Vector tube

A closed set $ \Phi $ of points of a space $ \Omega $ in which there has been specified a vector field $ \mathbf a ( M) $ such that the normal vector $ \mathbf n $ is orthogonal to $ \mathbf a $ everywhere on its boundary surface $ S $. The vector tube $ \Phi $ consists of vector lines $ \Gamma $ of the field $ \mathbf a $, i.e. curves in $ \Omega $ at each point of which the tangent direction coincides with the direction of $ \mathbf a $. A line $ \Gamma $ is completely contained in $ \Phi $ if one point of $ \Gamma $ is contained in $ \Phi $. If $ \mathbf a $ is the field of velocities of a stationary liquid flow, then $ \Gamma $ is the trajectory of the liquid particles, while $ \Phi $ is the part of $ \Omega $" swept along" by the motion of a given amount of liquid particles.

The intensity $ I $ of the tube $ \Phi $ in the cross-section $ S ^ \prime $ is the flux (cf. Vector analysis) of $ \mathbf a $ across $ S ^ \prime $:

$$ I( S ^ \prime ) = \int\limits \int\limits S ^ \prime ( \mathbf a , \mathbf n ) d \sigma , $$

where $ \mathbf n $ is the unit normal vector to $ S ^ \prime $. If the field $ \mathbf a $ is solenoidal ( $ \mathop{\rm div} \mathbf a = 0 $), the law of preservation of the intensity of the vector tube holds:

$$ I( S ^ \prime ) = I( S ^ {\prime\prime} ) . $$

Let $ a _ {1} ( x, y, z), a _ {2} ( x, y, z), a _ {3} ( x, y, z) $ be the orthogonal Cartesian coordinates of the vector $ \mathbf a = \mathbf a ( M), $ and let $ x, y, z $ be the coordinates of the point $ M $. Then the boundary of $ \Phi $ is locally defined by an equation $ F( x, y, z) = \textrm{ const } $, where $ F( x, y, z) $ satisfies the partial differential equation

$$ ( \mathbf a , \nabla F ) = $$

$$ = \ a _ {1} ( x, y, z) \frac{\partial F }{\partial x } + a _ {2} ( x, y, z) \frac{\partial F }{\partial y } + a _ {3} ( x, y, z) \frac{\partial F }{\partial z } = 0. $$

References

[a1]	A.P. Wills, "Vector analysis with an introduction to tensor analysis" , Dover, reprint (1958) pp. Sect. 45