If we want to write the continuity equation for a moving control volumes instead of a fixed one, we simply apply the Leibnitz theorem twice: once for the control volume itself,
Hence from the physics:
Similarly the momentum equation becomes:
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Example:
Problem: Write the equation of motion of a balloon in terms of the exit area and the relative exit velocity of the air from the balloon.
Solution: The control volume is the balloon. This is an arbitrary region.
Mass conservation:
Assuming the relative air velocity at the exit is normal to it and constant:
Let
be the mass flux out of the exit:
then
Momentum equation (with
)
Substract
times the continuity equation:
Ignoring the difference between average velocity and the boundary velocity of the exit and assuming one-dimensional motion
where D is the drag force and
the thrust force due to outflow. This can be solved along with