What Is Viscous Drag / Stokes' Law for a Sphere in a Fluid?

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• $F_d$ is the viscous drag force in newtons (N)
• $\eta$ is the dynamic viscosity in pascal-seconds (Pa s)
• $r$ is the sphere's radius in metres (m)
• $v$ is the sphere's speed relative to the fluid in metres per second (m s$^{-1}$)

For example, a sphere of radius $1.0 \times 10^{-3}\,\text{m}$ moving at $0.20\,\text{m s}^{-1}$ through water with $\eta = 0.89 \times 10^{-3}\,\text{Pa s}$ experiences

Stokes' law applies when the sphere is small, the flow is laminar, and the speed is low enough that turbulence is negligible.

The linear relationship with speed distinguishes Stokes' drag from the approximately quadratic drag often used at higher speeds. The condition is commonly expressed using a low Reynolds number, meaning viscous effects dominate over inertial effects and the fluid flows smoothly around the sphere.

For a sphere falling through a fluid, its speed initially increases. This increases viscous drag until drag plus buoyancy equals weight. The resultant force then becomes zero, so acceleration becomes zero and the sphere moves at constant terminal speed.

A common misconception is that drag always equals weight. It does not: this equality occurs only when the resultant force is zero, after buoyancy has also been included.

Exam Technique

For an IB calculation, draw a free-body diagram, show drag opposite the motion, include buoyancy where relevant, convert radius to metres, and substitute into Stokes' law. In terminal-speed questions, explicitly state that the resultant force is zero.