Stokes' Law

Viscosity and Stokes' Law

When a solid object moves through a fluid, the fluid exerts a resistive force that opposes the motion. This force arises from viscosity — the internal friction between layers of the fluid. For a small sphere moving slowly through a viscous fluid, this resistive force takes a particularly simple form.

The Origin of the Viscous Drag

Consider a sphere of radius $r$ moving with a constant velocity $v$ through a fluid of viscosity $\eta$. The fluid layers in contact with the sphere's surface are dragged along, while layers farther away are at rest. This velocity gradient creates shear stresses that produce a net retarding force on the sphere.

The exact calculation of this force requires solving the Navier-Stokes equations of fluid dynamics — a complex mathematical problem. However, for the special case of slow, steady motion (where inertial forces are negligible compared to viscous forces), the British physicist George Gabriel Stokes derived a remarkably simple result.

Stokes' Law

The viscous drag force $F$ on a sphere moving through a fluid is directly proportional to:

• The viscosity $\eta$ of the fluid
• The radius $r$ of the sphere
• The velocity $v$ of the sphere

The constant of proportionality is $6\pi$, making the complete expression $F = 6\pi\eta r v$.

Terminal Velocity of a Falling Sphere

When a sphere is released in a viscous fluid, three forces act on it:

1. Weight acting downward: $W = mg = \frac{4}{3}\pi r^3 \rho g$, where $\rho$ is the density of the sphere material
2. Buoyant force acting upward: $F_b = \frac{4}{3}\pi r^3 \sigma g$, where $\sigma$ is the density of the fluid
3. Viscous drag acting upward: $F_d = 6\pi\eta r v$

Initially, the sphere accelerates downward because its weight exceeds the sum of buoyancy and drag. As velocity increases, the drag force grows proportionally. Eventually, the net force becomes zero and the sphere continues with constant velocity — the terminal velocity.

At terminal velocity $v_t$:

Solving for $v_t$:

This is the terminal velocity of a sphere falling through a viscous fluid under gravity.

Applications of Stokes' Law

Determining viscosity: By measuring the terminal velocity of a small sphere falling through a fluid of known density, the viscosity can be calculated using Stokes' law. This is the principle behind the falling-sphere viscometer.

Sedimentation: The settling of particles in liquids (sedimentation) follows Stokes' law. This is used in:

• Purifying water by allowing suspended particles to settle
• Separating minerals in ore processing
• Understanding the deposition of sediments in rivers and oceans

Motion of microorganisms: Many bacteria and single-celled organisms live in the low-Reynolds-number regime where Stokes' law applies. Their swimming mechanisms are fundamentally different from those of larger organisms because viscous forces dominate. …