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 molecular 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 attached 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 (ηsdil) and shear (ηssh) surface viscosities. The particle intercepts a circular disc in the interface, whose size depends on the particle penetration inside the VDB. The three-dimensional fluids inside and outside the interface may be different. The analysis holds in the low Reynolds number and low capillary number regime. A toroidal (x1,x2,φ) coordinate system is introduced, which considerably simplifies the geometry of the problem. Then the hydrodynamic equations and boundary conditions are written in x1,x2,φ. The solution is searched for the first-order Fourier component of the velocity field in the radial angle φ. Reformulating the equations in “two-vorticity-one-velocity” representation, one basically ends up with a set of equations in x1,x2 only. This set is numerically solved by means of the Alternating-Direction-Implicit method. Numerical results show that the particle friction is influenced both by the viscosity and by the finiteness of the VDB volume. Finite-size effects have two origins: a recirculation effect when a/R is not very small, and an overall rotation of the VDB-particle complex when ηs is very large. In principle, the theory allows for a quantitative determination of ηs whatever a/R, including the limit a/R=0 (flat interface).