  Considering turbulent clouds containing small inertial particles, we
investigate the effect of particle collision, in particular
collision-coagulation, on particle clustering and particle relative motion. We
perform direct numerical simulation (DNS) of coagulating particles in isotropic
turbulent flow in the regime of small Stokes number ($St=0.001-0.54$) and find
that, due to collision-coagulation, the radial distribution functions (RDFs)
fall-off dramatically at scales $r \sim d\,\,$ (where $d$ is the particle
diameter) to small but finite values, while the mean radial-component of
particle relative velocities (MRV) increase sharply in magnitudes. Based on a
previously proposed Fokker-Planck (drift-diffusion) framework, we derive a
theoretical account of the relationship among particle collision-coagulation
rate, RDF and MRV. The theory includes contributions from
turbulent-fluctuations absent in earlier mean-field theories. We show
numerically that the theory accurately accounts for the DNS results (i.e.,
given an accurate RDF, the theory could produce an accurate MRV). Separately,
we also propose a phenomenological model that could directly predict MRV and
find that it is accurate when calibrated using fourth moments of the fluid
velocities. We use the model to derive a general solution of RDF. We uncover a
paradox: the past empirical success of the differential version of the theory
is theoretically unjustified. We see a further shape-preserving reduction of
the RDF (and MRV) when the gravitational settling parameter ($S_g$) is of order
$O(1)$. Our results demonstrate strong coupling between RDF and MRV and imply
that earlier isolated studies on either RDF or MRV have limited relevance for
predicting particle collision rate.
