Streaming potential generated by a drop moving along the centreline of a capillary

The electrical streaming potential generated by a two-phase pressure-driven Stokes flow in a cylindrical capillary is computed numerically. The potential difference Delta Phi between the two ends of the capillary, proportional to the pressure difference Delta p for single-phase flow, is modified by the presence of a suspended drop on the centreline of the capillary. We determine the change in Delta Phi caused by the presence of an uncharged insulating neutrally buoyant drop at a small electric Hartmann number, i.e. when the perturbation to the flow field caused by electric stresses is negligible. The drop velocity and deformation, and the consequent changes in the pressure difference Delta p and streaming potential Delta Phi, depend upon three independent parameters: the size a of the undeformed drop relative to the radius R of the capillary; the viscosity ratio lambda between the drop phase and the continuous phase; and the capillary number Ca which measures the ratio of viscous to capillary forces. We investigate how the streaming potential depends on these parameters: purely hydrodynamic aspects of the problem are discussed by Lac & Sherwood (J. Fluid Mech., doi: 10.1017/S0022112009991212). The potential on the capillary wall is assumed sufficiently small so that the electrical double layer is described by the linearized Poisson-Boltzmann equation. The Debye length characterizing the thickness of the charge Cloud is taken to be small compared with all other length scales, Including the width of the gap between the drop and the capillary wall. The electric potential satisfies Laplace's equation, which we solve by means of a boundary integral method. The presence of the drop increases vertical bar Delta Phi vertical bar when the drop is more viscous than the Surrounding fluid (lambda > 1), though the change in vertical bar Delta Phi vertical bar can take either sign for lambda < 1. However, the difference between Delta Phi and Delta p (suitably non-dimensionalized) is always positive. Asymptotic predictions for the streaming potential in the case of a vanishingly small spherical droplet, and for large drops at high capillary numbers, agree well with Computations.