Electrolyte flows under magnetic fields: Manning-like counterion condensation in one dimension

We present a theoretical framework for unidirectional electromagnetohydrodynamic flow of dilute electrolytes under perpendicular magnetic fields. Starting from the Navier-Stokes equation coupled with the Poisson-Nernst-Planck formulation, we show that the problem admits a sequential decoupling: the Stokes equation is solved first to obtain the velocity profile, which defines a hydrodynamic potential entering the Nernst-Planck description of ions. This Lorentz-force-induced potential competes with electrostatic attraction and significantly alters ionic distributions. We analyze this mechanism in two canonical geometries. In planar Couette shear, it produces a Manning-Oosawa-like condensation transition in one dimension, a phenomenon absent in classical electrostatics. We derive an eigenvalue equation predicting a sharp threshold between counterion enrichment and depletion at the charged wall. In cylindrical Taylor-Couette flow, the same effect shifts the classical Manning criterion by a magnetic parameter, enabling tunable control of condensation. These findings extend Manning-Oosawa phenomenology to driven, nonequilibrium systems and provide a basis for magnetic manipulation of screening in electrolytes, with implications for microfluidics, electrochemical systems, and nonlinear boundary-value theory.