Sliding of a liquid spherical droplet in an external insoluble liquid at low Reynolds numbers

TL;DR

This paper investigates the motion of liquid droplets in an external viscous liquid at low Reynolds numbers, introducing a generalized Hadamard-Rybczynski equation by applying the Navier slip condition at the interface, which accounts for partial slip effects.

Contribution

The study applies the Navier boundary condition to the liquid-liquid interface, deriving a generalized HR equation that includes slip length, extending the classical model for droplet motion.

Findings

The generalized HR equation reduces to the classical form when slip length is zero.

The model accounts for partial slip at the interface, influencing droplet velocity.

It provides a new framework for understanding droplet motion with non-zero slip conditions.

Abstract

The experiment shows that small liquid droplets under the action of gravity and the Archimedes force move in the external viscous liquid practically according to the Stokes drag force equation, and not in accordance with the Hadamard-Rybczynski (HR) formula, which was specially developed to describe the motion of a liquid droplet in an external viscous liquid. Various mechanisms are proposed to explain this: increased viscosity at the interface between two liquids and the presence of unaccounted surfactants. However, there is another fundamental mechanism that has not been taken into account. It can be expected that the velocities of such liquids, insoluble in each other, may not equalize at the boundary of the droplet. No slip condition may be may be unnatural at the droplet interface. In this paper, the Navier condition is applied to the liquid-liquid boundary for the first time. A generalized HR equation is obtained. If slip length {\lambda}=0 that equation transforms into the usual HR equation. At certain {\lambda}, we arrive at a model with continuity of the components of the viscous stress tensor at the interface of two fluids. For infinite viscosity of the drop, it becomes a well-known relation generalizing the Stokes drag force for a solid sphere, taking into account the boundary condition of partial slip.