engineering fundamentals Fluid Mechanics: Bernoulli
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Bernoulli Equation
A non-turbulent, perfect, compressible, and barotropic fluid undergoing steady motion is governed by the Bernoulli Equation:
where g is the gravity acceleration constant (9.81 m/s2; 32.2 ft/s2), V is the velocity of the fluid, and z is the height above an arbitrary datum. C remains constant along any streamline in the flow, but varies from streamline to streamline. If the flow is irrotational, then C has the same value for all streamlines.

The function is the "pressure per density" in the fluid, and follows from the barotropic equation of state, p = p(r).

For an incompressible fluid, the function simplifies to p/r, and the incompressible Bernoulli Equation becomes:

Derivation from Navier-Stokes
The Navier-Stokes equation for a perfect fluid reduce to the Euler Equation:
Rearranging, and assuming that the body force b is due to gravity only, we can eventually integrate over space to remove any vector derivatives,
If the fluid motion is also steady (implying that all derivatives with respect to time are zero), then we arrive at the Bernoulli equation after dividing out by the gravity constant (and absorbing it into the constant C),
Note that the fluid's barotropic nature allowed the following chain rule application,
with the "pressure per density" function defined as,
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