## Expression for Momentum Density

It is useful to describe the equation for momentum . Eulerian derivative of momentum density is rewritten as

 (A.12)

where we used equations (A.7) and (A.11). In the Cartesian cordinate , this becomes
 (A.13)

Using the momentum stress tensor
 (A.14)

the above equation is written as
 (A.15)

where represents the Kronecker's delta as for and for .

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

 (A.16)

However, this is incorrect, because there exists the pressure force in the fluid and thus the momentum of the fluid element is not conserved.

Kohji Tomisaka 2012-10-03