Kinetic Energy

(Book: 5.10)

The energy equation derived above tells you the combined changes in internal energy and kinetic energy of the fluid. Often, you are interested in the changes in those quantities separately instead of combined. After all, if you are interested in the motion, the kinetic energy is important, while if you are interested in the state of the fluid, the internal energy is.

Fortunately, we can get an equation for kinetic energy by itself in terms of the forces by multiplying the momentum equation,

by v_i and summing over i: It is seen that kinetic energy of the fluid elements changes due to work done by the gravity and stress gradient forces.

Converted back to conservation form it becomes:

Converted back to integral form:

In words, the kinetic energy changes due to work done by gravity and the work done by stress-induced forces on the fluid.

The kinetic energy equation is not a new, additional, equation, but a consequence of momentum.

You should now be able to do questions 5.7