2 0 Ja n 20 04 Hybrid molecular-continuum fluid dynamics

The flow of complex fluids near interfaces is governed by a subtle interplay between the fast microscopic dynamics within a small localised region of the system close to the interface and the slow dynamics in the bulk fluid region. This scenario is encountered in a wide variety of applications ranging from nanotechnology (nanofluidics) and other industrial processes (such as wetting, droplet formation, critical fluids near heated surfaces or crystal growth from a fluid phase) to biological systems (for example, membranes or biomolecules near interfaces). The dynamics of these systems depends on the intimate connection of many different spatio-temporal scales: from the nanoscale to the microscale and beyond. Realistic simulations of such systems via standard classical molecular dynamics (MD) are prohibitive, while continuum fluid dynamics (CFD) cannot describe the important details within the interfacial region. In view of this fact, the field of computer simulation is now faced with the need for new techniques, which bridge a wider range of time and length scales with the minimum loss of information. A hybrid particle-continuum approach provides a resolution to this dilemma. A hybrid algorithm retains all the atomistic detail within the relevant localized domain and couples this region to the continuum hydrodynamic description of the remainder of the system. Indeed hybrid algorithms for liquids can be expected to provide a powerful tool for the fast growing field of nanofluidics in micro electro-mechanical systems (MEMS) and our ongoing contributions have been recognized by the nanoscience community (R. Delgado-Buscalioni and Coveney 2003a) as offering a promising simulation technique with nanotechnological applications. Hybrid algorithms for solids (Abraham et al. 1998) and gases (Garcia et al 1999) were the first to be fully developed in the literature. As expected in most theoretical descriptions of matter, the hybrid description of the liquid state is the most challenging one. The general procedure is to connect the particle domain (P) and the continuum domain (C) within an overlapping region comprised of two buffers: C→P and P→C (see figure 1). Within the P→C buffer the particle dynamics are coarse-grained to extract the boundary conditions for the C-region. The most complicated part of any hybrid scheme is the C→P coupling where the microscopic dynamics need to be reconstructed to adhere to the prescriptions given by the continuum variables. Moreover, in doing so the unphysical artifacts thereby introduced should be minimized (following Occam’s razor). In this paper we provide an overview of the state-of-the-art of the hybrid modelling of liquids. In §II we start by presenting an overview of the hybrid scheme and some preliminary topics such as the inherent constraints on the continuum time step and the spatial-grid size. Section III discusses several implementations of the temporal-coupling. The C→P coupling scheme is then explained in §IV for the general case of mass, momentum and energy. We illustrate this important part of the scheme by reproducing the three hydrodynamic modes (shear, sound and heat) governing the relaxing flows in an infinite medium. Section V is devoted to the P→C coupling, based on a finite volume method solving the flow within the C domain. Some comments on the effect of noise on the accuracy of the scheme are made. The full method is used in §VI to solve the problem of shear flow driven by oscillatory wall motion in a nano-slot. Finally, conclusions and future directions for this research are described in §VII.