Arbitrary Regions

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,

where is the velocity of the boundary of the control volume, and once for the material region that at the considered time coincides with the control volume,

where v is the fluid velocity. Comparing

Hence from the physics:

The bottom line is that for a moving control volume, the factor becomes .

Similarly the momentum equation becomes:

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

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