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Difference between revisions of "Frozen-in integral"

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The integral of the induction equation of a magnetic field for the limiting case of an ideally-conducting medium. The physical meaning of the frozen-in integral is that by moving the liquid, the force lines of the magnetic field are moved together with the liquid particles.
 
The integral of the induction equation of a magnetic field for the limiting case of an ideally-conducting medium. The physical meaning of the frozen-in integral is that by moving the liquid, the force lines of the magnetic field are moved together with the liquid particles.
  
For the motion of an ideally-conducting medium the magnetic field strength <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041830/f0418301.png" /> is described by the equation:
+
For the motion of an ideally-conducting medium the magnetic field strength $  \mathbf H $
 +
is described by the equation:
 +
 
 +
$$
  
<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/f/f041/f041830/f0418302.png" /></td> </tr></table>
+
\frac{d}{dt }
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041830/f0418303.png" /> is the density and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041830/f0418304.png" /> is the rate of motion of the medium. A change in the line element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041830/f0418305.png" /> of a force line of the magnetic field is described by the equation
+
\left ( {
 +
\frac{\mathbf H} \rho
 +
} \right )  = \
 +
\left ( {
 +
\frac{\mathbf H} \rho
 +
} , \nabla \right ) \mathbf v ,
 +
$$
  
<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/f/f041/f041830/f0418306.png" /></td> </tr></table>
+
where  $  \rho $
 +
is the density and  $  \mathbf v $
 +
is the rate of motion of the medium. A change in the line element  $  d \mathbf l $
 +
of a force line of the magnetic field is described by the equation
  
The vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041830/f0418307.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041830/f0418308.png" /> are collinear:
+
$$
  
<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/f/f041/f041830/f0418309.png" /></td> </tr></table>
+
\frac{d}{dt }
 +
d \mathbf l  = ( d \mathbf l , \nabla ) \mathbf v .
 +
$$
 +
 
 +
The vectors  $  \mathbf H $
 +
and  $  d \mathbf l $
 +
are collinear:
 +
 
 +
$$
 +
{
 +
\frac{\mathbf H} \rho
 +
= \textrm{ const } \cdot d \mathbf l .
 +
$$
  
 
The following equation, which goes by the name of frozen-in integral, is valid:
 
The following equation, which goes by the name of frozen-in integral, is valid:
  
<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/f/f041/f041830/f04183010.png" /></td> </tr></table>
+
$$
 +
 
 +
\frac{\mathbf H  d \mathbf l _ {0} } \rho
 +
  = \
 +
 
 +
\frac{\mathbf H _ {0} }{\rho _ {0} }
 +
  d \mathbf l ,
 +
$$
  
 
where the index  "0"  refers to the values of the parameters at the initial moment of time.
 
where the index  "0"  refers to the values of the parameters at the initial moment of time.

Latest revision as of 19:40, 5 June 2020


The integral of the induction equation of a magnetic field for the limiting case of an ideally-conducting medium. The physical meaning of the frozen-in integral is that by moving the liquid, the force lines of the magnetic field are moved together with the liquid particles.

For the motion of an ideally-conducting medium the magnetic field strength $ \mathbf H $ is described by the equation:

$$ \frac{d}{dt } \left ( { \frac{\mathbf H} \rho } \right ) = \ \left ( { \frac{\mathbf H} \rho } , \nabla \right ) \mathbf v , $$

where $ \rho $ is the density and $ \mathbf v $ is the rate of motion of the medium. A change in the line element $ d \mathbf l $ of a force line of the magnetic field is described by the equation

$$ \frac{d}{dt } d \mathbf l = ( d \mathbf l , \nabla ) \mathbf v . $$

The vectors $ \mathbf H $ and $ d \mathbf l $ are collinear:

$$ { \frac{\mathbf H} \rho } = \textrm{ const } \cdot d \mathbf l . $$

The following equation, which goes by the name of frozen-in integral, is valid:

$$ \frac{\mathbf H d \mathbf l _ {0} } \rho = \ \frac{\mathbf H _ {0} }{\rho _ {0} } d \mathbf l , $$

where the index "0" refers to the values of the parameters at the initial moment of time.

It follows from the frozen-in integral that the magnetic flux of a field across any surface, encircled by a contour of liquid particles, is independent of time.

References

[1] T.G. Cowling, "Magneto-hydrodynamics" , Interscience (1957)
[2] L.D. Landau, E.M. Lifshitz, "Electrodynamics of continous media" , Pergamon (1960) (Translated from Russian)
[3] A.G. Kulikovskii, G.A. Lyubimov, "Magnetic hydrodynamics" , Moscow (1962) (In Russian)
How to Cite This Entry:
Frozen-in integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Frozen-in_integral&oldid=46996
This article was adapted from an original article by A.P. Favorskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article