A continuum and molecular dynamics hybrid multiscale method is
developed to simulate micro- and nano-fluid flows. The method uses the
continuum Navier-Stokes equations for one flow region and atomistic
molecular dynamics for another. The spatial coupling between continuum
equations and molecular dynamics is achieved through constrained
dynamics in an overlap region which can be obtained through finding
the extremum of the integral of the Lagrangian under the constraint of
velocity matching.  To validate the numerical method, the proposed
multiscale method is used to simulate sudden-start Couette flow and
channel flow with nano-scale rough walls, showing quantitative
agreement with results from analytical solutions and full molecular
dynamics simulations for different parameter regimes. A hybrid
numerical method is then used to study corner singularities in driven
cavity flow. Continuum equations with no-slip boundary conditions
predict singular stresses at the corners between moving and static
walls. Molecular dynamics simulations are used to resolve these
singular regions, and the flow field in the remainder of the cavity is
obtained from the Navier-Stokes (NS) equation. This hybrid solution
agrees well with fully atomistic simulations on small systems. The
multiscale method allows five order of magnitude speed up in
calculations on larger systems and is able to access length scales
ranging from milimeter to single molecular when the multiscale method
is combined with adaptive grid method.  By comparing the hybrid
solution to NS solutions at different Reynolds number, we show that
slip occurs near the corners and that it can not be described by the
Navier slip condition. At high wall velocities, the fluid also
exhibits non-Newtonian behavior within five atomic diameters from the
corner. The numerical algorithms pertinent to multiscale time and fast
convergence to steady state flows will be presented. The applications
of the proposed multiscale method for multiphase flows, thermal flows
and electrokentic flows will be discussed.