The physics of charge transport by a fluid flowing in a channel subject to electrostatic cross-field modulation is analyzed as a two-dimensional time-transient problem. The coupled non-linear system of equations is comprised of the Poisson and current continuity equations. The fluid flow is assumed to be decoupled from the electrostatics and a simple potential flow model is currently used. A numerical procedure is described for the solution of the non-dimensionalized equations. The procedure computes electrostatic potential and field using the boundary element method (BEM), and charge density using the method of characteristics. Time transient results for turn-on and shut-off of the ion stream are discussed.