  We compute the joint distribution of relative velocities and separations of
identical inertial particles suspended in randomly mixing and turbulent flows.
Our results are obtained by matching asymptotic forms of the distribution. The
method takes into account spatial clustering of the suspended particles as well
as singularities in their motion (so-called 'caustics'). It thus takes proper
account of the fractal properties of phase space and the distribution is
characterised in terms of the corresponding phase-space fractal dimension D_2.
The method clearly exhibits universal aspects of the distribution (independent
of the statistical properties of the flow): at small particle separations R and
not too large radial relative speeds |V_R|, the distribution of radial relative
velocities exhibits a universal power-law form \rho(V_R,R) \sim |V_R|^{D_2-d-1}
provided that D_2 < d+1 (d is the spatial dimension) and that the Stokes number
St is large enough for caustics to form. The range in V_R over which this power
law is valid depends on R, on the Stokes number, and upon the nature of the
flow. Our results are in good agreement with results of computer simulations of
the dynamics of particles suspended in random velocity fields with finite
correlation times. In the white-noise limit the results are consistent with
those of [Gustavsson and Mehlig, Phys. Rev. E84 (2011) 045304].
