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# Bernoulli's Theorem

In its most general form, Bernoulli's theorem--which was discovered by Daniel Bernoulli (1700-1783)--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.

The proof is straightforward. Consider the body of fluid bounded by the cross-sectional 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, cross-sectional 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 cross-sections 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 force-field, 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.

Next: Vortex Lines, Vortex Tubes, Up: Incompressible Inviscid Fluid Dynamics Previous: Streamlines, Stream Tubes, and
Richard Fitzpatrick 2012-04-27