Droplet Deformation and Emulsion Rheology in Two-Dimensional Odd Stokes Flow

TL;DR

This paper derives an analytical model for droplet deformation in two-dimensional odd Stokes flow, revealing how odd viscosity influences emulsion rheology and deformation beyond classical results.

Contribution

It provides the first analytical solution for droplet shape and flow in odd viscosity fluids, highlighting the effects of odd viscosity differences on emulsion rheology.

Findings

Steady-state droplet deformation remains classical at leading order.

Odd viscosity differences influence droplet deformation beyond leading order.

Flow depends only on the difference in odd viscosity between droplet and medium.

Abstract

We study the deformation of a two-dimensional viscous droplet in simple shear in the presence of odd viscosity. We derive an analytical solution for the droplet shape and surrounding flow field within the framework of odd Stokes flow, allowing for differences in both even and odd viscosity between the droplet and the surrounding fluid. This solution yields closed-form expressions for the macroscopic apparent even and odd viscosities of a dilute emulsion. We show that, provided all viscosity differences remain moderate, the steady-state Taylor deformation parameter satisfies $D_T^\infty = \text{Ca} + \mathcal{O}(\text{Ca}^2)$ so that the leading-order droplet deformation is unchanged from the classical (even-viscous) result. Nevertheless, pronounced effects emerges beyond leading order, where our direct numerical simulations reveal odd-viscous differences to the droplet deformation. In addition, we show that the flow is influenced only by the difference in odd viscosity between the droplet and the medium and not on their individual values. Our analysis clarifies how odd viscosity might modify the effective rheology of dilute emulsions and provides a framework for interpreting droplet-based measurements of odd-viscous response. Key words: odd viscosity $|$ droplets $|$ emulsions $|$ surface tension $|$ chiral fluids