Viscous drag of a solid sphere straddling a spherical or flat surface

The aim of this paper is to compute the friction felt by a solid particle, of radius a, located across a flat or spherical interface of radius R, and moving parallel to the interface. This spherical interface can be a molecul ar film around an emulsion or aerosol droplet, the membrane of a vesicle or the soap film of a foam bubble. For simplicity, the acronym VDB is used to refer to either vesicle, drop, or bubble. The theory is designed as a tool to interpret surface viscosimetry experiments involving spherical probes a ttached to films or model membranes, taking care of the finite-size effects when the film encompasses a finite fluid volume. The surface of the VDB is a two-dimensional fluid, characterized by dilational (eta(s)(dil)) and she ar (eta(s)(sh)) surface viscosities. The particle intercepts a circular dis c in the interface, whose size depends on the particle penetration inside t he VDB. The three-dimensional fluids inside and outside the interface may b e different. The analysis holds in the low Reynolds number and low capillar y number regime. A toroidal (x(1),x(2),phi) coordinate system is introduced , which considerably simplifies the geometry of the problem. Then the hydro dynamic equations and boundary conditions are written in x(1),x(2),phi. The solution is searched for the first-order Fourier component of the velocity field in the radial angle phi. Reformulating the equations in "two-vortici ty-one-velocity" representation, one basically ends up with a set of equati ons in x(1),x(2) only. This set is numerically solved by means of the Alter nating-Direction-Implicit method. Numerical results show that the particle friction is influenced both by the viscosity and by the finiteness of the V DB volume. Finite-size effects have two origins: a recirculation effect whe n a/R is not very small, and an overall rotation of the VDB-particle comple x when eta(s) is very large. In principle, the theory allows for a quantita tive determination of eta(s) whatever a/R, including the limit a/R=0 (flat interface). (C) 2000 American Institute of Physics. [S1070-6631(00)01210-1] .