Streaming potential generated by two-phase flow in a capillary

Abstract

The streaming potential generated by pressure-driven two-phase flow in a circular capillary differs from that generated by single-phase flow. Three model problems are considered, in which the dispersed phase consists of either (i) a rigid spherical particle (possibly charged), (ii) an uncharged spherical bubble, and (iii) a long, uncharged Bretherton bubble. In all three cases, the particle or bubble is assumed to lie on the center line of the capillary tube, so that the problem is axisymmetric, and is assumed to be of almost the same diameter as the internal diameter of the capillary, so that lubrication theory can be used. The electrical potentials on the surface of the particle and on the walls of the capillary are ζp and ζc, respectively, and the Debye length is assumed much smaller than the gap between the particle and the walls of the capillary. If the flow rate is held constant, the presence of the rigid particle increases the pressure drop between the ends of the capillary, and also changes the streaming potential by an amount proportional to ζc−ζp. This change in potential will in general be small compared to the total streaming potential developed between the two ends of a long capillary. However, if the capillary is filled with a large number of rigid particles, not only will the changes in pressure drop and streaming potential between the two ends of the capillary be large, but there will be a significant change in the coefficient of proportionality between pressure drop and streaming potential. The presence of an uncharged spherical bubble or Bretherton bubble changes the pressure drop between the ends of the capillary (for a given flow rate) but does not change the linear relation between pressure drop and streaming potential. However, the linear relation between flow rate and streaming potential is modified for the spherical bubble, and becomes nonlinear when a Bretherton bubble is present.