  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.
