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Re: [Phys-l] question about Bernoulli



John M wrote:

The Bernoulli equation (for incompressible fluids) says that the sum of the
kinetic energy density, the gravitational potential energy density, and the
pressure is constant along a streamline so that if the speed DECREASES from
point A to point B along a horizontal streamline, the pressure must be HIGHER
at point B than at point A.

But in the rest frame of the fluid at point A, the speed is zero at point A
and, thus, necessarily is higher at point B so that, in THAT frame, the
pressure must be LOWER at point B than at point A.

What's up with that?

----------

Hmm, that is a good question. My first thought goes as follows: I suppose point A to be inside a constriction and point B to be outside (let's say downstream and call that the +x direction) of the constriction. I'll take the fluid to be incompressible and to have zero viscosity.

Start with everything at rest and visualize the tube to be shaped as follows, extending to + and - infinity and filled with fluid:

------
/
/
---

A B

---
\
\
------

Now to turn on the fluid motion. But in A's frame, the fluid is at rest. But the pipe moves to the right. That pushes on fluid in region B, so that it starts to move, but obviously somewhat slower than the pipe. It is the diagonal portion of the pipe (where it is increasing in diameter) that sweeps out volume that must push fluid in B forward. We will thus get precisely that the speed of fluid in B equals the ratio of area of region B to area of region A, as continuity requires. We now have some kind of "ramjet" arrangement, recorded as a pressure back on the diagonal portion of the pipe and hence as a pressure in region B. But there is no pressure in region A as the pipe moves past the viscosity-free (superfluid) liquid.

Am I on the right track in the kind of thinking you were expecting for a solution? Carl

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Carl E Mungan, Assoc Prof of Physics 410-293-6680 (O) -3729 (F)
Naval Academy Stop 9c, 572C Holloway Rd, Annapolis MD 21402-5002
mailto:mungan@usna.edu http://usna.edu/Users/physics/mungan/