3 The coefficient of viscosity (η)

To keep the upper plate moving at constant velocity v, a force F must be applied equal to the viscous force. Experiments show that F is

• directly proportional to the velocity v of the moving plate,

• directly proportional to the area A of the moving plate (in contact with the liquid), and

• inversely proportional to the distance d between the plates.

  F ∝ A v / d → F = η A v / d

The constant of proportionality η (eta) is the coefficient of viscosity:

η = F d / (A v)

Definition. The coefficient of viscosity of a liquid is the tangential force acting on a unit area that produces a velocity difference of one unit between two layers of the liquid separated by a unit distance (perpendicular to the flow).

• Unit: N·s/m² (= Pa·s) = kg/(m·s).
• Dimensional formula: [F][d]/([A][v]) = MLT⁻² · L / (L² · LT⁻¹) = ML⁻¹T⁻¹.
• η depends on the type of liquid and on its temperature (it falls as a liquid is heated; engine oil flows more easily when hot).

Example 1 — finding η. A plate of area 0.04 m² moves at 0.1 m/s over a layer of oil 2 mm thick when a force of 0.5 N is applied. η = Fd/(Av) = 0.5 × 0.002 / (0.04 × 0.1) = 0.001/0.004 = 0.25 N·s/m².

Example 2 — finding the force. A liquid of η = 0.8 N·s/m² lies between two plates 4 mm apart; the upper plate has area 0.02 m² and moves at 0.05 m/s. F = ηAv/d = 0.8 × 0.02 × 0.05 / 0.004 = 0.2 N.

Example 3 — effect of doubling. If the distance d between the plates is doubled while the same v and area are used, the force is halved (F ∝ 1/d). If the velocity is doubled, the force is doubled.

Example 4 — comparing liquids. To move the same plate at the same velocity needs a force of 0.3 N in oil A and 1.2 N in oil B. Then η_B/η_A = 4: oil B is four times as viscous.