Sliding of a liquid spherical drop in an external fluid: a generalization of the Hadamard-Rybczynski equation

TL;DR

This paper derives a generalized Hadamard-Rybczynski equation for the slow movement of a liquid droplet in another fluid, incorporating partial slip at the interface, and applies it to experimental data relevant to emulsions and industry.

Contribution

It introduces a generalized equation accounting for partial slip at the liquid-liquid interface, extending classical models to more realistic boundary conditions.

Findings

Derived a generalized Hadamard-Rybczynski equation with partial slip.

Applied the model to experimental droplet fall velocity data.

Identified conditions where the model best describes hydrophobic and hydrophilic interfaces.

Abstract

An analytical solution is obtained for the problem of the slow movement of a small drop of liquid in another immiscible liquid in an infinitely large reservoir with the boundary condition of partial slip at the liquid-liquid interface. That generalizes the conventional Navier condition of partial slip that given at the liquid-solid interface, since the solid (rigid) state can be considered as a liquid with infinite viscosity. A generalized Hadamard-Rybczynski equation (HRE) is obtained. If slip length {\lambda}=0 that equation transforms into the conventional HRE. For infinite viscosity of the droplet, generalized HRE becomes a well-known relation generalizing the Stokes drag force for a solid sphere, taking into account the boundary condition of partial slip. At certain {\lambda}, we arrive at a model with continuity of the all components of the viscous stress tensor at the interface of two fluids, including diagonal tensor components. Generalized HRE is applied to the interpretation of the experiment in which the velocity of falling of a spherical droplet of silicate oil in the castor oil is investigated. Streamlines have been built corresponding to both the Hadamard-Rybczynski model and the partial slip approach. Presumably, the best applicability of the generalized HRE should be expected for the interface of hydrophobic liquid and hydrophilic one (water - hydrocarbons, water - higher alcohols, in general: aqueous emulsions, water - lipophilic organic liquids and oils, etc.). These are quite important emulsions in practical terms, for example, for the oil industry and medicine. Experimental methods for determining the slip length are discussed.