  This paper is concerned with a kineitc-fluid model with random initial inputs
in the fine particle regime, which is a system coupling the incompressible
Navier-Stokes equations and the Vlasov-Fokker-Planck equations that model
dispersed particles of different sizes. A uniform regularity for random initial
data near the global equilibrium is established in some suitable Sobolev spaces
by using energy estimates, and we also prove the energy decays exponentially in
time by hypocoercivity arguments, which means that the long time behavior of
the solution is insensitive to the random perturbation in the initial data. For
the generalized polynomial chaos stochastic Galerkin method (gPC-sG) for the
model, with initial data near the global equilibrium and smooth enough in the
physical and random spaces, we prove that the gPC-sG method has spectral
accuracy, uniformly in time and the Knudsen number, and the error decays
exponentially in time.
