Terminal velocity: when the forces balance.

A sphere falls through a liquid; weight, buoyancy and 6πηav drag bars; v-t curve levels to v_t = 2a²(ρ−σ)g/9η.

Choose the radius of a steel ball so it falls through glycerine at 5 cm/s; then double the radius and predict v_t.

Terminal velocity grows linearly with radius; viscous force alone balances weight (buoyancy forgotten).

• Real fluids resist motion through a kind of internal friction between layers moving relative to one another. This is viscosity.
• Oil between two glass plates, lower plate fixed, upper plate moved at constant velocity v: honey in place of oil needs a bigger force for the same v, so honey is more viscous.
• The fluid touching a surface moves with that surface: the top layer at v, the bottom layer at rest, with speed rising uniformly between. Each layer is pulled forward by the one above and back by the one below; this is laminar flow, like pages of a book sliding when the cover is pushed.
• In a pipe the layer along the axis moves fastest and the speed falls to zero at the walls; the speed is constant over any cylindrical surface coaxial with the tube.
• A shape ABCD in the fluid becomes AEFD after a time Δt, a shear strain Δx/l that keeps growing. Unlike a solid, the stress depends on the strain rate Δx/(lΔt) = v/l, not on the strain itself.
• Coefficient of viscosity η = shearing stress ÷ strain rate = (F/A)/(v/l) = Fl/(vA). SI unit poiseuille (Pl), also N s m⁻² or Pa s; dimensions [ML⁻¹T⁻¹].
• Viscosities (mPl), thin liquids low and thick ones high; blood's viscosity relative to water stays constant from 0 °C to 37 °C: water 1.0 at 20 °C and 0.3 at 100 °C; blood 2.7 at 37 °C; machine oil 113 at 16 °C and 34 at 38 °C; glycerine 830 at 20 °C; honey 200; air 0.017 at 0 °C and 0.019 at 40 °C.
• The viscosity of a liquid falls as temperature rises (its molecules become more mobile); the viscosity of a gas rises (random motion increases).
• Block of area 0.10 m² on a 0.30 mm liquid film, pulled by a 0.010 kg hanging mass over an ideal pulley, slides at a steady 0.085 m s⁻¹: F = 9.8 × 10⁻² N, shear stress 0.98 N m⁻², strain rate 0.085/(0.30 × 10⁻³) s⁻¹, η = 3.46 × 10⁻³ Pa s.
• Stokes' law: a sphere of radius a moving at speed v through a fluid of viscosity η feels a drag F = 6πηav, opposite to its motion and proportional to its speed. Raindrops and swinging pendulum bobs meet this drag.
• A falling sphere speeds up until viscous drag plus buoyancy equals its weight; then it falls at a constant terminal velocity: 6πηa v_t = (4π/3)a³(ρ − σ)g, so v_t = 2a²(ρ − σ)g/(9η), with ρ the sphere's density and σ the fluid's. v_t grows as the square of the radius and falls inversely with the viscosity.
• Copper ball, radius 2.0 mm, falling through oil at 20 °C with terminal velocity 6.5 cm s⁻¹ (ρ = 8.9 × 10³, σ = 1.5 × 10³ kg m⁻³, g = 9.8 m s⁻²): η = 2a²(ρ − σ)g/(9v_t) = 9.9 × 10⁻¹ kg m⁻¹ s⁻¹.

Teaching this idea? Put the simulation on your own page or LMS with this code.