A highly accurate boundary integral equation method for surfactant-laden drops in 3D

TL;DR

This paper introduces a highly accurate boundary integral equation method for simulating 3D viscous drops with surfactants in Stokes flow, enabling precise modeling of drop deformation and interactions at microfluidic scales.

Contribution

It presents a novel boundary integral approach using spherical harmonics, advanced reparameterization, and specialized quadrature for accurate, stable simulations of surfactant-laden drops.

Findings

High accuracy in simulating drop deformation

Effective handling of close interactions and viscosity differences

Stable adaptive time stepping with implicit surfactant diffusion

Abstract

The presence of surfactants alters the dynamics of viscous drops immersed in an ambient viscous fluid. This is specifically true at small scales, such as in applications of droplet based microfluidics, where the interface dynamics become of increased importance. At such small scales, viscous forces dominate and inertial effects are often negligible. Considering Stokes flow, a numerical method based on a boundary integral formulation is presented for simulating 3D drops covered by an insoluble surfactant. The method is able to simulate drops with different viscosities and close interactions, automatically controlling the time step size and maintaining high accuracy also when substantial drop deformation appears. To achieve this, the drop surfaces as well as the surfactant concentration on each surface are represented by spherical harmonics expansions. A novel reparameterization method is introduced to ensure a high-quality representation of the drops also under deformation, specialized quadrature methods for singular and nearly singular integrals that appear in the formulation are evoked and the adaptive time stepping scheme for the coupled drop and surfactant evolution is designed with a preconditioned implicit treatment of the surfactant diffusion.