Drag force on spherical particles trapped at a liquid interface

TL;DR

This paper combines asymptotic and numerical methods to derive analytical expressions for the drag force on spherical particles at a deformable liquid interface, considering small deformations and contact angle deviations.

Contribution

It introduces a novel analytical approach to quantify hydrodynamic drag and capillary forces on particles at fluid interfaces, accounting for interface deformation effects.

Findings

Explicit formulas for drag coefficients as functions of contact angle and viscosity ratio.

Quantitative analysis of interfacial deformation around particles and pairs.

Calculation of capillary forces due to interface deformation.

Abstract

The dynamics of particles attached to an interface separating two immiscible fluids are encountered in a wide variety of applications. Here we present a combined asymptotic and numerical investigation of the fluid motion past spherical particles attached to a deformable interface undergoing uniform creeping flows in the limit of small Capillary number and small deviation of the contact angle from 90 degrees. Under the assumption of a constant three-phase contact angle, we calculate the interfacial deformation around an isolated particle and a particle pair. Applying the Lorentz reciprocal theorem to the zeroth-order approximation corresponding to spherical particles at a flat interface and the first correction in Capillary number and correction contact angle 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. In addition, the capillary force acting on the particles due to the interfacial deformation is calculated.