Page 404 - Introduction to Continuum Mechanics

388 Vorticity Vector

Using Eq. (6.19.1) and (ii) ,Eq. (i) becomes

Now, if n is an eigenvector of D, then

and

and Eq. (6.19.3) becomes

which is the desired result.
Eq. (6.19.6) and Eq. (6.19.1) state that the material elements which are in the principal
directions of D rotate with angular velocity a> while at the same time changing their lengths.
In rectangular Cartesian coordinates,

Conventionally, the factor of 1/2 is dropped and one defines the so-called vorticity vector £
as

The tensor 2W is known as the vorticity tensor.
It can be easily seen that in indicial notation, the Cartesian components of? are

and in invariant notation,

In cylindrical coordinates (r,0,z)