Theory of magneto-electrokinetics in unidirectional electrolyte flow: hydrodynamic potential and Hall effect

We develop a theoretical framework for magneto-electrokinetic transport in dilute electrolytes under unidirectional flow and streaming current conditions. Unlike standard magnetohydrodynamics in liquid metals, where charge carriers are electrons and conductivity is typically uniform, electrolytes exhibit spatially varying ion densities and additional forces such as osmotic pressure. In steady state, the coupled Navier–Stokes, Nernst–Planck and Poisson equations become analytically tractable. We derive a Boltzmann-like expression for ion distributions that incorporates both the electrostatic potential and a novel hydrodynamic potential arising from the magnetic field. This potential depends explicitly on the flow profile and is not screened close to the bounding walls. Applying the theory to Poiseuille flow, we derive expressions for the induced Hall potential and demonstrate its near-linear dependence on flow velocity and magnetic field strength, while remaining largely independent of salt concentration. To validate this one-dimensional framework, we analyse the two-dimensional spatial transients near the channel entrance. We demonstrate a separation of length scales: the Hall electric field and space charge adjust to the magnetic field almost instantaneously, whereas the macroscopic salt concentration wake develops over a much longer Graetz length dictated by the Péclet number. We also examine a similar flow with counterions only and no added salt, determining the resulting potential and ion distributions. This work offers new theoretical tools for understanding ion transport in microfluidic and electrochemical systems.