Confused by Continuity Equation

Question

I thought I was aware of it. My question is about the derivative of time. Sometimes it is the total derivative

and sometimes it is partial, we can ignore the $\phi$ here

Does mass change with time or position?

Answer 1

In these equations, $\rho$ is the density. It should not be surprising that density can depend on time and position.

When we integrate over the entire system, we do get the mass. Having integrated over the whole system, there is no possibility of position dependence. However, there can easily be time dependence if mass is being added to or removed from the modeled system. (Typical examples: chemical reactions produce gasses which escape the system, nuclear interactions convert mass to energy or vice versa, evaporation from a free boundary is being modeled, a bullet is fired into a semi-elastic medium, et al.)

When we integrate over a modeling cell, position dependence becomes possible. For instance evaporation can occur from a cell meeting the free boundary, but not from an interior cell.

Answer 2

The total derivative is made of two pieces: the partial and the convective. You are following an individual particle in your fluid. It’s density can change because of local compression/rarefaction or because of the particle moving to a different location. If you write the continuity eqn in integral terms the time derivative should be total since it only depends on time once you fix your control volume. When you localize the equation you obtain $$ \frac{d\rho}{dt}+\rho {\rm div}v=0 $$ With total time derivative which equals $$ \frac{\partial\rho}{dt}+v\cdot\nabla\rho $$ When you replace the total time derivative by this expression you can put together the second and third terms which amounts to taking the density inside the divergence.