black), a = 0.10 (red), a = 0.05 (blue), and a = 0 (green, this case corresponds to the approximation). Inset: Efficiency versus parameter a for d = 0.20 (open symbols) and d = 0.25 (black symbols).
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