The constancy of H when is a famous result, and has many simple qualitative consequences. It states that the pressure in a fluid is smaller when the velocity is larger. The classical example of this is flow through segments of pipe of varying diameter. Whenever the radius in the pipe drops, the pressure in the pipe increases. This has little to do with global rotations of the fluid.
Shear flows have nonzero vorticity.
Recall our discussion of the derivation of viscosity. Viscous stresses depended on only the symmetric component of the gradient of viscosity. The antisymmetric component is in fact essentially the vorticity.