Many-body microhydrodynamics of colloidal particles with active boundary layers

TL;DR

This paper introduces a boundary integral method using tensorial spherical harmonics to efficiently compute many-body hydrodynamic interactions among active colloidal particles with boundary layers, enabling large-scale simulations.

Contribution

It develops a novel boundary integral approach with propulsion matrices for active particles, reducing computational complexity for simulating large suspensions.

Findings

Efficient computation of many-body hydrodynamic interactions.

Linear relation between rigid body motion and boundary flow coefficients.

Scalable simulations of hundreds of thousands of active particles.

Abstract

Colloidal particles with active boundary layers - regions surrounding the particles where nonequilibrium processes produce large velocity gradients - are common in many physical, chemical and biological contexts. The velocity or stress at the edge of the boundary layer determines the exterior fluid flow and, hence, the many-body interparticle hydrodynamic interaction. Here, we present a method to compute the many-body hydrodynamic interaction between $N$ spherical active particles induced by their exterior microhydrodynamic flow. First, we use a boundary integral representation of the Stokes equation to eliminate bulk fluid degrees of freedom. Then, we expand the boundary velocities and tractions of the integral representation in an infinite-dimensional basis of tensorial spherical harmonics and, on enforcing boundary conditions in a weak sense on the surface of each particle, obtain a system of linear algebraic equations for the unknown expansion coefficients. The truncation of the infinite series, fixed by the degree of accuracy required, yields a finite linear system that can be solved accurately and efficiently by iterative methods. The solution linearly relates the unknown rigid body motion to the known values of the expansion coefficients, motivating the introduction of propulsion matrices. These matrices completely characterize hydrodynamic interactions in active suspensions just as mobility matrices completely characterize hydrodynamic interactions in passive suspensions. The reduction in the dimensionality of the problem, from a three-dimensional partial differential equation to a two-dimensional integral equation, allows for dynamic simulations of hundreds of thousands of active particles on multi-core computational architectures.