Bernoulli integral

of the equations of hydrodynamics

An integral which determines the pressure $p$ at each point of a stationary flow of an ideal homogeneous fluid or a barotropic gas $(p=F(\rho))$ in terms of the velocity $\mathbf v$ of the flow at that point and the body force function per unit mass $u(x,y,z)$:

\begin{equation}\int\frac{dp}\rho=C-\frac12|\mathbf v|^2+u.\label{1}\end{equation}

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

\begin{equation}\int\frac{dp}\rho=\frac{\partial\phi}{\partial t}-\frac12|\mathbf v|^2+u+f(t),\label{2}\end{equation}

where

$$\mathbf v=\grad\phi(x,y,z,t),$$

and $f(t)$ is an arbitrary function of time.

For an incompressible liquid the left-hand sides of equations \eqref{1} and \eqref{2} are converted to the $p/\rho$ form; for a barotropic gas $(p=F(\rho))$ to the form

$$\int\frac{dp}\rho=\int F'(\rho)\frac{dp}\rho.$$

The integral was presented by D. Bernoulli in 1738.

References

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