Sinking in Syrup
Stokes' law is a mathematical equation that calculates the drag force exerted on a small spherical object moving slowly through a viscous fluid. Derived by George Gabriel Stokes in 1851, the law is essential for understanding how small particles behave when settling out of a suspension.
When a particle falls through a fluid, gravity pulls it down, while buoyancy and viscous drag push it back up. As the particle accelerates, the drag force increases until it perfectly balances the downward gravitational pull. At this point, the particle stops accelerating and falls at a constant speed, known as the terminal velocity.
Industrial and Natural Applications
- Raindrops and Fog: Stokes' law explains why tiny water droplets in fog remain suspended in the air almost indefinitely (their terminal velocity is near zero), while larger, heavier raindrops fall quickly to the ground.
- Centrifuges: Laboratories use centrifuges to artificially increase the 'g-force' on suspended biological cells, allowing them to overcome viscous drag and separate out of the liquid plasma much faster than gravity alone would permit.
- Brewing: When beer ferments, the yeast cells slowly sink to the bottom of the vat over weeks. Brewers use Stokes' law to predict how long 'flocculation' will take to clear the beer.
The Formula
Example Calculation
Calculate the drag force on a tiny glass bead with a radius of $0.001 , ext{m}$ (1 mm) falling at $0.05 , ext{m/s}$ through thick motor oil (viscosity $\mu = 0.5 , ext{Pa}\cdot\text{s}$).
- Multiply $6 \cdot \pi \cdot \mu \cdot r \cdot v$: $6 \cdot \pi \cdot 0.5 \cdot 0.001 \cdot 0.05 \approx 0.00047 , ext{N}$.
The thick oil exerts a gentle drag force of $0.00047 , ext{Newtons}$ upward against the sinking bead. If this force equals the bead's weight (minus buoyancy), it has reached its terminal velocity.