Comparison between the analytical model (solid lines) with the numerical simulations from Appelgren et al. (2023; dashed lines) at different times. We assume Z₀ = 0.008 and to match the simulation, and St= 0.03 in the three analytical expressions. Left: assuming constant metallicity Z(r, t) = Z₀ significantly overestimates the pebble flux over the lifespan of the disk. Center: assuming constant St_χ (see Eq. (32)) gives a relatively good match to the overall behavior of the pebble flux. Right: assuming constant St gives an accurate description over the first ~1 Myr, but it severely underestimates the flux at late times, since the value of St is limited by the growth timescale (see Fig. 2) in the outer disk in the simulations.