  In this paper, we prove some a priori estimates for a system of partial
differential equations arising in the nonstationary flow of a nonhomogeneous
incompressible asymmetric fluid in a bounded domain with smooth boundary. The
unknowns of the system are the velocity field of the fluid particles, the
angular velocity of rotation of the fluid particles, the mass density of the
fluid and the pressure distribution. For the density functions we consider the
application of the Helmholtz decomposition.
