Origin of the streaming current
Adjacent to the channel walls, the charge-neutrality of the liquid is violated due to the presence of an electrical double layer: a thin layer of counterions attracted by the charged surface. The transport of these counterions along with the pressure-driven fluid flow gives rise to a net charge transport: the streaming current. The opposite effect, generating a fluid flow by applying a potential difference, is called electroosmotic flow.
Measurement method
A typical setup to measure streaming currents consists of two electrodes placed on either side of a fluidic geometry across which a known pressure difference is applied. When both electrodes are held at the same potential, the streaming current is measured directly as the electrical current flowing through the electrodes. Alternatively, the electrodes can be left floating, allowing a streaming potential to build up.
References:
1) J. Lyklema, Fundamentals of Interface and Colloid Science
2) F.H.J. van der Heyden et al., Phys. Rev. Lett. 95, 116104 (2005)
3) C. Werner et al., J. Colloid Interface Sci. 208, 329 (1998)