According to Surface tension, water molecule don't get the force from outside and get little bit outward. Is one reason viscosity? Let's look at the water in a fully filled glass. No part is outside the surface.
Now I pour more water, due to friction from lower level, water doesn't flow towards the side and stays there, the more water we add the more surface is build above 0 level. As the level increases, force of friction decreases, and at certain point, force of friction is overcome by water flow and water flows out.
So there are some water left above the surface of fully filled glass and one of the reason reason is viscosity?