Dear PeterH,

this is a really good question. The answer is not so trivial and longer than you think. I am assuming that T=time, M=mass, L=length. With this in mind, I end up with

[G] = L^3 * M^-1

which is the inverse of the units of the density. And this actually fits both equations: the interaction strength equation and the pressure equation.

First important thing is that the shan chen force is actually a volume force, thus, its units are Newton/meter^3 or in your notation M^1 * L^-2 * T^-2. Now, the force equation reads

F = -G \psi sum_i w_i psi(x + e_i) e_i

where the sum is a finite difference approximation for grad(psi). The problem now is that you can find two sets of weighting coefficients w_i in the literature. Take the following discretisation as an example:

e_i = (1,0) (0,1) (-1,0) (0,-1) (1,1) (-1,1) (-1,-1) (1,-1)

then usually the corresponding weights are

1/9, 1/9, 1/9, 1/9, 1/36, 1/36, 1/36, 1/36

However, these actually contain the squared lattice speed of sound! In the derivation of the finite difference stencils, the weights are usually normalized by the squared speed of sound, i.e.,

1/3, 1/3, 1/3, 1/3, 1/12, 1/12, 1/12, 1/12

(For the derivation, see e.g. Shan 2006 Phys Rev E 73, 047701 (2006) or Sbragaglia et al. Phys Rev E 75, 026702 (2007).

Depending on the weights you use, the weighting coefficients actually would have units.

Normalized -> w_i is dimensionless

Not normalized -> w_i has dimensions L^2*T^-2

Whats the consequence of this? If you use NORMALIZED weights, your finite difference formula will approximate

F = -G psi grad(psi)

On the other hand, if you use the non-normalized weights (1/9, 1/9, 1/9, 1/9, 1/36, 1/36, 1/36, 1/36), you will approximate

F = -G psi c_s^2 grad(psi)

This formula is actually used to derive the pressure equation in your post (p = c_s^2 rho + c_s^2*G*0.5*psi^2). If I now use the force equation with the squared speed of sound in it to obtain the units of G, I will end up with my result. Using the pressure equation, I will end up at the same result. The problem is that (depending on the weights) there will be a c_s^2 appearing in front of G in the pressure or not. This leads to the inconsistency.

I hope this is understandable.

Regards,

Knut