  We introduce a new heterogeneous multi-scale method for the simulation of
flows of non-Newtonian fluids in general geometries and present its application
to paradigmatic two-dimensional flows of polymeric fluids. Our method combines
micro-scale data from non-equilibrium molecular dynamics (NEMD) with
macro-scale continuum equations to achieve a data-driven prediction of complex
flows. At the continuum level, the method is model-free, since the Cauchy
stress tensor is determined locally in space and time from NEMD data. The
modelling effort is thus limited to the identification of suitable interaction
potentials at the micro-scale. Compared to previous proposals, our approach
takes into account the fact that the material response can depend strongly on
the local flow type and we show that this is a necessary feature to correctly
capture the macroscopic dynamics. In particular, we highlight the importance of
extensional rheology in simulating generic flows of polymeric fluids.
