Fluid mechanics: continuity equation

In order to overcome the problem in FLIP solver, I looked into fluid mechanics.

It turns out the condition for incompressibility is not correct, or to say, based on an incorrect assumption.

For continuous fluid, the governing equation is called Continuity Equation

I'll skip the proving part of this equation, lots of reading materials could be found online discussing proving of this equation. In symbolic form, this equation could be expressed as:

Compare to the incompressible confinement, we could see, it's just assuming material derivative for density is 0. Yet this assumption is not correct if fluid cells are marked by the particles: temporay incoherence would lead to a non-zero density material derivative.

One big problem associated with adding this term to the confinement is, we'll have to deal with both time derivative and spacial gradient in the Eulerian grid.

So one of the possible solutions would be: combine SPH and FLIP together in this step. The density carried by each particle provides sufficient information for the material derivative. I'll keep on reading and thinking about other possible solutions.