As the electrolyte solution flows down a pressure gradient, the cloud of counter-ions is advected by the flow and a streaming current is established. If no short-circuit is present between the two ends of the capillary, accumulation of charge sets up a potential difference along the channel, termed the “streaming potential”. Such potential, or equivalently electric field, opposes the mechanical transfer of charge by creating a reverse conduction current through the bulk solution such that the total net electric current is zero. This induced axial electric field scales with the applied pressure gradient and leads to the creation of an induced electroosmotic back-flow which effectively slows down the fluid motion in the capillary: a smaller flow rate for a given pressure drop is obtained than in the regular Poiseuille case, as if the liquid had a higher shear viscosity than expected. Consequently this effect is usually referred to as the primary “electroviscous effect” [@burgeen; @rice; @levine; @hunter; @probstein].

Let us consider the pressure-driven flow in a channel of height $2h$ and width $w\gg h$ of the electrolyte solution with electrostatic potential given by equation . We calculate below the value of the steady-state streaming electric field $E_S {\bf e}_x$ induced 