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) |
Bernoulli integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bernoulli_integral&oldid=43482