Re: [Phys-L] Bernoulli's equation

[John Denker <jsd@av8n.com>]

On 01/25/2013 09:36 AM, Anthony Lapinski wrote:

> Teaching fluids now. Is there an "easy/conceptual" way to teach/derive
> Bernoulli's equation?
>
> P + 0.5pv2 = pgh = constant                  [1]

That's a good question.  AFAICT the easy ways are not
good, and the good ways are not easy.
 
> Using conservation of energy and other formulas, this is the most
> tedious/complicated derivation. I'm just looking for a different approach.

1) Several people have suggested the "energy" approach,
but alas it is not correct.  The constant in equation [1] 
has the same dimensions as energy per unit volume, but 
it is *not* equal to E/V.  In fact it is the /enthalpy/
per unit volume.

  2) It must also be emphasized that equation [1] is
  only a simplified first-order approximation to a 
  more-general result.  Don't bother looking for an
  exact derivation of equation [1], because there
  cannot possibly be one.

I know of two ways of deriving Bernoulli's equation, one
using enthalpy balance, and the other using force balance.
Both are presented and discussed at
  http://www.av8n.com/physics/bernoulli.htm

I'm not claiming that either derivation is "easy".  As a 
general rule, there is nothing easy about fluid dynamics.

As for "conceptual", I would say that both derivations
are reasonably "conceptual", strictly speaking:
 -- The concept of force balance is simple enough;  on the 
  other hand applying the concept to this situation  is
  tedious, insofar as it requires keeping careful track
  of many different forces.
 -- Meanwhile, applying the "concept" of enthalpy is 
  straightforward, and somewhat less tedious ... provided
  the students have a good understanding of what enthalpy 
  is, which they probably don't.
 -- The perfectly reasonable "concept" of first-order 
  approximations is required in order to reach equation [1].

===========

There is a proverb that says that if a difficult calculation
produces a simple result, it's a sign that you don't really
understand what's going on.  I keep hoping to find a simple 
(yet correct) derivation of equation [1].  If you come
across one, please let me know.

OTOH I don't lose toooo much sleep over equation [1], because
it is only a first-order approximation, and there are lots
of situation where the higher-order terms need to be taken
into account.