We consider Bianchi VI spacetime, which also can be reduced to Bianchi types VI0-V-III-I. We initially consider the most general form of the energy-momentum tensor which yields anisotropic stress and heat flow. We then derive an energy-momentum tensor that couples with the spatial curvature in a way so as to cancel out the terms that arise due to the spatial curvature in the evolution equations of the Einstein field equations. We obtain exact solutions for the universes indefinitely expanding with constant mean deceleration parameter. The solutions are beriefly discussed for each Bianchi type. The dynamics of the models and fluid are examined briefly, and the models that can approach to isotropy are determined. We conclude that even if the observed universe is almost isotropic, this does not necessarily imply the isotropy of the fluid (e.g., dark energy) affecting the evolution of the universe within the context of general relativity.