Re: [Phys-l] question about Bernoulli

[John Mallinckrodt <ajm@csupomona.edu>]

Just in case anyone may be wondering what I was on about with my Bernoulli question ...

I 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?


The question could be rephrased as simply noting that Bernoulli's equation is manifestly NOT invariant wrt Galilean transformations because kinetic energy density is frame-dependent while pressure is not.  To appreciate the apparent strangeness of the result it is particularly useful to imagine a flowing fluid far from any boundaries like pipes or airfoils, that is, in a region where there is a lack of reference *objects* and the frame-dependence of the fluid velocity becomes more obvious.  Since the Bernoulli equation clearly gives wrong answers unless the correct ("preferred") reference frame is chosen, the question is, "What determines that preferred reference frame?"  

The answer is that Bernoulli's eqn is derived under the assumption that the flow is "steady" (i.e., not time-dependent) and is, therefore, only applicable to steady flows. But, except in boring cases, steady flows are only steady in *one* reference frame.  Of course, to have a non-boring steady flow, one does also have to have "stationary" boundaries like pipes or river beds or mountaintops or airfoils, which then also determine the preferred reference frame, but that is not the fundamental requirement.

This result is somewhat similar to the equation describing the classical Doppler shift, which is only valid in the rest frame of the medium supporting the waves, but I think also slightly more subtle.

I thank Carl Mungan for carrying on an off-list conversation with me about this.

John Mallinckrodt
Cal Poly Pomona