First Helmholtz vortex theorem

Values of angular-velocity vector and sectional area may vary along the given vortex core, however vortex core consumption along its whole length remains constant. This is a content of the first Helmholtz theorem.

This theorem is only of kinematic type and is valid for any continuum provided that the field of velocities is a continuous function of coordinates. We shall prove this theorem using Zhukovsky method. Starting from equations (2.34), we shall write partial derivatives with respect to coordinates , and for angular velocity

Having summed them we obtain the equation:

(2.42)or

.(2.43)

Last equation is similar to the continuity equation (2.25) if we should assume that incompressible fluid moves within the vortex tube with vector . Thus, the equation (2.42) is the continuity equation for vector .By analogy with the flow consumption equation (2.29) for vortex filament for which angular rate may be considered as constant value over the section, it is possible to write down

.(2.44)

For all cross-sections of vortex line strength is a constant value, therefore at reduction of cross-sectional area angular rate will increase and on the contrary. If at angular rate that is physically impossible. Thus, the vortex line can not be needle point at its end in a fluid, it only can lean against its solid boundaries, or on a free surface or to swing in a ring.