Rotation in Soft Lubricated Hertzian Contacts

Forces associated with relative sliding of lubricated soft surfaces are relatively well understood. However, experiments show that the surface deformation can also produce an elastohydrodynamic torque, which has only been characterized in the limit of non-conforming surfaces (Saintyves et al. 2020). In this work, we analyze this torque on a lubricated cylinder sliding parallel to the surface of an elastic substrate, for the full range of sliding velocities from low-velocity Hertzian contact to high-velocity non-conforming contact. We numerically solve Reynolds lubrication theory in the thin fluid layer, coupled with the equilibrium equations of the elastic solid, both for a thin elastic layer and a thick elastic substrate. We then use these solutions to calculate the resulting rotation rate when the cylinder is freely suspended in the fluid. In the limit of small sliding velocities, which corresponds to a vanishingly thin film of entrained fluid, we show that the cylinder rotates as if it were purely rolling. The dimensionless rotation rate decreases at larger sliding speeds due to an increase in the fluid layer thickness and non-conformality of the surfaces. We find agreement of the rotation rate with experiments and develop analytical approximations in the limits of small and large sliding velocities. Our results suggest opportunities to control elastohydrodynamic lubrication through the application of external torques.