The effect of interfacial slip on the motion and deformation of a droplet in an unbounded arbitrary Stokes flow

TL;DR

This paper derives general expressions for how slip at the interface affects the motion and deformation of a droplet in unbounded Stokes flow, extending previous specific case results to arbitrary flows.

Contribution

It introduces a unified framework to analyze the impact of interfacial slip on droplet dynamics in any Stokes flow, generalizing prior specific flow case studies.

Findings

Derived expressions for slip effects on droplet velocity and shape deformation.

Extended Faxén's law to include interfacial slip effects.

Validated results against classical shear and Poiseuille flow cases.

Abstract

The motion and deformation of a droplet suspended in an unbounded fluid with an arbitrary, but Stokesian, imposed flow is investigated when there is a slip at the interface between the two liquids. The boundary condition at the interface is accounted by means of a simple Navier slip condition. Expressions are derived considering the effect of slip on the velocity and the shape deformation of the droplet for any arbitrary imposed flow field, and results are presented for the specific cases of shear flow and Poiseuille flow with the results of Hetsroni and Haber (J. Fluid Mech., 1970, vol. 41(04), pp. 689-705); and Ramachandran and Leal (J. Rheol., 2012, vol. 56(6), pp. 1555-1587) as the limiting cases of our general expressions. The modification to Fax\'en's law is also presented in the above perspective.