  We study how hydrodynamic interactions affect the collective behaviour of
active particles suspended in a fluid at high concentrations, with particular
attention to lubrication forces which appear when the particles are very close
to one another. We compute exactly the limiting behaviour of the hydrodynamic
interactions between two spherical (circular) active swimmers in very close
proximity to one another in the general setting in both three and (two)
dimensions. Combining this with far-field interactions, we develop a novel
numerical scheme which allows us to study the collective behaviour of large
numbers of active particles with accurate hydrodynamic interactions when close
to one another. We study active swimmers whose intrinsic flow fields are
characterised by force dipoles and quadrupoles. Using this scheme, we are able
to show that lubrication forces when the particles are very close to each other
can play as important a role as long-range hydrodynamic interactions in
determining their many-body behaviour. We find that when the swimmer force
dipole is large, finite clusters and open gel-like clusters appear rather than
complete phase separation. This suppression is due to near-field lubrication
interactions. For swimmers with small force dipoles, we find surprisingly that
a globally polar ordered phase appears because near-field lubrication rather
than long-range hydrodnamics dominate the alignment mechanism. Polar order is
present for very large system sizes and is stable to fluctuations with a finite
noise amplitude. We explain the emergence of polar order using a minimal model
in which only the leading rotational effect of the near-field interaction is
included. These phenomena are also reproduced in two dimensions.
