Fig. 5: Channel geometry and numerical solutions to the reaction–diffusion–advection Eqs. (3)–(7).

Illustration of the flow channel as used for the numerical calculations in COMSOL with resulting concentration profiles. The center of the flow channel is at the origin \((x=0,\,r=0)\). The channel length and diameter are \(2L=40\) mm and \(2R=4.8\) mm, respectively, and the dissolving surface length is \(2H=25\) mm. The color represents the fluoride concentration. In a, the result of the numerical calculations in the full three-dimensional model geometry can be seen in flow-off conditions. Panel (b) is a zoom-in on the channel under flow-off conditions (top) and flow-on conditions (bottom) at 1 mL·min⁻¹ in the positive x-direction as also indicated by the parabolic Poiseuille flow profile (white arrows). With flow, the concentration in the center of the channel essentially equals that in the reservoir. Nonetheless, a significant diffuse boundary layer remains at the dissolving surface, with a boundary layer thickness of \(\delta (0)\approx 0.5\) mm. The thickness of the boundary layer scales as \(\delta \propto {(x+H)}^{1/3}\). The boundary layer, which is much smaller than the channel radius, reduces the effective advection rate by orders of magnitude, as the fluid velocity is low near the channel wall.