  In the present work, we first introduce a general framework for modelling
complex multiscale fluids and then focus on the derivation and analysis of a
new hybrid continuum-kinetic model. In particular, we combine conservation of
mass and momentum for an isentropic macroscopic model with a kinetic
representation of the microscopic behaviour. After introducing a small scale of
interest, we compute the complex stress tensor by means of Irving-Kirkwood
formula. The latter requires an expansion of kinetic distribution around an
equilibrium state and a successive homogenization over the fast in time and
small in space scale dynamics. For a new hybrid continuum-kinetic model the
results of linear stability analysis indicates a conditional stability in the
relevant low-speed regimes and instability for high speed regimes for higher
modes. Extensive numerical experiments confirm that the proposed multiscale
model can reflect new phenomena of complex fluids not being present in standard
Newtonian fluids. Consequently, the proposed general technique can be
successfully used to derive new interesting systems combining the macro and
micro structure of a given physical problem.
