Thermodynamics

(Book: 5.10, 12)

Now we want to get an equation for the thermodynamic properties only (no kinetic energy.)

This can be done by substracting the kinetic energy equation

from the energy equation

to give:

To clean up even more, split the stress tensor T_ji into the thermodynamical pressure and the viscous stress, :

Then use the continuity equation,

to make the pressure term also fully thermodynamical. Taking it to the left hand side, we get

Finally, compare the left hand side to the second law to get a very simple and important final result:

Assuming that the stress tensor is symmetric, the first term is also equal to and is called the dissipation.

Exercise:

Comment on the properties of reversible processes in fluid mechanics. Also comment on the sign of the viscous stress term.

The entropy equation is not an additional equation, but can be used as an alternative to the energy equation. Obviously, it is a lot simpler.