On 11/18/2003 04:12 PM, Bob LaMontagne wrote:
> Bernoulli's Equation is an energy equation.
> Higher v does not 'cause' a
> lower pressure any more than potential energy decrease 'causes' the
> kinetic energy of a falling ball to increase. The energy changes are
> simply correlated.
I agree with at least 98% of the sentiments there.
Let me pursue a couple of nitpicky tangents that may
be of interest.
1) When trying to be careful about cause-and-effect,
it is often convenient to say "A is calculated from B"
and duck the question of whether A might be caused by B,
unless the latter question has some actual significance.
1a) Given v^2 we can calculate P and vice versa. There's
a nice symmetry there, reminiscent of F=ma and ma=F.
1b) But there's a catch: Given v we can calculate P,
but knowing P at one given point is not sufficient for
calculating v at that point ... we can get |v| but not
v itself. So depending on what you take as your
"fundamental" variables, the Bernoulli equation is not
quite as symmetric as F=ma.
Calculating P throws away information about the
direction of v. The computation is irreversible.
Some entropy is produced.
(If you know P everywhere in a region you can calculate
v in that region using the equation of motion ... but
you'll need more than just the Bernoulli equation and
you'll need more than just one isolated observation of P.)
========================
2) The Bernoulli equation comes in various forms.
Like any important equation such as E=mc^2 or Ohm's
law, it is commonly abused by people who write down
the equation without much idea what the symbols
mean, nor much idea of the limits of validity of
the equation.
In general, to get the pressure as a function of
velocity, you need
-- the energy principle and
-- the equation of state of the fluid.
The funny thing is, if all you want is a first-order
approximation, the result is in some sense independent
of the equation of state. That is, all fluids are
alike to first order, as far as Bernoulli is concerned.
All you need to know is the pressure. This isn't
particularly magical or mysterious; in some sense
the definition of pressure *is* the equation of state
to first order.
(Here first order means that the dynamic pressure is
small compared to the total pressure, or, equivalently,
that the speed is small compared to the speed of sound.)
I bring this up (a) because some zealots make a huge
celebration around the fact that you don't need to
know anything except the energy principle to derive
the Bernoulli equation. You shouldn't get too
excited about this, since it's only true to first order.
Conversely (b) some pessimists argue that Bernoulli's
equation is worthless because it only applies to
"incompressible" fluids, and there are no incompressible
fluids. Fear not, even the baby first-order version of
Bernoulli's equation deals with compressibility just
fine. Compressibility is another "first order"
property of the equation of state, and it is fully
accounted for in the baby version of the B. equation.
Meanwhile the full-grown version takes into account
more details of the equation of state, so it works
even for large speeds and large dynamic pressures.