Accelerated boundary integral method for multiphase flow in non-periodic geometries

TL;DR

This paper introduces an accelerated boundary integral method for simulating multiphase Stokes flow in non-periodic geometries, achieving linear or near-linear computational complexity suitable for large-scale particle simulations.

Contribution

The paper develops a novel boundary integral formulation and implements an efficient GGEM-based acceleration for multiphase flow in arbitrary geometries, especially planar slit geometries.

Findings

Achieves O(N) or O(N log N) computational complexity.

Validated with simulations of particles and drops in slit geometries.

Accurately predicts particle velocities, deformations, and trajectories.

Abstract

An accelerated boundary integral method for Stokes flow of a suspension of deformable particles is presented for an arbitrary domain and implemented for the important case of a planar slit geometry. The computational complexity of the algorithm scales as O(N) or $O(N\log N$), where $N$ is proportional to the product of number of particles and the number of elements employed to discretize the particle. This technique is enabled by the use of an alternative boundary integral formulation in which the velocity field is expressed in terms of a single layer integral alone, even in problems with non-matched viscosities. The density of the single layer integral is obtained from a Fredholm integral equation of the second kind involving the double layer integral. Acceleration in this implementation is provided by the use of General Geometry Ewald-like method (GGEM) for computing the velocity and stress fields driven by a set of point forces in the geometry of interest. For the particular case of the slit geometry, a Fourier-Chebyshev spectral discretization of GGEM is developed. Efficient implementations employing the GGEM methodology are presented for the resulting single and the double layer integrals. The implementation is validated with test problems on the velocity of rigid particles and drops between parallel walls in pressure driven flow, the Taylor deformation parameter of capsules in simple shear flow and the particle trajectory in pair collisions of capsules in shear flow. The computational complexity of the algorithm is verified with results from several large scale multiparticle simulations.