I have a very basic question. Which model (Regular LBM or Incompressible LBM (He, Luo 1997)) do you suggest for Multiphase (Shan-Chen model) and why? I am looking for some kind of discussion here. Any inputs will be valuable for me.

From what I understand from the Shan-Chen model, each phase is tracked by the density (so the density also acts a bit like the composition or volume fraction). Therefore I’m not sure if the incompressible LBE of He and Luo will be able cope with phase changes and/or separation because density is constant (or at least the density fluctuations are an order smaller). Of course there is still pressure variation in the incompressible LBE, which in principle incorporates an equation of state, but since the Shan-Chen model is a “one-fluid” model I doubt if the two can be successfully combined. I have never used the Shan-Chen model so perhaps someone else could confirm or correct me.

There are other models which use an incompressible-type LBE together with another equation for tracking density/composition, such as He Chen and Zhang (1999), Lee and Lin, Inumaro for example (I can’t remember the years of those last two). Some such approaches consider forces (attraction) between fluids, but they usually also consider free-energies for phase separation. In other words, they are different from the original Shan-Chen approach (but also different from the original free-energy models of Swift and Yeomans et al).

In short, if you want to use Shan and Chen then I think you need to use the standard LBE. If you want to use the incompressible LBE then you need to consider how you are going to distinguish between phases and maintain phase separation and surface tension effects. Moreover, you need to consider which multiphase model is most suitable for your purpose.