FCV conversion

Most of the time in fluids, we want the time changes for fixed control volumes:

Unfortunately, the continuity equation for a material region does not apply to a fixed control volume:

Also,

Trick: find out what happens to M_FCV from what happens to the mass of material region that coincides with the control volume at any arbitrary chosen time t. At the chosen time

The time derivative of the material region's mass is the derivative of an integral with a moving boundary. We can rewrite it using the Leibnitz rule:

• The Leibnitz rule in one dimension:

  

• The Leibnitz rule in three dimensions: The interval [a,b] becomes a volume ; the endpoints a and b become the surface S of that volume, and the sum over the two end points becomes an integral over the surface:

  

Since the material regions are the only ones we know anything about, we should apply this to the integral of the mass of the material region:

The first term is the rate of mass change of the fixed control volume we were looking for.

Bottom line: since we do not have a material region, in fluid mechanics we get a surface integral in addition to what we have in physics.

Exercise:

Evaluate if , , and R is the unit cube .

Instead of using the Leibnitz rule, we can derive the mass equation for a fixed control volume from more physical arguments. This is done on the class web page under Reynolds Transport Theorem.

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