Scheme 1.

Characterizing nonlinear EOF at the nanopipette tip. When c_in ≠ c_ex, a spatially varying Debye length (dashed yellow line) induces a nonuniform zeta potential. An electric field exacerbates this nonlinearity through induced charge polarization (indicated by sizes and signs of black symbols). Together, the effects drive amplified EOF near sidewalls (u_ICEO). In the presence of fast sidewall flow, hydrostatic back flows (u_HBF) develop to preserve fluid continuity. Vortex flows (red) connect ICEO and back flow streamlines. Far from the tip, EOF is linear (Figure S9).