Streaming potential generated by a small charged drop in Poiseuille flow

The streaming potential generated by a charged spherical drop of viscosity αμe suspended in fluid of viscosity μe flowing through a circular capillary is considered. The spherical drop is neutrally buoyant and has radius a much smaller than the capillary radius R0. The drop viscosity is arbitrary, and the drop becomes a rigid sphere in the limit α→∞. The drop is assumed to be a perfect conductor, with no exchange of charge between the drop and the external fluid. This model has been used in the past to study the electrophoresis of a charged mercury drop suspended in water, and may perhaps offer insight into the streaming potential caused by a water droplet immersed in oil. When the electrical potential ζd at the surface of the drop is small, the streaming potential is determined for charge cloud thicknesses κ−1 that are small compared to the capillary radius R0, but are arbitrary compared to the drop radius a. A second analysis is given for thin double layers (aκ⪢1) at higher potentials. Streaming potentials generated by a viscous charged drop are predicted to be larger than for a rigid particle. The effect of a rigid sphere at potential ζd in a charged capillary with surface potential ζc is also considered. When potentials are small and the charge clouds are thin, the change in streaming potential caused by the presence of the sphere is proportional to ζc−ζd.