View source for Bernoulli integral

''of the equations of hydrodynamics''

An integral which determines the pressure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015610/b0156101.png" /> at each point of a stationary flow of an ideal homogeneous fluid or a barotropic gas <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015610/b0156102.png" /> in terms of the velocity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015610/b0156103.png" /> of the flow at that point and the body force function per unit mass <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015610/b0156104.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015610/b0156105.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

The constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015610/b0156106.png" /> has a specific value for each flow line and varies from one flow line to another. If the motion is potential, the constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015610/b0156107.png" /> 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:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015610/b0156108.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015610/b0156109.png" /></td> </tr></table>

and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015610/b01561010.png" /> is an arbitrary function of time.

For an incompressible liquid the left-hand sides of equations (1) and (2) are converted to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015610/b01561011.png" /> form; for a barotropic gas <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015610/b01561012.png" /> to the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015610/b01561013.png" /></td> </tr></table>

The integral was presented by D. Bernoulli in 1738.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.M. Milne-Thomson,   "Theoretical hydrodynamics" , Macmillan  (1950)</TD></TR></table>