Drag force on particles at a fluid interface in creeping flows

The problem of particles attached to an interface between two immiscible fluids has been extensively studied and has a wide variety of engineering and medical applications. Here we present an asymptotic/numerical investigation of the fluid motion past spherical particles attached to a deformable interface between two immiscible fluids undergoing uniform creeping flows in the limit of small Capillary number. Under the assumption of a constant three-phase contact angle, we analytically obtain the interfacial deformation around a single particle and numerically the two-particle deformation. Applying the Lorentz reciprocal theorem to the zeroth-order approximation for spherical particles at a flat interface and to the first correction in Capillary number allows us to obtain explicit analytical expressions for the hydrodynamic drag in terms of the zeroth-order approximations and the correction deformations. The drag coefficients are computed as a function of the three-phase contact angle, the viscosity ratio of the two fluids, the Bond number, and the separation distance between the particles.