Viscosity Turns a Velocity Difference into Shear Stress
Fill the gap between two plates with fluid and slide the top plate sideways. Drag the top plate's speed to watch how the fluid gets dragged along.
Slide the top plate and the fluid layer just under it is dragged along, that layer drags the next, and so on down. Drag the top plate's speed. The layers slide over each other, faster the higher you go. This flow of layers shearing past one another is called Couette flow.
Draw the speed at each height as an arrow and you get a straight triangle. Zero at the bottom, rising at a steady rate to U at the top plate. Drag the top speed and this linear profile scales as a whole. The rate of change of speed with height, du/dy, is the same everywhere.
One key assumption: the fluid does not slip at a wall. The layer touching the bottom is at rest (0); the layer touching the top plate moves exactly with it, at U. Drag the gap to narrow it. The end speeds stay pinned to the walls, but the closer the plates, the steeper the gradient du/dy.
As an equation, τ = μ du/dy. Shear stress is viscosity μ times the velocity gradient. Drag the gradient and τ grows in proportion. The faster you make the layers slide past each other, the stronger the viscous friction resisting it.
The larger the viscosity μ, the larger the shear stress for the same gradient. Switch between water and honey. Water is thin and barely resists, but honey is sticky, so stirring it at the same speed takes far more force. That difference is exactly viscosity.