In this talk, we consider a model of a binary mixture of compressible viscous and macroscopically immiscible fluids, based on the diffuse interface approximation,which we called it as the Navier-Stokes-Cahn-Hilliard system. This system has strong nonlinear, degenerative and singularity. We will mainly study the generalized Navier boundary value problem for 3D barotropoic incompressible Navier-Stokes-Cahn-Hilliard system with several rigid bodies. The global existence of weak solutions is proved for the problem of the motion of  several rigid bodies immersed in two immiscible incompressible shear-thinning non-Newtonian viscous fluid.  The viscosity depends on  the shear rate and the concentration of the fluids.  The Navier-Stokes-Cahn-Hilliard system with the Generalized Navier oundary Conditions is considered. The fictitious domain method and the pressure localization method are  used.