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Conservation of Energy

We use the assumption of conservation of energy to model the internal energy
of the system. Again we see the generic model apply as the rate of change in the
internal energy within the volume is directly related to the rate of internal
energy conducted into the volume. As sources we find the energy increase due to
compression (reversible), and viscous dissipation2.9 (irreversible). The rate of change in
internal energy per volume e, with local velocity u, pressure p, and stress tensor is:

Here we have introduced the double dot product

see [7]
A.4-28.

Now since is symmetric, we have

and hence

Using , the definition , and indices in the kinematic viscosity , we find:

Finally we need to find an expression for the temperature to relate back to
our original problem. The temperature can be determined using the equation of
state for ideal gas, with the volume V, pressure p, density , temperature T, gas constant R, number of
moles n,

pV=nRT,

and the relation

where M is the mole mass, we can find the temperature,

We see that to determine the temperature we need to find
the pressure p. Again using the equation of state we find the temperature
T:

We now insert this expression into the equation of
adiabatic change in internal energy of an ideal gas as found in [12],
§13.2 and determine the internal energy U:

For the final necessary relation we now utilize the
definition of ratio of heat capacities, again from [12],
and find:

Letting e designate the internal energy per volume
we can now find the pressure p by inserting into the latter relation:

We have now found an appropriate description of the energy,
and determined the how to compute temperature and pressure.

Peter Søndergaard & Jan Vittrup Hansen