  The object of this paper is twofold. Firstly, we study a class of generalized
Newtonian fluid related to "power law ". For the corresponding non-Newtonian
Navier-Stokes problems, the existence of a weak and periodic solutions is
proved in the large for a bounded domain in $\mathbb{R}^3$. Further,
variational inequalities and local-in-time well-posedness of the
initial-boundary value problem are investigated. Secondly, we deduce a
generalization of the Graffi-Kazhikhov-Smagulov model based on an
advective-diffusion process in the context of multiphase theory. Local-in-time
well-posedness of the initial-boundary value problem is investigated
