Molecular to fluid dynamics: the consequences of stochastic molecular motion

The derivation of fluid dynamic equations from molecular equations is considered. This is done on the basis of a stochastic model for the molecular motion which can be obtained by a projection of underlying deterministic equations. The stochastic model is used to derive fluid dynamic equations where the molecular stress tensor and heat flux appear as unknowns. However, the stochastic model also implies transport equations for these quantities. Combined with the assumption of a local equilibrium state, these transport equations can be used to derive a hierarchy of algebraic expressions for the molecular stress tensor and heat flux. A scaling analysis then explains the range of applicability of the Navier-Stokes model. The latter is relevant, for example, to simulations of high-Mach-number turbulent flows.