Viscosity is a property of a fluid, not an object in and of itself, just like 'hard' is a property of an object not something which can exist in its own right.

Vorticity is vector, viscosity is a scalar, utterly different, though you have gotten vectors and scalars mixed up on the past.....

Its possible to reformulate the Navier Stokes equations to include explicit vorticity terms or to turn off viscosity but these can be done independently as they are two *different* terms in the equations. Viscosity is something you can set *before* solving the equations while vorticity is much harder to set, if at all possible, as it is explicitly dependent on the flow field $$\mathbf{v}$$, which you obtain by solving the equations. Inviscid flows can have vorticity and irrotational flows can have viscosity.

The fact you refer to properties of objects as objects (ie vorticity is a property of an object, the fluid, not an object itself) suggests you haven't quite wrapped your head around whatever it is you're looking at. For a change I have a *slight* amount of sympathy for you because I've been spending the last month pretty much trying to learn particular bits of fluid mechanics from scratch (half a lecture course 6 years ago isn't much to go on) and the physics of objects in fluids is a bitch. Just as GR is simplified if you assume a small test mass doesn't alter the gravitational field it moves through fluid mechanics is a lot easier if you assume objects in the flow don't alter the flow. If you can't make that assumption its a whole lot more unpleasant...