# Uniform / rigid rotation in fluids, velocity gradients and resulting friction

In the derivation of the Navier-Stokes equation e.g. in Landau & Lifshtiz Volume 6 on fluid mechanics it is stated that the viscous stress tensor

$$\sigma_{ij}^{\prime}$$ must also vanish when the whole fluid is in uniform rotation, since it is clear that in such a motion no internal friction occurs in the fluid.

Moreover, a uniform / rigid rotation leads to velocity gradients due to the change of direction. I fail to understand why these velocity gradients are not leading to frictional forces?

How does angular momentum conservation play in here?

Any pointers to literature / explanation / derivation?

We see this by considering the relative motion between close points (see the standard book from Batchelor from which I took most of the following). Let us consider the velocity $$\vec{u}$$ at position $$\vec{x}$$ as well as the neighboring velocity $$\vec{u}+\delta \vec{u}$$ at $$\vec{x}+\vec{r}$$ at time $$t$$. Then to first order we can write $$\delta u_i = r_j \frac{\partial u_i}{\partial x_j}\,.$$
Moreover we remember that we can split the gradient of the velocity field into symmetric and anti-symmetric part. We do this for the difference in velocities between the two neighboring points $$\delta u_i = \delta u_i^{(s)} + \delta u_i^{(a)} = r_i e_{ij} + r_i \xi_{ij} = r_j e_{ij} - \frac{1}{2} r_j \epsilon_{ijk} \omega_k$$ with vorticity $$\vec{\omega}$$. In vector notation $$\delta\vec{u}^{(a)} = \frac{1}{2} \vec{\omega} \times \vec{r} \,.$$ Thus, $$\delta \vec{u}^{(a)}$$ is the velocity due to rigid-body rotation at $$\vec{r}$$ with angular velocity $$\frac{1}{2} \vec{\omega}$$.
The crucial point is to look at the temporal change of the displacement vector $$\vec{r}$$ or its square $$\frac{1}{2} \frac{\mathrm{D}}{\mathrm{D}t} r_i^2 = r_i \delta u_i = r_i \left( r_j e_{ij} - \frac{1}{2} r_j \epsilon_{ijk} \omega_k \right) = r_i r_j e_{ij}$$ because the second term vanishes once we expand the sum. Thus rotations do not affect the distance between neighboring points and thus do not cause a relative motion between the. However relative motion between neighboring fluid particles is needed for friction.