The balance of mass can be expressed as:
where is the current mass density, is the material time derivative of ρ, and is the velocity of physical particles in the body Ω bounded by the surface .
We can show how this relation is derived by recalling that the general equation for the balance of a physical quantity is given by
To derive the equation for the balance of mass, we assume that the physical quantity of interest is the mass density . Since mass is neither created or destroyed, the surface and interior sources are zero, i.e., . Therefore, we have
Let us assume that the volume Ω is a control volume (i.e., it does not change with time). Then the surface has a zero velocity (u_{n} = 0) and we get
Using the divergence theorem
we get
or,
Since Ω is arbitrary, we must have
Using the identity
we have
Now, the material time derivative of ρ is defined as
Therefore,
