3.4 Particle-based mesoscopic models
Earlier, we outlined the connection between the Boltzmann equation (a
particular case of the master equation) and the continuum transport
equations. However, at the nano to mesoscopic lengthscales, neither the
molecular description using molecular dynamics nor a continuum
description based on the Navier-Stokes equation are optimal to study
nanofluid flows. The number of atoms is too large for MD to be
computationally tractable. The microscopic-level details, including
thermal fluctuations, play an essential role in demonstrating the
dynamic behavior, an effect which is not readily captured in continuum
transport equations. Development of particle-based mesoscale simulation
methods overcomes these difficulties, and the most common coarse-grained
models used to simulate the nanofluid flows are Brownian dynamics (BD)
and multi-particle collision dynamics (MPCD) methods. The general
approach used in all these methods is to average out relatively
insignificant microscopic details in order to obtain reasonable
computational efficiency while preserving the essential
microscopic-level details.
3.4.1 Brownian dynamics (BD) simulations: The physical system of
nanofluids contains relatively small solvent molecules and relatively
larger nanoparticles, which move much more slowly due to their larger
size. A broad range of time scales, from short time steps for the fast
motion to very long runs for the evolution of the slower mode, needs to
be accommodated by any simulation method as applied to nanofluids,
making the process time-consuming. However, in the BD simulation
technique, explicit solvent molecules are replaced by a stochastic
force, and the hydrodynamic forces mediated by them are accounted for
through a hydrodynamic interaction (HI) kernel. The BD equation thus
replaces Newton’s equations of motion in the absence of inertia: