  This paper deals with the finite-time blow-up phenomena of classical
solutions for Vlasov/Navier-Stokes equations under suitable assumptions on the
initial configurations. We show that a solution to the coupled kinetic-fluid
system may be initially smooth, however, it can become singular in a finite
period of time. We provide a simple idea of showing the finite time blow up of
classical solutions to the coupled system which has not been studied so far. We
also obtain analogous results for related systems, such as isentropic
compressible Navier-Stokes equations, two-phase fluid equations consisting of
pressureless Euler equations and Navier-Stokes equations, and thick sprays
model.
