It is useful to describe the equation for momentum
Eulerian derivative of momentum density
is rewritten as
where we used equations (A.7)
In the Cartesian cordinate
, this becomes
Using the momentum stress tensor
the above equation is written as
where represents the Kronecker's delta as
If there is no external force, rhs of equation (A.15)
is equal to zero.
If the momentum density
were to obey the continuity equation,
equation (A.15) would be
However, this is incorrect, because there exists the pressure force
in the fluid and thus the momentum of the fluid element is not conserved.