In this thesis we first develop a hybrid atomistic-continuum scheme for simulating micro- and nano-flows with heat transfer. The approach is based on spatial "domain decomposition" in which molecular dynamics (MD) is used in regions where atomistic details are important, while classical continuum fluid dynamics is used in the remaining bulk regions. The two descriptions are matched in a coupling region where we ensure continuity of mass, momentum, energy and their fluxes. We demonstrate our scheme in simulating 1-D and 2-D, steady and transient heat transfers in channel flows. Good agreement between hybrid results and analytical or pure MD results is achieved. We then develop a multi-scale time algorithm to further accelerate hybrid simulations. The time algorithm is based on the temporal separation between the continuum time step and MD time step, and the assumption that the molecular dynamics in one continuum time step can be treated as quasi-steady state. We demonstrate our time algorithm by simulating isothermal channel flow driven by a sinusoidally moving wall. The results with our time algorithm converge to the original hybrid simulation results when the temporal separation is large enough, meanwhile the computational time is tremendously reduced.

We develop a highly accurate and efficient molecular approach to simulate micro and nano electrokinetic flows. We calculate the long range Coulombic interactions using the Particle-Particle Particle-Mesh (P3M) approach. The Poisson equation for the electrostatic potential is solved in physical space using an iterative multi-grid technique. We first demonstrate our approach by simulating electro-osmotic channel flow with nano roughness on the walls. By comparing with traditional pressure driven flows, our results indicate that in electro-osmotic flow the roughness affects the flow through altering the charge density distribution in the electrical double layer (EDL). We then apply our approach for simulating electrowetting on dielectric (EWOD) and for the first time we test Lippmann's prediction of the relation between contact angle and applied voltage using atomistic simulations. We also suggest a possible hybrid scheme for electrokinetic flows based on "domain decomposition" and "constrained dynamics".

From recent theoretical work on models for binary particulate systems, an interspecies stress term was derived [Zhang, Ma, and Rauenzahn, Phys. Rev. Lett. 97, 048301 (2006)] in a two-equation formulation. This term is usually neglected in other models. However this term plays the key role in recovering the averaged equations for disperse two-phase flow from binary particulate systems. As a starting point, we study the behavior of stress terms in this two-equation formulation through numerical simulations of the simplest possible granular system. We find the interspecies stress is of the same order of magnitude as other stresses for dense systems. For systems with relative motion between two species, we find that both intra and interspecies stresses are quadratically proportionally to the relative velocity. The results are important in the attempt to seek a unified mathematical description that is valid for problems ranging from molecular scale mixing to disperse multiphase flows.