Flux boundary in multiphase (Chen/He/Zhang)

In Chen multiphase model, how should we impose the B.C’s to the unknown ‘f’ ,‘g’ and ‘phi’ knowing velocity on the inlet or outlet?
As far as I know from the standard Zou/He flux B.C ,a set of equations should be solved to obtain the unknown quantities ( f, rho), but in this case the number of equations are less than unknowns (f, g, phi, P) and I don’t exactly know how to assume a new relations (like bounce-back assumption done in standard Zou/He).

The model is presented here:
He XY, Chen SY, Zhang RY. A lattice Boltzmann scheme for incompressible multiphase ¯ow and its application in simulation of Rayleigh±Taylor instability. J Comp Phys 1999;152:642±63.



I’m working with another model. However, BB usually helps with everything - even for the second distribution. There are two modifications of the bounce-back - pressure anti bounce back - can help you with imposing Dirichlet boundary condition or BB with velocity imposed. Another alternative is to use Inamuro boundary conditions “Lattice Boltzmann simulations for flow and heat/mass transfer problems in a three-dimensional porous structure”. However, both of the alternatives depend on what you impose as a physical boundary condition for a phase field.

Hopefully it will help,