does viscosity change with different shear rate?

Question

in what direction do you see the change to be at higher shear rate?

Answer 1

Try googling "non-newtonian fluid". These materials increase viscosity with increased deformation rate. You can easily make such a material from cornstarch and water. Just mix them into a paste. The ref. web page is a hilarious video of a couple of guys running across a pool of the stuff.
There also materials whose viscosity decreases with deformation rate; many household paints are deliberately formulated this way so they can be applied like a liquid but drip and flow more slowly after application. But as the previous answer says, most liquids are just plain, boring relatively-constant viscosity. With these the viscous force increases with shear rate of course, but not the viscosity.

Answer 2

yes it can - it depends upon the liquid

For liquids like water it does not, but for things like tomato ketchup they become thinner with increased shear (and is also depends how long it is subject to shear). Though there are other fluids that behave the opposite way

Generally speaking a pure liquid will have a boring set of properties that don't vary with shear rate or time, and those which are mixtures, or have organic content will

It's a few years since I studied it so I'm a bit hazy on the details...