Compressible Navier-Stokes
 

Is it possible to solve compressible NS equations with non homogeneous divergence (grad . V = f, f is some function not equal to zero) using FDM?

Your question is not very clear to me... you are simply stating that you want to solve the compressible form wherein the density equation writes as

Div V = f = -(1/rho) (d rho/dt + V . Grad rho)

why are you asking about FDM? The equations can be solved by FDM, FVM, FEM ....

My problem is flow of a fluid mixture. Water + particles. There is a source of particles and hence the density changes and continuity has a source term. Fluid is not compressible but the density changes with time. I used f for the function for particle source. Thanks.

Hi ckwollongong,

This is a hard task but to tackle this problem you should use the local averaged Navier-Stokes equations which introduce the void fraction.
The continuity equation which is derived is d(eps)/dt+ div(eps.U)=0
(d/dt is here the time partial derivative, div the divergence operator and eps is the void fraction).
You should also compute the interactions between particles (collisions + lagrangian Newton's law solver) and from their positions the void fraction at each time step and for each cell. There is also a source term in the averaged NS equations which is the resultant of the drag force on the particles. This force accounts for the interactions acting on the fluid due to the presence of particles.
You may found a lot of references in the litterature.

Thanks all