I tried to answer the question only using the definitions and the Navier-Stokes equation:

$$\rho \frac{Dv}{Dt} = -\nabla P +\rho g -\mu[\nabla \times(\nabla \times v)] $$

In my opinion if the vorticity is zero, then the fluid is irrotational, regardless of presence of the viscous forces, thus $\mu$ can have a non-zero value which implies the existence of viskeuze forces, while the $\nabla \times v = 0$.