[FMDenaro]

Consider a pipe filled with water (single phase) being at the rest. This case is still compatible to produce a non-vanishing gradient of the density rho since


drho/dt +v .Grad rho = - rho div v



is satisfied for a zero velocity field. You can also write Drho/Dt=0 that means that a particle of water retain its density along the trajectory that, in this case, is just the position at the rest. Of course, the representation by means of the discrete particles depends on the number of finite particles of water you use.


You can do the same reasoning for water moving in the pipe, being now


drho/dt +v .Grad rho = Drho/Dt =0