Capillary torque on a rolling particle in the presence of a liquid film at small capillary numbers

Abstract

A theoretical analysis was developed for the capillary torque acting on a spherical particle rolling on a flat surface in the presence of a thin liquid film. The capillary number (the ratio of viscous force to surface tension force) is assumed to be sufficiently small that the liquid bridge has a circular cross-section. The theory identifies two mechanisms for capillary torque. The first mechanism results from the rearward shift of the liquid bridge in the presence of particle rolling, which causes the line of action of the pressure force within the liquid bridge to be located behind the particle centroid, inducing a torque that resists particle rolling. The second mechanism results from the contact angle asymmetry on the advancing and receding sides of the rolling particle, which leads to a net torque on the particle arising from the tangential component of the surface tension force. Estimates for these two types of capillary torque are obtained using experimental data, and correlations for both torques are obtained in the form of power-law fits as functions of the capillary number. When combined with a standard expression for viscous torque on a rolling particle, the capillary torque expressions are found to yield predictions for particle terminal velocity that are in good agreement with experimental data for a particle rolling down an inclined surface.