28 CHAPTER 3. FULL THERMODYNAMIC THEORY
Similarly, momentum conservation implies that the rate of change @t(v) of the momentum
density in the same test volume is given by the stress forces acting on the
surface (with normal n), and the inux (direction ��n) of the momentum current density
vv. In dierential form, neglecting body forces, this becomes
@t(v) = r
�� p 1 �� vv
: (3.10a)
Applying the expansion r (vv) = (v r)v + v(r (v)) and the continuity equation
(3.9a), the momentum conservation equation (3.10a) can be written in Lagrangian form,
(@t + v r)v = r
�� p 1
; (3.10b)
known as the NavierStokes equation, which was treated adiabatically in Chapter 2. When
devided by the density , this equation states that the material derivative @t + (v r) of
the velocity equals the force per mass due to viscous stresses and pressure, and it is thus
the analogous to Newtons second law for uids.
Finally, energy conservation implies that the rate of change @t
��
"+ 1
2v2
of the energy
density (internal plus kinetic), is given by the power of the stress forces v on the surface
(direction n), and the inux (direction ��n) of both heat conduction power ��kthrT and
energy current density (" + 1
2v2)v. In dierential form, neglecting heat sources in the
volume, this becomes
@t
��
" + 1
2v2
= r
v �� p v + kthrT �� (" + 1
2v2)v
: (3.11a)
The following rewrite of the energy conservation equation to substitute terms involving
" by terms involving T and p is quite extensive. To make the rewrite less cumbersome,
we initially leave out the terms @t( 1
2v2), r (v ), r (kthrT), and r ( 1
2v2)v as
they are not rewritten, and then reintroduce
them in the end. In
the meanwhile they
are represented by X = ��@t( 1
v2) +r
v + kthrT �� ( 1
v2)v
. Letting the timederivative
22and divergence operators work on the product terms leads to
"@t + @t" = ��pr v �� v rp �� "r (v) �� v r" + X: (3.11b)
The rst term on the left and third term on the right cancel due to continuity Eq. (3.9a).
Next, @t" and r" is substituted by Eq. (3.5a), leading to
cp@tT ��pT@tp+ p
@t = ��prv��(vr)p��v
cprT �� pTrp + p
+X: (3.11c)
r
The third term on the left side and the rst and fth terms on the right side cancel due to
continuity Eq. (3.9a). Resubstituting X, the energy conservation equation becomes
@t
��1
2v2
+cp@tT��pT@tp+cp(vr)T+(1��pT)(vr)p = r
v + kthrT �� ( 1
2v2)v
:
(3.11d)
Equation (3.11d) is the nal form of the energy conservation equation, which will be used
in the perturbation expansion.