  We present a model for the relative velocity of inertial particles in
turbulent flows. Our general formulation shows that the relative velocity has
contributions from two terms, referred to as the generalized acceleration and
generalized shear terms, because they reduce to the well known acceleration and
shear terms in the Saffman-Turner limit. The generalized shear term represents
particles' memory of the flow velocity difference along their trajectories and
depends on the inertial particle pair dispersion backward in time. The
importance of this backward dispersion in determining the particle relative
velocity is emphasized. We find that our model with a two-phase separation
behavior, an early ballistic phase and a later tracer-like phase, as found by
recent simulations for the forward (in time) dispersion of inertial particle
pairs, gives good fits to the measured relative speeds from simulations at low
Reynolds numbers. In the monodisperse case with identical particles, the
generalized acceleration term vanishes and the relative velocity is determined
by the generalized shear term. At large Reynolds numbers, our model gives a
$St^{1/2}$ dependence of the relative velocity on the Stokes number $St$ in the
inertial range for both the ballistic behavior and the Richardson separation
law. This leads to the same inertial-range scaling for the two-phase separation
that well fits the simulation results. Our calculations for the bidisperse case
show that, with the friction timescale of one particle fixed, the relative
speed as a function of the other particle's friction time has a dip when the
two timescales are similar. We find that the primary contribution at the dip is
from the generalized shear term, while the generalized acceleration term is
dominant for particles of very different sizes.
