Page 470 - Introduction to Continuum Mechanics

454 Integral Formulation of General Principles

Including that due to the left arm, the integral has the value of -2a)Qpr 0e^. Therefore, from
the moment of momentum principle for a moving control volume, we have

from which

7.10 Principle of Conservation of Energy

The principle of conservation of energy states that the time rate of change of the kinetic
energy and internal energy for a fixed part of material is equal to the sum of the rate of work
done by the surface and body forces and the heat energy entering the boundary surface. That
2
is, if v denotes vv, u the internal energy per unit mass, and q the rate of heat flow vector
across a unit area, then the principle states:

the minus sign in the last term is due to the convention that n is an outward unit normal vector
and therefore -q-n represents inflow.
Again, using the Reynold's transport theorem Eq. (7.4.2), we have

In Example 7.2.3 we obtained that

Also, the divergence theorem gives

Thus, using Eqs. (i)(ii) and (iii), Eq. (7.10.1) becomes