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The Poynting vector is a representation of the energy flux in electromagnetics, showing the amount and direction of power flow at different points in space. In electric circuits, the energy is not carried inside the wires (meaning the vector is just 0 inside them?), but by the electric and magnetic fields surrounding the wires. The DC circuit is the simplest case:

enter image description here

Is there an equivalent concept of energy flux for an equivalent hydraulic circuit?

enter image description here

Does the energy flow inside the pipes in this case? I'm guessing the energy flux has some kind of parabolic profile inside the pipes proportional to the flow rate?

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There is such a diagram--- it is the flux of energy in the linearized gravitational field (if you use full GR, you get complications with defining the energy). Unlike the electromagnetic case, where the electric field carry the bulk of the energy and the momentum of the charge carriers is negligible, in the gravitational case, it is the opposite.

You can also imagine electromagnetic circuits in which you accelerate very massive spheres which are very lightly charged, and use these as current carriers, and in this case, the momentum of the current carriers will not be negligible.

EDIT: To clarify, there are gravitational fields created by moving water which surround the pipe, like the electric and magnetic fields surround the current-carrying wire. There is an energy flow in these gravitational fields, which carries energy, just like the Poynting flux does. These effects are negligible for ordinary materials at ordinary density. Nearly all the energy flux (all but the tiny negligible fraction in the gravitational field) is carried by the water in the pipe, but the momentum in the water is not analogous to the Poynting vector, it is analogous to the electron momentum, which also carries a small amount of energy in a current carrying wire.

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Downvote? What is the issue with the answer? It is correct that the gravitational field Poynting-type vector are the direct analog for bulk mechanical flows and pressures of the Poynting vector in EM for bulk flow of charges. – Ron Maimon Nov 17 '11 at 16:27
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@Georg: The question asked for the mechanical analog of the Poynting vector. There is such an analog only in the gravitational field, and in no other way. This is the proper answer--- there is no better one. – Ron Maimon Nov 17 '11 at 17:44
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@endolith: there are gravitational fields created by moving water which surround the pipe, like the electric and magnetic fields surround the current-carrying wire. There is an energy flow in these gravitational fields, which carries energy, just like the Poynting flux does. These effects are negligible for ordinary materials at ordinary density. Nearly all the energy flux (all but the tiny negligible fraction in the gravitational field) is carried by the water in the pipe, but this flow is not analogous to the Poynting vector, it is analogous to the electron momentum. – Ron Maimon Nov 17 '11 at 18:32
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@endolith: The momentum and energy in electric circuits is not carried by electromagnetic waves, but by a static electromagnetic field (not a wave). It is a stationary energy flow, which is set up at the speed of light when you first switch on the current. The gravitational system is exactly the same. There is a static momentum flow in the gravitational field which is set up quickly when you first start the flow, and it is there all the time, because it is a momentum flow in a static field. – Ron Maimon Nov 17 '11 at 19:06
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@ColinK: I don't think GR is all that complicated, and this question was begging for a GR answer. It is condescending to assume that the questioner can't understand something as basic as "energy-momentum sources gravity". I don't respect the labels "elementary" and "advanced", because they are just warning labels to the public, "learn this" and "don't learn this". Any adult of sound mind can learn General Relativity, especially in the linear regime. – Ron Maimon Nov 18 '11 at 21:08

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