Application to moving Control Volumes

The Continuity and the Momentum Equations we have derived can be extended to cases where the control volume is not fixed in space. One such case is when the control volume is moving with a constant velocity, say an aeroplane or a ship moving at a constant speed. Note that the equations we have derived assume that the speeds are all referred to the control volume. So it becomes a simple matter to consider a control volume moving at a constant speed, V_cv. Define

$$V_{rel}=V~-~V_{cv}$$ (3.52)

which now is the speed relative to the control volume. The equation for Reynolds Transport theorem, Eqn.3.27 gets altered as

$$\left. {{dN} \over {dt}} \right)_s~=~~{\partial \over \partial t}... ... \forall~~+~~ \int_{CS}~\eta \rho \overrightarrow{V_{rel}}.d\overrightarrow{A}$$ (3.53)

(c) Aerospace, Mechanical & Mechatronic Engg. 2005
University of Sydney