1. Concept Overview
When an object moves through a fluid, it experiences a resistive force that opposes its motion. This force comes from the fluid’s viscosity and is called the viscous force. Stokes’ law gives a simple mathematical expression for this force when a small spherical object moves slowly through a highly viscous fluid.
This principle helps explain how raindrops reach terminal velocity, how tiny particles settle in water, and how blood cells move through plasma.
2. Viscous Force
2.1. Meaning of Viscous Force
Viscous force is the frictional force exerted by a fluid on a body moving through it. It always acts opposite to the direction of motion.
2.2. Dependence on Fluid Motion
The stronger the viscosity of the fluid, the larger the resisting force on the moving object. Slow, smooth motion in viscous fluids is where viscous force becomes highly noticeable.
3. Understanding Stokes’ Law
3.1. Conditions for Stokes’ Law
Stokes’ law applies only under specific conditions:
- The object must be spherical.
- The motion must be slow (low Reynolds number).
- The fluid should be viscous and flow should be laminar.
3.2. Mathematical Form of Stokes’ Law
According to Stokes:
\( F_v = 6 \pi \eta r v \)
Where:
- \(F_v\): viscous force
- \(\eta\): coefficient of viscosity
- \(r\): radius of the sphere
- \(v\): velocity of the sphere
3.3. Meaning of the Formula
The viscous force increases with:
- the viscosity of the fluid (thicker fluids resist more)
- the size of the particle
- the speed of motion
4. Terminal Velocity
4.1. What is Terminal Velocity?
When an object falls through a viscous fluid, the downward weight and the upward viscous force plus buoyant force act on it. As speed increases, viscous force increases until it balances the downward forces completely. At this point, the object moves with constant speed—this is terminal velocity.
4.2. Expression for Terminal Velocity
For a sphere falling through a viscous fluid:
\( v_t = \dfrac{2 r^2 (\rho_s - \rho_f) g}{9 \eta} \)
Where:
- \(v_t\): terminal velocity
- \(\rho_s\): density of the sphere
- \(\rho_f\): density of the fluid
- \(r\): radius of the sphere
- \(\eta\): viscosity of the fluid
5. Physical Interpretation
5.1. Competition of Forces
At the start, weight is greater than viscous force, so the object accelerates. As velocity increases, so does viscous force. Eventually, forces balance and acceleration stops.
5.2. Influence of Radius and Density
Larger objects reach higher terminal velocities because they have more weight for the same viscous resistance. Objects denser than the fluid sink quickly; low-density objects take longer or may float.
6. Real-Life Examples
6.1. Raindrops Reaching Steady Speed
Raindrops do not keep accelerating indefinitely. Air’s viscosity slows them until they reach terminal velocity, preventing them from falling dangerously fast.
6.2. Sedimentation of Tiny Particles
Dust or pollen settling in water moves slowly because viscous forces dominate. Stokes’ law describes their motion accurately.
6.3. Motion of Cells in Blood
Blood plasma is viscous. Red blood cells move through it with forces similar to those predicted by Stokes’ law, especially at low speeds.