Reynolds Transport Theorem

The ReynoldsTransportTheorem is a way of analyzing ConservationLaw? problems.

The ReynoldsTransportTheorem lets you apply ConservationLaw?s to OpenSystem?s (where your conserved quantity can move in and out of the ControlVolume?), not just ClosedSystem?s (where your conserved quantity is not allowed to enter or escape).

Newton's Third Law of Mechanics is an example of the ReynoldsTransportTheorem.

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In simple language, the ReynoldsTransportTheorem says that what goes in either stays in, or comes out.

In mathematical language:

 For a quantity that is conserved:

 Let the ControlSurface? be the surface enclosing the ControlVolume?.
 The ControlVolume? can change shape over the course of the problem, 
 and the materials in the problem can move in and out of the ControlVolume?.

 The SurfaceIntegral? over the ControlSurface?
 of (the velocity of the material passing out of the ControlSurface?
                 (relative to the ControlSurface?)
     dot-producted with the outward pointing unit vector of the ControlSurface?)
 equals the rate of decrease (with respect to time)
 of the VolumeIntegral? of the quantity inside the ControlSurface?.

EditHints:

• Include on page a picture of mathematical notation.
• Also add a picture of a ControlSurface? and a ControlVolume?, showing stuff flowing through.

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The ReynoldsTransportTheorem can be applied to any ConservationLaw?. For example:

• Conservation of Mass and Energy
  • aka the First Law of Thermodynamics
  • aka Kirchhoff's Voltage Law
  • aka "You can't win"
• Conservation of Momentum
  • aka Newton's Laws of Mechanics
• Conservation of Angular Momentum
• Conservation of Charge
  • aka Kirchhoff's Current Law

With a subtle modification, it can be applied to quantities that tend to increase, not decrease:

• Entropy
  • aka the Second Law of Thermodynamics
  • aka "You can't break even"

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I've never heard of this as a theorem before, let alone a named one. Most mathematical and physical texts I've seen take it as the definition of a conserved quantity. What do you prove it in terms of?

I've never heard it used as the definition. Noetherian invariants are usually the root of conservation laws.

The theorem is that the statement "in mathematical language" above matches the behavior of a conserved quantity.

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Please note the spelling: "Kirchhoff" rather than "Kirchoff". The double h is because it is a combination of two words: Kirch+Hoff.

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CategoryMath