Inclusion of a force term in RK color gradient model using single relaxation time


I am using the RK color gradient model based on suggestion of Huang et al. book on multiphase lattice boltzmann. I could get the laplace law correctly, but am having a trouble in getting two phase layered Poiseullie flow under gravity correctly. Most of the publications on this benchmark do not explicitely describe the inclusion of a force term when used with single relaxation time scheme. I see some publications which are described with MRT scheme but for the moment, I would like contine with SRT scheme.

Now I am using force scheme given by Guo et al. as follows,

$$f^k_i = \Omega^3_i(\Omega^2_i+\Omega^1_i)$$

Now I tried to add a force term in $\Omega^1_i$ with Guo et al. scheme, but it does not work as the velocity of fluid shoots above one (in lattice units).

I think the reason is f^eq,k_i should be also modified in the presence of external forces. Can anybody direct me towards a source describing the inclusion of force term?

Many thanks in advance.

If anybody is still interested, I found the description of force inclusion in single relaxation time case in the article

[url=]the link to artile[

Hello, I could not get the link, could you please share it again? Thank you.