  The Rayleigh-Plateau instability occurs when surface tension makes a fluid
column become unstable to small perturbations. At nanometer scales, thermal
fluctuations are comparable to surface energy densities. Consequently, at these
scales, thermal fluctuations play a significant role in the dynamics of the
instability. These microscopic effects have previously been investigated
numerically using particle-based simulations, such as molecular dynamics, and
stochastic partial differential equation based hydrodynamic models, such as
stochastic lubrication theory. In this paper we present an incompressible
fluctuating hydrodynamics model with a diffuse-interface formulation for binary
fluid mixtures designed for the study of stochastic interfacial phenomena. An
efficient numerical algorithm is outlined and validated in numerical
simulations of stable equilibrium interfaces. We present results from
simulations of the Rayleigh-Plateau instability for long cylinders pinching
into droplets for Ohnesorge numbers of Oh = 0.5 and 5.0. Both stochastic and
perturbed deterministic simulations are analyzed and ensemble results show
significant differences in the temporal evolution of the minimum radius near
pinching. Short cylinders, with lengths less than their circumference, were
also investigated. As previously observed in molecular dynamics simulations, we
find that thermal fluctuations cause these to pinch in cases where a perturbed
cylinder would be stable deterministically. Finally we show that the
fluctuating hydrodynamics model can be applied to study a broader range of
surface-tension driven phenomena.
