Page 465 - Introduction to Continuum Mechanics

Control Volume Fixed with respect to a Moving Frame 449

and

Thus,

Now, let FI be an inertial frame so that the momentum principle reads

Using Eq. (7.7.6), the momentum principle [Eq. (7.7.7)] becomes

Equation (7.7.8) shows that when a moving frame is used to compute momentum and its
time rate of change, the same momentum principle for an inertial frame can be used provided
we add those terms given inside the bracket in the right-hand side of Eq. (7.7.8) to the surface
and body force terms.

7.8 Control Volume Fixed with respect to a Moving Frame

If a control volume is chosen to be fixed with respect to a frame of reference which moves
relative to an inertial frame with an acceleration a 0, an angular velocity to and angular
acceleration i», the momentum equation is given by Eq. (7.7.8). If we now use the Reynold's
transport theorem for the left-hand side of Eq. (7.7.8), we obtain