Fig. 7: Numerical and analytic results in the long-channel limit.
From: Liquid flow reversibly creates a macroscopic surface charge gradient
a Numerical (dashed) and analytic (solid) results for the radially averaged fluoride concentration profile in the channel as a function of the distance from the center for Peclet numbers ranging from \({{{\rm{Pe}}}}=0.31\) (average flow velocity \(3.2\times {10}^{-8}\) m·s−1, green) to 3100 (average flow velocity \(3.2\times {10}^{-4}\) m·s−1, pink) in steps of a factor 10. b Radially averaged concentration at the center of the channel as a function of the Peclet number, comparing analytic (solid lines) and numerical (symbols) results for geologically realistic values (\({k}_{d{{{\rm{is}}}}}={10}^{-11}\) mol·m−2·s−1, \(L={10}^{-2}\) m, 10< average flow velocity <10−4 m·s−1\(,\,D\) = 10−9 m2·s−1, \(\eta\) = 10−3 Pa·s). Note that the considered geometry has length \(L=1\,{{{\rm{cm}}}}\) and radius \(R=0.1\,{{{\rm{mm}}}}\) ensuring that the radius is much smaller than the boundary layer thickness, which is not the case for the experimental geometry.
