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We propose a method for multi-scale hybrid simulations of molecular dynamics (MD) and computational fluid dynamics (CFD). In the method, usual lattice-mesh based simulations are applied for CFD level, but each lattice is associated with a small MD cell which generates a "local stress" according to a "local flow field" given from CFD instead of using any constitutive functions at CFD level. We carried out the hybrid simulations for some elemental flow problems of simple Lennard-Jones liquids and compared the results with those obtained by usual CFDs with a Newtonian constitutive relation in order to examine the validity of our hybrid simulation method. It is demonstrated that our hybrid simulations successfully reproduced the correct flow behavior obtained from usual CFDs as far as the mesh size ∆x and the time-step ∆t of CFD are not too large comparing to the system size l MD and the sampling duration t MD of MD simulations performed at each time step of CFDs. Otherwise, simulations are affected by large fluctuations due to poor statistical averages taken in the MD part. Properties of the fluctuations are analyzed in detail. PACS numbers: 31.15.xv 46.15.-x

The flow behaviors of polymer melt composed of short chains with ten beads between parallel plates are simulated by using a hybrid method of molecular dynamics and computational fluid dynamics. Three problems are solved: creep motion under a constant shear stress and its recovery motion after removing the stress, pressure-driven flows, and the flows in rapidly oscillating plates. In the creep/recovery problem, the delayed elastic deformation in the creep motion and evident elastic behavior in the recovery motion are demonstrated. The velocity profiles of the melt in pressure-driven flows are quite different from those of Newtonian fluid due to shear thinning. Velocity gradients of the melt become steeper near the plates and flatter at the middle between the plates as the pressure gradient increases and the temperature decreases. In the rapidly oscillating plates, the viscous boundary layer of the melt is much thinner than that of Newtonian fluid due to the shear thinning of the melt. Three different rheological regimes, i.e., the viscous fluid, viscoelastic liquid, and viscoelastic solid regimes, form over the oscillating plate according to the local Deborah numbers. The melt behaves as a viscous fluid in a region for omegatauR < approximately 1 , and the crossover between the liquidlike and solidlike regime takes place around omegataualpha approximately equal 1 (where omega is the angular frequency of the plate and tauR and taualpha are Rouse and alpha relaxation time, respectively).