A moving fluid interface. Part 2. The removal of the force singularity by a slip flow

If the no-slip condition is used to determine the flow produced when a fluid interface moves along a solid boundary, a non-integrable stress is obtained. In part 1 of this study (Hocking 1976), it was argued that, when allowance was made for the presence of irregularities on the solid boundary, an effective slip coefficient could be found, which might remove the difficulty. This paper examines the effect of a slip coefficient on the flow in the neighbourhood of the contact line. Particular cases which are solved in detail are liquid–gas interfaces at an arbitrary angle, and normal contact of fluids of arbitrary viscosity. The contribution of the vicinity of the contact line to the force on the boundary is obtained. The inner region, near the contact line, must be matched with an outer flow, in which the no-slip condition can be applied, in order to obtain the total value of the force on the boundary. This force is determined for the flow of two fluids between parallel plates and in a pipe, with a plane interface. The enhanced resistance produced by the presence of the interface is calculated, and it is shown to be equivalent to an increase in the length of the column of fluid by a small multiple of the pipe radius.