What rigid body rotation causes rotational viscosity? According to the book Non-Equilibrium Thermodynamics by S. R. De Groot and P. Mazur page 309, there are some rigid body rotations causing rotational viscosity. What kind of fluid is that? How can this kind of rotational viscosity be imagined?
Excerpt from chapter XII, § I, page 309:

If on the other hand the fluid motion is like the rotation of a rigid body
$$ \tag{30} \textbf{v} = \textbf{b} \wedge \textbf{r}, \ \ ({\textbf b}\ constant\ vector), $$
then (Grad ${\bf v})^{s}$ and div ${\bf v}$ vansih, but
$$ \tag{31} rot \ {\bf v} = 2 {\bf b} ,$$
so that only the rotational viscosity could play a role.

(Grad ${\bf v})^s$ is the symmetric part with trace of $\nabla {\bf v}$.
 A: Partially fill a cylindrical cup with water, place it at the center of a turntable, and make the turntable rotate at a constant rotation rate. The constant rotation rate coupled with skin friction and viscosity will eventually make the water equilibrate into a shape that rotates the same way as would an identically shaped rigid body.
The water of course is not a rigid body. Rigid bodies respond instantaneously and uniformly to disturbances, while the water will act as a viscous fluid in response to disturbances.
A constant rotation rate yields some very nice properties, one of which is that the surface of the fluid is a circular paraboloid. Use mercury as the fluid and you have the basis for a relatively low cost zenith-looking telescope. Use a curable liquid as the fluid and you have the basis for a higher cost but more useful telescope.
A: Rigid body rotations in continua consisting of particles do not really rotate as rigid bodies since the motion of the particles between collisions do not participate in the rigid body rotation. Seen from the rotating system the particles between collisions are affected by Coriolis forces having as a consequence to counteract the rotation and slowing it down in a similar way to viscosity.
