Re: [Phys-L] rotating cans of water

[John Denker <jsd@av8n.com>]

On 01/05/2016 06:02 AM, Carl Mungan wrote:

> The expression for p looks just like what Bernoulli’s equation 
> predicts. But it isn’t clear why Bernoulli should apply.

It's 100% clear that Bernoulli does *not* apply.

Bernoulli describes the pressure of a particular parcel as it
flows along a streamline.  It cannot be used to compare one
streamline to another, unless you have some a_priori reason
to think they both started out at the same pressure, which
in this case they didn't.

Here's another reason why you know it's wrong:  It's got the
wrong sign!  There's a minus sign in Bernoulli's equation 
that isn't there in the rotating fluid.

> Bernoulli comes from work-energy

It's not relevant to the present situation, for reasons 
discussed above ... but for next time, note that work/energy
is not the best way to think about Bernoulli's equation, for
multiple reasons:

 A) You can understand Bernoulli's equation in terms of
  /balance of forces/ without mentioning energy (or enthalpy)
  at all.

 B) The equation has /dimensions/ of energy ... but there is
  more to physics than dimensional analysis.  Lot of things 
  have dimensions of energy but aren't actual energy, e.g. 
  torques, Lagrangians, enthalpies, et cetera.

 In fact the thing on the RHS of Bernoulli's equation, called
 the Bernoulli /head/, is *enthalpy*.  If you are dealing
 with a compressible fluid, enthalpy gives the right answer
 and energy doesn't.

Both viewpoints (enthalpy and force balance) are worked out
and discussed at https://www.av8n.com/physics/bernoulli.htm

========================================

> Although the surface can no longer adopt a parabolic shape, there is
> a radial distribution of the normal force from the top surface of the
> can down onto the “centrifugally” spun flat water surface. But I’m
> not sure I’m right about p(r,z). Can anyone confirm or correct me?

If you remove the scare quotes from "centrifugally", then the 
question answers itself.  The analysis is trivial if you choose
the rotating frame and balance pressure against the centrifugal
force.

This gets back to my message from yesterday:
  -- The centrifugal field exists in the rotating frame and
   not otherwise.
  -- The centrifugal field is as real as the gravitational field.

For things that are stationary relative to the rotating frame,
using the rotating frame is super easy (no Coriolis terms to
worry about).