  We introduce a full-Lagrangian heterogeneous multiscale method (LHMM) to
model complex fluids with microscopic features that can extend over large
spatio-temporal scales, such as polymeric solutions and multiphasic systems.
The proposed approach discretizes the fluctuating Navier-Stokes equations in a
particle-based setting using Smoothed Dissipative Particle Dynamics (SDPD).
This multiscale method uses microscopic information derived on-the-fly to
provide the stress tensor of the momentum balance in a macroscale problem,
therefore bypassing the need for approximate constitutive relations for the
stress. We exploit the intrinsic multiscale features of SDPD to account for
thermal fluctuations as the characteristic size of the discretizing particles
decrease. We validate the LHMM using different flow configurations (reverse
Poiseuille flow, flow passing a cylinder array, and flow around a square
cavity) and fluid (Newtonian and non-Newtonian). We showed the framework's
flexibility to model complex fluids at the microscale using multiphase and
polymeric systems. We showed that stresses are adequately captured and passed
from micro to macro scales, leading to richer fluid response at the continuum.
In general, the proposed methodology provides a natural link between variations
at a macroscale, whereas accounting for memory effects of microscales.
