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# Re: [Phys-L] Bernoulli's equation

Thanks to all who responded. This is for my (high school) honors physics
class (non calculus).

Of all the equations we derive (range equation, work-energy theorem,
nonconservative work, impulse-momentum theorem, etc.), this one is the
toughest. So that's why I wanted something "simple."

Phys-L@Phys-L.org writes:
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.

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