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Bernoulli Equation

In an ideal inviscid, incompressible flow, we have, by conservation of energy,

$\displaystyle p + \frac{1}{2}\rho u^2 + \rho g h =$   constant

where

\begin{eqnarray*}
p &=& \mbox{pressure (newtons/m$^2$\ = kg /(m s$^2$))}\\
u &=& \mbox{particle velocity (m/s)}\\
\rho &=& \mbox{volume density of air (kg/m$^3$)}\\
g &=& \mbox{Newton's gravitational constant (m/s$^2$)}\\
h &=& \mbox{Height of flow's center-of-mass axis (m)}\\
\mbox{\lq\lq Inviscid''} &=& \mbox{\lq\lq Frictionless'', \lq\lq Lossless''}
\end{eqnarray*}

This basic energy conservation law was published in 1738 by Daniel Bernoulli in his classic work Hydrodynamica.

From §B.7.3, we have that the pressure of a gas is proportional to the average kinetic energy of the molecules making up the gas. Therefore, when a gas flows at a constant height $ h$ , some of its ``pressure kinetic energy'' must be given to the kinetic energy of the flow as a whole. If the mean height of the flow changes, then kinetic energy trades with potential energy as well.


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-06-11 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA