viscosity and no-slip condition
Fill a fluid between two plates and push the top plate sideways. See how the fluid is dragged along by dragging the speed of the top plate. When the top moves and the bottom is held, the fluid between them shears layer by layer.
Push the top plate and the fluid layer just under it is dragged, and that layer drags the next, and the next. Drag the speed of the top plate and the layers slide on each other, and you see the speed grow as you go up. A flow in which layers shear and slide on each other is called Couette flow. It is a picture of one layer holding and pulling its neighbor.
Draw the speed at each height as an arrow and it makes a straight triangle. The floor is 0, and it grows at a steady rate until it is U at the top plate, and when you drag the speed of the top plate this linear distribution grows whole. That is because the rate of change of speed with height, du/dy, is the same everywhere. The tips of the arrows lie on one straight line.
There is one key assumption. The fluid does not slip at a wall, so the layer touching the floor is at rest, 0, and the layer touching the top plate has the same speed U as the plate. Drag the gap between the plates to narrow it and the speeds at the two ends stay stuck to the walls, while the narrower the gap the steeper the slope du/dy. Pack the same difference of speed into a shorter height and the slope stands up.
In symbols, τ = μ du/dy, and the shear stress is the viscosity times the velocity gradient. Drag the slope and the stress grows in proportion, which means that the faster you make the layers shear, the stronger the viscous friction that resists it. The slope is the size of the friction.
The larger the viscosity, the larger the shear stress for the same slope. Switch water and honey and water is thin and resists only a little, while honey is sticky and stirring it at the same speed takes a much larger force, and that difference is the viscosity. The same shear, and the sticky side resists more.