  A new lattice Boltzmann method for simulating multiphase flows is developed
theoretically. The method is adjusted such that its continuum limit is the
Navier-Stokes equation, with a driving force derived from the Cahn-Hilliard
free energy. In contrast to previous work, however, the bulk and interface
terms are decoupled, the former being incorporated into the model through the
local equilibrium populations, and the latter through a forcing term. We focus
on gas-liquid phase equilibria with the possibility to implement an arbitrary
equation of state. The most novel aspect of our approach is a systematic
Chapman-Enskog expansion up to the third order. Due to the third-order gradient
in the interface forcing term, this is needed for full consistency with both
hydrodynamics and thermodynamics. Our construction of a model that satisfies
all conditions is based upon previous work by Chen, Goldhirsch, and Orszag (J.
Sci. Comp. 34, 87 (2008)), and implies 59 and 21 velocities in three and two
dimensions, respectively. Applying the conditions of positivity of weights,
existence of a two-phase region in the phase diagram, and positivity of the
bulk viscosity, we find substantial restrictions on the permitted equation of
state, which can only be lifted by an even more refined model. Moreover, it
turns out that it is necessary to solve a self-consistent equation for the
hydrodynamic flow velocity, in order to enforce the identity of momentum
density and mass current on the lattice. The analysis completely identifies all
spurious terms in the Navier-Stokes equation, and thus shows how to
systematically eliminate each of them, by constructing a suitable collision
operator. (continued - see article PDF)

Statistical Mechanics