Lee Lin Multiphase Method -- Obtaining Cahn Hilliard...

Hi all. I’ve been studying the Lee-Lin multiphase method recently (Phys Rev E 67 056703 2003, J Comp Phys 206 16-47 2005), and I have a problem reproducing their macroscopic equations. I can reproduce the equation for pressure and momentum exactly, but I cannot reproduce their Cahn-Hilliard-like continuity equation for density (i.e. continuity equation with diffusive source term). In particular, I find that my density satisfies a continuity equation with no source term.

Beginning from the Lattice Boltzmann equation with both the collision term and arbitrary forcing term S_i discretized using the trapezoidal method, I derive equations for zeroth and first moments of the distribution function. In the case of their ‘f’ distribution, the zeroth moment is density, and it satisfies a continuity equation with a source term equal to the zeroth moment of the forcing term S_i. This is a standard result.

This is where the problem arises. The forcing term for the ‘f’ distribution takes the form

S_i = (e_ik - u_k) * F_k * f^(eq)_i

where e_ik is a lattice vector component, u_k is macroscopic velocity, F_k is the body force, and f^(eq)_i is the equilibrium distribution function. The zeroth moment of this expression is obviously zero. Moreover, this is true for arbitrary body force F_k. This means that the source term in the continuity equation vanishes, but it must not if I am to reproduce Lee and Lin’s result.

In their 2003 paper, Lee and Lin make an attempt to address this problem, however I find their explanation lacking. In particular, they say that the form of the body force F_k determines whether or not the zeroth moment of the forcing term vanishes. I’ve thought about this over and over, and I simply disagree – as long as F_k is not a function of the lattice direction i (which it isn’t, and anyway that would be bizarre), then the zeroth moment vanishes.

Has anyone performed the Chapman-Enskog expansion on Lee-Lin? If so, did you come across a similar problem? I appreciate any help you can give.