On the slow translation of a solid submerged in a fluid with a surfactant surface film—II

In Shail & Gooden (1982) the problem of a solid particle translating in a semi-infinite fluid, whose surface is contaminated with a surfactant film, was examined in the quasi-steady Stokes flow régime. Various linearised models governing the variation of film concentration were considered, but the analysis was approximate in that the fluid motion generated was represented by that due to a Stokeslet situated at the centre of the particle. In this paper we remove the latter restriction and treat two specific solids, namely a rigid flat circular disk and a sphere, which move axisymmetrically perpendicular to the fluid surface. This surface is assumed to remain plane throughout the motion. The velocity field in the translating-disk problem is represented in terms of harmonic functions, and the resulting mixed boundary-value problems are reduced, for each of the film behaviours examined, to the solution of sets of simultaneous Fredholm integral equations of the second kind. These equations are solved both iteratively and numerically, and the drag on the disk is computed. For the sphere a stream-function formulation in bispherical coordinates is used. Application of the boundary conditions at the sphere and film results in infinite sets of simultaneous linear equations for the coefficients in the eigenfunction expansion of the stream function. These equations are solved by the method of truncation, and the drag on the sphere is determined.