Streaming potential generated by a long viscous drop in a capillary

The streaming potential generated by motion of a long drop Of Viscosity mu(d) = lambda mu in a uniform circular capillary filled with fluid of viscosity mu is investigated by means of a model previously used to study electrophoresis of a charged mercury drop in water. The capillary wall is at potential c relative to the bulk fluid within it, and the Surface of the drop is at potential 4 Potentials are assumed to be sufficiently small so that the charge cloud is described by the linearized Poisson-Boltzmann equation, and the Debye length characterizing the thickness of the charge cloud is assumed to be thin compared with the gap h(0) between the drop and the capillary wall. tons in the external fluid are not allowed to discharge at the surface of the drop, and the wall of the capillary has a nonzero surface conductivity or, The drop is assumed to be sufficiently long so that end effects can be neglected. Recirculation of fluid within the drop gives rise to an enhanced streaming current when xi(d) is nonzero, leading to an anomalously high streaming potential. This vanishes as the drop viscosity becomes large. If V is the velocity of the drop and gamma is the coefficient of interfacial tension between the two fluids, then the capillary number is Ca = mu V/gamma, and the gap varies as h(0) infinity Ca-2/3. When Ca is small, the gap h(0) is small and electrical conduction along the narrow gap is dominated by the surface conductivity sigma(c) of the capillary wall, which is constant. The electrical current convected by flowing fluid is proportional to Ca, as is the change in streaming potential caused by the presence of the drop. If sigma(c) = 0, then the electrical conductance of the gap depends on its width h(0) and on the bulk fluid conductivity a and becomes small as h(0) approximate to Ca-2/3 -> 0. The streaming potential required to cancel the O(Ca) convection current therefore varies as Ca-1/3. If sigma(c) 0 and the drop is rigid (lambda -> infinity), then the change in streaming potential over and above that expected due to the change in pressure gradient is proportional to the difference in potentials xi(c) - xi(d).