Frozen-in integral

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)