So, this, in turn, leads us to the concept of a control volume.

So, for example, in this case, this region might be defined by

a control volume which I'll denote by cv around the beam.

And we are concerned with what happens inside this control volume.

So the shape of the control volume depends on the particular problem

that you're analyzing, and

a good choice of the control volume allows a much simpler solution to the problem.

So in the first case here, it could be as simple as the flow in a pipe,

in which case the control volume is just a cylindrical shape.

Or, if it's more complex like a jet engine here,

the control volume might completely encompass the jet engine.

So, those are fixed control volumes.

But the control volume could also be moving or it could be deforming.

For example, in the case of the balloon here which is collapsing,

the control volume might follow the interior surface of the balloon,

so as the balloon contracts, the control volume also contracts.

In other words, it deforms.

The particular, peculiar properties if you like of a control volume is that generally

speaking for us it will be fixed in space but it has the peculiar properties that

it allows fluid and matter and mass to pass freely through its boundaries.

The most generally statement of the Momentum Theorem in fluid mechanics

as applied to a controlled volume is this equation right here,

which I'll describe that more in a little bit more detail.

This equation here, the Momentum Theorem, we can cons,

think of as a fluid mechanical version of Newton's second law.