Page 405 - Introduction to Continuum Mechanics

Newtonian Viscous Fluid 389

In spherical coordinates (r,Q,<p)

Example 6.19.1

Find the vorticity vector for the simple shearing flow:

Solution. We have

and

That is,

We see that the angular velocity vector (= 5 / 2) is normal to the jcj x^ plane and the minus
sign simply means that the spinning is clockwise looking from the positive side ofx$.

Example 6.19.2
Find the distribution of the vorticity vector in the Couette flow discussed in Section 6.15.
Solution. Withv r = v z — 0 andv# = Ar+(B/r). It is obvious that the only nonzero vorticity
component is in the 2 direction.
From Eq. (6.19.11),