Viscosity

Honey pours slowly; water pours easily. The difference is viscosity: internal friction between layers of a fluid that slide over each other. In a river, the layer touching the bed is at rest and each layer above moves a little faster; viscosity tries to stop the layers slipping past each other.

For a fluid, the shearing stress depends on how fast the layers are being sheared (the strain rate), not on how far. That gives the coefficient of viscosity:

• η = (F/A)/(v/l): shear stress ÷ rate of shear strain, where v/l is the speed difference across a layer of thickness l.
• Unit: Pa s (also called the poiseuille); 1 Pa s = 10 poise. Dimensions [ML⁻¹T⁻¹].

Example: a plate of area 0.05 m² slides at 0.2 m/s over a 1 mm layer of oil with η = 0.8 Pa s. The force needed is F = ηAv/l = 0.8 × 0.05 × 0.2/0.001 = 8 N.

Stokes' law: a small sphere of radius a moving at speed v through a fluid feels a viscous drag F = 6πηav. The drag grows with speed.

So a falling sphere speeds up only until the drag plus the upthrust balance its weight. After that, the net force is zero and it falls at a steady terminal velocity:

• v_t = 2a²(ρ – σ)g/(9η), where ρ is the sphere's density and σ the fluid's.
• It grows with the square of the radius: big raindrops fall faster than small ones.
• Example: a water droplet of radius 0.1 mm falling in air (η = 1.8 × 10⁻⁵ Pa s) has v_t = 2 × (10⁻⁴)² × 1000 × 9.8/(9 × 1.8 × 10⁻⁵) ≈ 1.2 m/s, which is why fine drizzle drifts down so slowly.

Temperature matters, in opposite ways. Liquids flow more easily when warm (hot oil is thinner), so their viscosity falls. Gases become more viscous as they warm, because their molecules move faster and carry more momentum between layers.