I've got a problem regarding tensors.
Premise: we are considering a fluid particle with a velocity $\mathbf{u}$ and a position vector $\mathbf{x}$; $S_{ij}$ is the strain rate tensor, defined in this way:
$\displaystyle{S_{ij}=\frac{1}{2}\left(\frac{\partial u_i}{\partial x_j} +\frac{\partial u_j}{\partial x_i}\right)}.$
OK, the problem is in this paragraph taken from Fluid Mechanics. Fifth Edition, P. K. Kundu, I. M. Cohen, D. R. Dowling, 2011, p. 78:
Here we also note that $S_{ij}$ is zero for any rigid body motion composed of translation at a spatially uniform velocity $\mathbf{U}$ and rotation at a constant rate $\mathbf{\Omega}$ (see Exercise 3.17).[*] Thus, $S_{ij}$ is independent of the frame of reference in which it is observed, even if $\mathbf{U}$ depends on time and the frame of reference is rotating.
My question is: why is $S_{ij}$ independent of the frame of reference in which it is observed? Sure, it is zero in every frame in which the fluid particle translates with constant linear-velocity $\mathbf{U}$ and rotates with constant angular-velocity $\mathbf{\Omega}$, but this doesn't explain why it should be the case "even if $\mathbf{U}$ depends on time and the frame of reference is rotating."
[*] This is the Exercise 3.17:
For the flow field $\mathbf{u} = \mathbf{U} + \mathbf{\Omega} \times \mathbf{x}$, where $\mathbf{U}$ and $\mathbf{\Omega}$ are constant linear- and angular-velocity vectors, use Cartesian coordinates to a) show that $S_{ij}$ is zero, and b) determine $R_{ij}$.
[$R_{ij}$ is the rotation tensor: $\displaystyle{\frac{\partial u_i}{\partial x_j} -\frac{\partial u_j}{\partial x_i}}$.]