  The incessant swimming motion of microbes in dense suspensions can give rise
to striking collective motions and coherent structures. However, theoretical
investigations of these structures typically utilize either computationally
demanding numerical simulations, or simplified continuum models. Here we
analytically investigate the collective dynamics of a dense array of steady,
spherical squirmers. We first calculate the forces and torques acting on two
closely-separated squirmers, through solving the Stokes equations to
second-order in the ratio of mean spacing to squirmer radius. This lubrication
analysis is then used to assess the stability of a dense, vertical, planar
array of identical three-dimensional squirmers. The system of uniformly-spaced,
vertically-oriented squirmers is stable if the gravitational torque experienced
due to bottom-heaviness is sufficiently strong, and an intercellular repulsive
force is included. The predictions of instability and possible long time
behavior is qualitatively similar for monolayers confined between two parallel
rigid planes as for unconfined monolayers. The predictions compare favorably
with published numerical simulations, and reveal the existence of additional
dynamic structures not previously observed; puller-type squirmers show a
greater range of structures than pushers. The use of pairwise lubrication
interactions provides an efficient means of assessing stability of dense
suspensions usually tackled using full numerical simulations.
