Bernoulli's Principle is really just a statement of the conservation
of energy. Bernoulli's Principle is embodied in the Bernoulli
equations, which are special cases of the Navier-Stokes equations, the
fundamental equations that describe flow in fluids. There are various
forms of the Bernoulli equations (i.e., for steady incompressible
flow, steady compressible flow, time-dependent compressible or
incompressible flow, etc.), depending on which of various possible
simplifying assumptions are made in deriving them. The common form
that you are probably thinking of, and that is presented in most
introductory textbooks, applies to laminar, steady-state flow of an
incompressible fluid with constant density and zero viscosity:
p/D + 1/2*v^2 + g*z = constant
where p is the pressure, D is the density, v is the fluid speed, g is
the gravitational acceleration, and z is the height.
At low speeds, the assumption that a fluid like air is
incompressibleis pretty good; however as the speed increases, this
assumption becomes increasingly bad. At a Mach number of about 0.3,
the density of the fluid will change by several percent relative to
the reference (zero speed) density. This problem is probably what you
are referring to in your question.
To account for this, one has to relax the requirement that the fluid
be incompressible and derive a Bernoulli equation for compressible
fluids. A common way to do this is to assume instead that the
*entropy* of a parcel of fluid is constant as it travels along its
path (i.e., that the fluid follows an adiabatic path). Doing this
results in the most common form of the Bernoulli equation for
compressible flows:
h + 1/2*v^2 + g*z = constant
where h is the enthalpy of the fluid, and the other variables have the
same meaning as before.
A good reference for all this stuff is Professor M.S. Cramer's website
on fluid dynamics at http://www.navier-stokes.net/nsintro.htm |