Hydrodynamic fluctuations in a particle-continuum hybrid for complex fluids

A previously-developed hybrid particle-continuum method [J. B. Bell, A. Garcia and S. A. Williams, SIAM Multiscale Modeling and Simulation, 6:1256-1280, 2008] is generalized to dense fluids and two and three dimensional flows. The scheme couples an explicit fluctuating compressible Navier-Stokes solver with the Isotropic Direct Simulation Monte Carlo (DSMC) particle method [A. Donev and A. L. Garcia and B. J. Alder, J. Stat. Mech., 2009(11):P11008, 2009]. To achieve bidirectional dynamic coupling between the particle (microscale) and continuum (macroscale) regions, the continuum solver provides state-based boundary conditions to the particle subdomain, while the particle solver provides flux-based boundary conditions for the continuum subdomain; see [A. Donev, J.B. Bell, A. Garcia, and B. Alder, SIAM Multiscale Modeling and Simulation, 8:871-911, 2010.] for details. This paper summarizes two important numerical tests: First, the equilibrium diffusive (Brownian) motion of a large spherical bead suspended in a particle fluid is examined, demonstrating that the hybrid method correctly reproduces the velocity autocorrelation function of the bead but only if thermal fluctuations are included in the continuum solver. Second, the new scheme is applied to the well-known adiabatic piston problem and we find that it correctly reproduces the slow non-equilibrium relaxation of the piston toward thermodynamic equilibrium but, again, only if the continuum solver includes stochastic (white-noise) flux terms. These two fundamental examples clearly demonstrate the need to include fluctuations in continuum solvers employed in hybrid multiscale methods.