[Mathematician]

My problem is to simulate an incompressible fluid with certain cinematic viscosity in a staggered grid with ghost cells, and the computational domain looks like this; Staggered-Grid-MAC-Marker-And-Cell.png


but there are ghost cells too.

My problem is the following; if I have the gravity source term in the navier stokes equation, and I suppose that the fluid is in the first row of the computational domain, do I have to solve only on the i,j cells where I suppose where the fluid is(at the top, the first row) and then maybe after I updated velocity update position and allow to solve only where the fluid goes? or maybe I have to diffuse and advect even outside the fluid? Do I need something external to solve the problem(intermolecular interactions)?
Info;chorin projection method (split local NS with additional timestep and then discretize) and SOR for pressure.