Not yet reviewed by Pith; the record is open.

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.