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Bernoulli's Theorem
In its most general form, Bernoulli's theoremwhich was discovered by Daniel Bernoulli (17001783)states that, in the steady flow of an inviscid fluid, the quantity

(432) 
is constant along a streamline, where is the pressure, the density, and
the total energy per unit mass.
Figure 20:
Bernoulli's theorem.

The proof is straightforward. Consider the body of fluid bounded by the crosssectional areas and of the stream filament
pictured in Figure 20. Let us denote the values of quantities at and by the suffixes and
, respectively. Thus, , , , , are the pressure, fluid speed, density, crosssectional
area, and total energy per unit mass, respectively, at , etc. Suppose that, after a short time interval , the body of fluid has moved such
that it occupies the section of the filament bounded by the crosssections and , where
and
. Since the motion is steady, the mass of the fluid between
and is the same as that between and , so that

(433) 
Let denote the total energy of the section of the fluid lying between and . Thus, the increase in energy
of the fluid body in the time interval is

(434) 
In the absence of viscous energy dissipation, this energy increase must equal the net work done by the fluid pressures at
and , which is

(435) 
Equating expressions (434) and (435), we find that

(436) 
which demonstrates that
has the same value at any two points on a given stream filament, and is therefore constant along the filament. Note that Bernoulli's theorem has only been proved for the case of the steady motion of an inviscid fluid. However,
the fluid in question may either be compressible or incompressible.
For the particular case of an incompressible fluid, moving in a conservative forcefield, the total energy per unit mass is the
sum of the kinetic energy per unit mass, , and the potential energy per unit mass, , and
Bernoulli's theorem thus becomes

(437) 
If we focus on a particular streamline, 1 (say), then Bernoulli's theorem states that

(438) 
where is a constant characterizing that streamline. If we consider a second streamline, 2 (say), then

(439) 
where is another constant. It is not generally the case that . If, however, the fluid motion is
irrotational then the constant in Bernoulli's theorem is the same for all streamlines (see Section 5.7), so
that

(440) 
throughout the fluid.
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Up: Incompressible Inviscid Fluid Dynamics
Previous: Streamlines, Stream Tubes, and
Richard Fitzpatrick
20120427