Viscosity and Stokes' law

Inner friction

Pour honey and water side by side: the honey creeps. Every fluid resists its layers sliding past each other, some far more than others. This inner friction is viscosity.

Picture a liquid between a fixed plate and a moving plate. The layer touching each plate moves with it, and the layers between slide over each other.

The force needed to keep the top plate moving is

where $A$ is the plate's area, $v$ its speed and $d$ the thickness of the film. $\eta$ is the coefficient of viscosity, in $\text{Pa s}$: water $1.0 \times 10^{-3}\ \text{Pa s}$, glycerine $0.80\ \text{Pa s}$, air $1.8 \times 10^{-5}\ \text{Pa s}$.

Warm a liquid and its viscosity falls (warm honey runs); warm a gas and its viscosity rises.

Stokes' law and terminal speed

A small sphere of radius $r$ moving slowly at $v$ through a fluid feels a drag

Drop a ball into a tall jar of oil. It speeds up, the drag grows, and soon drag plus buoyancy balance its weight. It then falls at a steady terminal speed:

where $\rho$ is the ball's density and $\sigma$ the fluid's.

You should find that twice the radius gives four times the speed, and a ball as dense as the liquid does not fall at all.

Watch out. $v_t$ goes as $r^2$, not $r$. That is why fine mist floats while big raindrops fall fast.

Steady or turbulent?

The Reynolds number, $Re = \dfrac{\rho vd}{\eta}$, tells whether a flow stays smooth. Below about $1000$ it is steady; above about $2000$ it becomes turbulent.

A worked example

A glass ball ($2500\ \text{kg/m}^3$) of radius $1.0\ \text{mm}$ falls through glycerine ($1260\ \text{kg/m}^3$, $0.80\ \text{Pa s}$). Find its terminal speed.

1. Formula. $v_t = \dfrac{2r^2(\rho - \sigma)g}{9\eta}$.
2. Numbers. $2 \times (1.0 \times 10^{-3}\ \text{m})^2 \times 1240\ \text{kg/m}^3 \times 9.8\ \text{m/s}^2 = 2.43 \times 10^{-2}\ \text{kg/s}^2$, and $9\eta = 7.2\ \text{Pa s}$.
3. Answer. $v_t = 3.38 \times 10^{-3}\ \text{m/s}$, between $3\ \text{mm}$ and $4\ \text{mm}$ each second.

Practise with new numbers

Try it in Python

The program steps the ball's speed forward in time and compares it with the formula.

Remember these

• $F = \eta A\dfrac{v}{d}$; $\eta$ in $\text{Pa s}$.
• Stokes' law: $6\pi\eta rv$; terminal speed $\dfrac{2r^2(\rho - \sigma)g}{9\eta}$, which goes as $r^2$.
• Warm liquids are thinner; warm gases are thicker.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Why is engine oil thinner when the engine is hot?

2. One ball has three times the radius of another of the same material. Compare their terminal speeds in the same oil.

3. A plate of $0.10\ \text{m}^2$ slides at $0.40\ \text{m/s}$ on oil $1.0\ \text{mm}$ thick, with $\eta = 0.25\ \text{Pa s}$. What force does it need?

4. Why does a parachute bring a jumper down at a gentle steady speed?