Bernoulli integral

of the equations of hydrodynamics

An integral which determines the pressure at each point of a stationary flow of an ideal homogeneous fluid or a barotropic gas in terms of the velocity of the flow at that point and the body force function per unit mass :

(1)

The constant has a specific value for each flow line and varies from one flow line to another. If the motion is potential, the constant is the same for the entire flow.

For a non-stationary flow, the Bernoulli integral (sometimes called the Cauchy–Lagrange integral) holds in the presence of a velocity potential:

(2)

where

and is an arbitrary function of time.

For an incompressible liquid the left-hand sides of equations (1) and (2) are converted to the form; for a barotropic gas to the form

The integral was presented by D. Bernoulli in 1738.

References

[1]	L.M. Milne-Thomson, "Theoretical hydrodynamics" , Macmillan (1950)