Vortex Lines, Vortex Tubes, and Vortex Filaments

The curl of the velocity field of a fluid, which is generally termed vorticity, is usually represented by the symbol $\omega$ , so that

$$\mbox{\boldmath$\omega$} = \nabla\times {\bf v}.$$ (4.71)

A vortex line is a line whose tangent is everywhere parallel to the local vorticity vector. The vortex lines drawn through each point of a closed curve constitute the surface of a vortex tube. Finally, a vortex filament is a vortex tube whose cross-section is of infinitesimal dimensions.

Figure 4.9: A vortex filament.

Consider a section $AB$ of a vortex filament. The filament is bounded by the curved surface that forms the filament wall, as well as two plane surfaces, whose vector areas are ${\bf S}_1$ and ${\bf S}_2$ (say), which form the ends of the section at points $A$ and $B$ , respectively. (See Figure 4.9.) Let the plane surfaces have outward pointing normals that are parallel (or anti-parallel) to the vorticity vectors, $\omega$ $_1$ and $\omega$ $_2$ , at points $A$ and $B$ , respectively. The divergence theorem (see Section A.20), applied to the section, yields

$$\oint \mbox{\boldmath$\omega$} \cdot d{\bf S} = \int \nabla\cdot \mbox{\boldmath$\omega$} \,dV,$$ (4.72)

where $d{\bf S}$ is an outward directed surface element, and $d V$ a volume element. However,

$$\nabla\cdot \mbox{\boldmath$\omega$} =\nabla\cdot\nabla\times {\bf v} \equiv 0$$ (4.73)

[see Equation (A.173)], implying that

$$\oint \mbox{\boldmath$\omega$} \cdot d{\bf S} =0.$$ (4.74)

Now, $\omega$ $\cdot d{\bf S}=0$ on the curved surface of the filament, because $\omega$ is, by definition, tangential to this surface. Thus, the only contributions to the surface integral come from the plane areas ${\bf S}_1$ and ${\bf S}_2$ . It follows that

$$\oint \mbox{\boldmath$\omega$} \cdot d{\bf S} = S_2\,\omega_2-S_1\,\omega_1 = 0.$$ (4.75)

This result is essentially an equation of continuity for vortex filaments. It implies that the product of the magnitude of the vorticity and the cross-sectional area, which is termed the vortex intensity, is constant along the filament. It follows that a vortex filament cannot terminate in the interior of the fluid. For, if it did, the cross-sectional area, $S$ , would have to vanish, and, therefore, the vorticity, $\omega$ , would have to become infinite. Thus, a vortex filament must either form a closed vortex ring, or must terminate at the fluid boundary.

Because a vortex tube can be regarded as a bundle of vortex filaments whose net intensity is the sum of the intensities of the constituent filaments, we conclude that the intensity of a vortex tube remains constant along the tube.