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d " u ! 2 % ! ! ( (K j
! $ e + ' = !uiF + ( ) u ) + !Q * (10.1.1)
ij i
dt # 2 & (x j (x j
where e is the internal energy, F is the sum of all the body forces, including the
centrifugal force, Q is the rate of heat addition by heat sources and K is the heat flux

"T
vector which we saw could be written in terms of the temperature as K = !k . The
j
"x
j
stress tensor is composed of a diagonal part we have associated with the pressure plus a

frictional deviatoric part, i.e.
! = " p# + $ (10.1.2)
ij ij ij

We suppose that the body force per unit mass can be written in terms of a force potential,

!
F = !"# (10.1.3)
and this is certainly true for the combined gravitational and centrifugal force that we

identify with effective gravity. Note that the dot product of the body force with the
velocity is

! ! ! d# $#
uiF = !ui"# = ! + (10.1.4)
dt $t

This allows us to rewrite (10.1.1),

✸
Once again we are indebted to “History of Hydraulics”, Hunter Rouse and Simon Ince,
1957. Dover Publications , New York. pp 269.

Chapter 10 2