  We deal with the flows of non-Newtonian fluids in three dimensional setting
subjected to the homogeneous Dirichlet boundary condition. Under the natural
monotonicity, coercivity and growth condition on the Cauchy stress tensor
expressed by a power index $p\ge 11/5$ we establish regularity properties of a
solution with respect to time variable. Consequently, we can use this better
information for showing the uniqueness of the solution provided that the
initial data are good enough for all power--law indexes $p\ge 11/5$. Such a
result was available for $p\ge 12/5$ and therefore the paper fills the gap and
extends the uniqueness result to the whole range of $p$'s for which the energy
equality holds.
