Viscosity and Surface Tension: Quick Revision

A brief note on Viscosity and Surface Tension. Read it before a test. Try each question at the end before you open its answer.

The big ideas

• Viscosity: $F = \eta A\dfrac{v}{d}$, with $\eta$ in $\text{Pa s}$. Warm liquids are thinner; warm gases are thicker.
• Stokes' law: $6\pi\eta rv$; terminal speed $v_t = \dfrac{2r^2(\rho - \sigma)g}{9\eta}$.
• Reynolds number: $Re = \dfrac{\rho vd}{\eta}$; small means smooth flow.
• Surface tension: force per unit length, or energy per unit area; drops $\dfrac{2S}{r}$, soap bubbles $\dfrac{4S}{r}$.
• Capillary rise: $h = \dfrac{2S\cos\theta}{\rho gr}$; wetting liquids rise, others fall.

The facts at a glance

A worked example

A grain of sand of radius $0.050\ \text{mm}$ (density $2650\ \text{kg/m}^3$) settles through still water ($\eta = 1.0 \times 10^{-3}\ \text{Pa s}$).

1. Terminal speed. $\dfrac{2r^2(\rho - \sigma)g}{9\eta}$, with $2r^2(\rho - \sigma)g = 8.09 \times 10^{-5}\ \text{kg/s}^2$ and $9\eta = 9.0 \times 10^{-3}\ \text{Pa s}$, gives $8.98 \times 10^{-3}\ \text{m/s}$.
2. Is Stokes' law fair? $Re = 1000\ \text{kg/m}^3 \times 0.009\ \text{m/s} \times \dfrac{1.0 \times 10^{-4}\ \text{m}}{1.0 \times 10^{-3}\ \text{Pa s}} = 0.9$, small, so yes.
3. Settling $2.0\ \text{m}$. $\dfrac{2.0\ \text{m}}{0.009\ \text{m/s}} = 223\ \text{s}$, nearly four minutes.

Practice with new numbers

A program to try

The program works out the grain of sand of the worked example.

Mistakes to avoid

• Thickness in millimetres in $\eta A\dfrac{v}{d}$. Change to metres.
• $r$ for $r^2$ in the terminal speed. It grows as the square of the radius.
• The ball's density alone. Use $\rho - \sigma$.
• One surface for a soap film or bubble. It has two.
• A positive rise for mercury. Its $\cos\theta$ is negative: it falls.

Check yourself

1. Two drops of oil are falling through air at their terminal speeds; one has $1.5$ times the radius. How do their speeds compare?

2. What is the excess pressure inside a soap bubble of radius $2.0\ \text{cm}$, with $S = 0.025\ \text{N/m}$?

3. Water rises $3.0\ \text{cm}$ in a tube. How high in a tube of a third of the radius?

4. Why does a detergent help clean a greasy plate?

5. Why does honey pour faster on a hot day?