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Re: [Phys-L] Bernoulli's equation

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

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.