John Bell (Lawrence Berkeley National Laboratory)

Between Kinetic Theory and Navier-Stokes - Modeling Fluids at the Mesoscale

Traditionally fluids are modeled using the Navier-Stokes equations, which are deterministic partial differential equations that specify the evolution of density, momentum and energy. However at small scales, thermodynamic fluctuations, which reflect the molecular nature of fluids, lead to a breakdown of these equations. For systems at equilibrium the role of thermal fluctuations is relatively benign but for systems out of equilibrium, correlations between fluctuations can result in macroscopic phenomena. One approach to modeling thermal fluctuations is fluctuating hydrodynamics. Fluctuating hydrodynamics (FHD) augments traditional continuum partial differential equations with stochastic fluxes designed to capture the effect of thermal fluctuations. Here we discuss how FHD models are constructed in the simplified context of heat conduction. We then describe the development and analysis of finite-volume methods for solving the resulting stochastic system. Finally, we discuss FHD models for several more complex examples. In each case we present numerical results that validate the methodology and illustrate the impact of fluctuations on systems out of equilibrium.