Motion of a Spherical Particle Attached to the Interface Between Two Viscous Fluids

The motion of small particles, attached to fluid interfaces, is important for the production of 2D-ordered micro- and nano-layers, which are applied for the production of solar panels, CCDs, and bio-memory chips. The problem was solved semi-analytically for water/air interface and three-phase contact angles α ≤ 90∘, using the Mehler–Fox transformation (Zabarankin, Proc R Soc A 463:2329–2349, 2007). We propose a numerical method, based on the gauge formulation of the Stokes equations for two viscous fluids, for calculating the velocity field, pressure, and drag force coefficient. The method is applicable for all values of α and fluid viscosities. The weak singularity of the solutions at the three-phase contact line is studied and the respective phase diagram is calculated. The isolation of the type of singularity helps us to construct an efficient second-order numerical scheme, based on the ADI approach. The problem is solved numerically for different particle positions at the interface and ratios of the fluid viscosities.