View source for 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.

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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  T.G. Cowling,   "Magneto-hydrodynamics" , Interscience  (1957)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.D. Landau,   E.M. Lifshitz,   "Electrodynamics of continous media" , Pergamon  (1960)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.G. Kulikovskii,   G.A. Lyubimov,   "Magnetic hydrodynamics" , Moscow  (1962)  (In Russian)</TD></TR></table>