Coupled Navier–Stokes molecular dynamics simulation: Theory and applications based on iterative operator-splitting methods

•We present a multi-scale model solved by iterative splitting methods and contribute its accuracy to standard schemes.•The model equations are based on coupled Navier–Stokes with molecular dynamics equations.•We combine relaxation and averaging methods to upscale the microscopic scales.•We apply colloidal damping and fluid–solid problems with non-Newtonian flow.•Numerical results present the benefit of the iterative methods with respect to time-dependent flows.

In this paper, we contribute a multi-scale method based on an iterative operator splitting method, which takes into account the disparity between the macro- and micro-scopic scales. We couple the Navier–Stokes equations with molecular dynamics equations, while taking into account their underlying scales. Combining relaxation methods and averaging techniques, we can optimize the computational effort. The motivation arose from modeling fluid transport under the influence of a multiscale problem that has to be solved with smaller time scales, e.g., a non-Newtonian flow problem. The application concerned colloidal damping or fluid–solid problems, where we study an area where the Navier–Stokes equations have too little information about the stream field and we need at least the Boltzmann equation to obtain information about the whole density field. A novel research field is for example that of carbon nanotubes, where we have to couple macro- and micro-models and obtain a fluid–solid area which uses the Lennard–Jones fluid model.The proposed method to solve such delicate problems enables simulations in which the continuum flow aspects of the flow are described by the Navier–Stokes equations at time-scales appropriate for this level of modeling, while the viscous stresses within the Navier–Stokes equations are the result of molecular dynamics simulations, with much smaller time-scales. The main benefit of the proposed method is that time-dependent flows can then be modeled with a computational effort which is significantly less than if the complete flow were to be modeled at the molecular level, as a result of the different time-scales at the continuum and molecular levels, enabled by the application of the iterative operator-splitting method.We carry out a convergence analysis for the splitting methods, see also [30].Finally, we present numerical results for the modified methods and applications to real-life flow problems.