Equations (53), (55), (59), (III-4) and (64) are all alternate forms of the same scalar relation derived directly from the equation of motion and hence from the principle of conservation of momentum. An independent equation will now be derived from the principle of conservation of total energy (including thermal energy as well as mechanical energy).
The formulation of the equation of conservation of energy requires the statement of several postulates.
, a function of position and
time within the fluid, which when added to the kinetic
energy per unit mass gives the total energy per unit mass
(exclusive of gravitational potential energy). THe
quantity is dependent upon the thermal state
and the chemical composition of the fluid mixture and is defined
as the internal energy per unit mass.
which represents the
non-advective flux of thermal energy by conduction
of heat and/or radiation. This vector is a function of position
and time.
such that 
represents the non-advective flux of chemical energy due to
diffusion. The quantity is dependent upon the thermal
state of the fluid mixture.
; (2) by the
efflux of internal chemical energy at its boundaries due to
diffusion; (3) by the work done on the fluid inside the volume
by gravity and tidal forces; and (4) by the work done on the fluid
due to surface stresses (including pressure) at the boundaries
of the volume.
It follows at once from the above postulates that for a volume V whose boundaries move with the fluid
where A is the surface area of the volume V and 
(or 
) is the outward unit normal as usual. Employing
Eq. (I-3) and the divergence theorem leads to the relation
Finally, since V is of arbitrary size, Eq. (68) can be satisfied only if
This is the differential equation expressing the conservation of combined thermal and mechanical energy.