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Difference between revisions of "Bernoulli integral"

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''of the equations of hydrodynamics''
 
''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" />:
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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 $u(x,y,z)$:
  
<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>
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\begin{equation}\int\frac{dp}\rho=C-\frac12|\mathbf v|^2+u.\label{1}\end{equation}
  
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.
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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:
 
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>
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\begin{equation}\int\frac{dp}\rho=\frac{\partial\phi}{\partial t}-\frac12|\mathbf v|^2+u+f(t),\label{2}\end{equation}
  
 
where
 
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>
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$$\mathbf v=\grad\phi(x,y,z,t),$$
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015610/b01561010.png" /> is an arbitrary function of time.
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and $f(t)$ 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
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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
  
<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>
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$$\int\frac{dp}\rho=\int F'(\rho)\frac{dp}\rho.$$
  
 
The integral was presented by D. Bernoulli in 1738.
 
The integral was presented by D. Bernoulli in 1738.

Latest revision as of 23:29, 24 November 2018

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)
How to Cite This Entry:
Bernoulli integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bernoulli_integral&oldid=18421
This article was adapted from an original article by L.N. Sretenskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article