  Hydrodynamic interactions in a suspension of spherical particles confined
between two parallel planar walls are studied under creeping-flow conditions.
The many-particle friction matrix in this system is evaluated using our novel
numerical algorithm based on transformations between Cartesian and spherical
representations of Stokes flow. The Cartesian representation is used to
describe the interaction of the fluid with the walls and the spherical
representation is used to describe the interaction with the particles. The
transformations between these two representations are given in a closed form,
which allows us to evaluate the coefficients in linear equations for the
induced-force multipoles on particle surfaces. The friction matrix is obtained
from these equations, supplemented with the superposition lubrication
corrections. We have used our algorithm to evaluate the friction matrix for a
single sphere, a pair of spheres, and for linear chains of spheres. The
friction matrix exhibits a crossover from a quasi-two-dimensional behavior (for
systems with small wall separation H) to the three-dimensional behavior (when
the distance H is much larger than the interparticle distance L). The crossover
is especially pronounced for a long chain moving in the direction normal to its
orientation and parallel to the walls. In this configuration, a large pressure
buildup occurs in front of the chain for small values of the gapwidth H, which
results in a large hydrodynamic friction force. A standard wall superposition
approximation does not capture this behavior.
