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d # u ! 2 & )" ) ) # )T &
! % e + +" = ! + - . * pu + + u , / + % k ( + !Q
(
i ij 0
dt $ 2 ' )t )x j i ij )x $ )x '
j
j
)" )u )p ) # )T &
= ! * p i * u + % u , + k ( + !Q
)t )x j )x )x $ i ij )x '
i j j j
(10.1.5)
)" p d! dp )p ) # )T &
= ! + * + + % u , + k ( + !Q
)t ! dt dt )t )x $ i ij )x '
j j

)" d # & )p ) # )T &
p
= ! * ! % ( + + % u , + k ( + !Q
$
)t dt !' )t )x $ i ij )x '
j j
The third step in the above derivation uses the equation of mass conservation and we
have allowed the potential Ψ to be time dependent, although it rarely is, to make a point

below. Combining terms and using our definition of the dissipation function and the

representation of the viscous forces in Chapter 6 e.g. see (6.114),

%
1 $u $u (
!" = # e = # i + j (10.1.6)
ij ij ij ' *
&
2 $x j $x )
i

we obtain ,

! 2
d # p u &
% e + + +" ( =
dt % ! 2 (
'
$
(10.1.7)
1 )p )" 1 ) * )T - 1 ! u )3
+ + Q + , k / + 0 + (2iu) + i ij
2
! )t )t ! )x + )x . ! ! )x
j j j

For the record, and because it will be useful, remember that the second law of
thermodynamics yields for the entropy, (6.1.22)

Chapter 10 3