  We introduce a mesoscale method for simulating hydrodynamic transport and
self assembly of inhomogeneous polymer melts in pressure driven and drag
induced flows. This method extends dynamic self consistent field theory (DSCFT)
into the hydrodynamic regime where bulk material transport and viscoelastic
effects play a significant role. The method combines four distinct components
as a single coupled system, including (1) non-equilibrium self consistent field
theory describing block copolymer self-assembly, (2) multi-fluid Navier-Stokes
type hydrodynamics for tracking material transport, (3) constitutive equations
modeling viscoelastic phase separation, and (4) rigid wall fields which
represent moving channel boundaries, machine components, and nano-particulate
fillers. We also present an efficient, pseudospectral implementation for this
set of coupled equations which enables practical application of the model in
periodic domains. We validate the model by reproducing well known phenomena
including equilibrium diblock meso-phases, analytic Stokes flows, and
viscoelastic phase separation of glassy/elastic polymer melts. We also
demonstrate the stability and accuracy of the numerical implementation by
examining its convergence under grid-size refinement.
