We study the system of PDEs describing unsteady flows of incompressible fluids with variable density and non-constant viscosity. Indeed, one considers a stress tensor being a nonlinear function of the symmetric velocity gradient, verifying the properties of $p$-coercivity and $(p-1)$-growth, for a given parameter $p>1$. The existence of Dirichlet weak solutions was obtained in [2], in the cases $p\ge 12/5$ if $d=3$ or $p\ge 2$ if $d=2$, $d$ being the dimension of the domain. In this paper, with help of some new estimates (which lead...