Stokes' Law

Stokes' law expresses the viscous force on a spherical object moving through a fluid. The law states that viscous force is directly proportional to the object's radius, velocity, and the fluid's viscosity coefficient. This relationship applies only for low Reynolds number fluids where flow is laminar and not turbulent, ensuring predictable drag behavior.

Viscous forces result from intermolecular friction that resists relative motion between fluid layers. When a solid body moves through a liquid, fluid layers near the object get dragged along, creating relative velocity between layers. This drag acts opposite to the object's motion, with magnitude depending on the object's shape, size, speed, and the liquid's viscosity coefficient.

Terminal velocity is the constant speed at which a falling sphere moves through a fluid when forces balance. Initially, weight and buoyancy accelerate the sphere downward. As velocity increases, viscous drag increases until it balances the weight and buoyant force, resulting in zero net force and constant velocity motion.

By measuring the terminal velocity of a sphere falling through a highly viscous liquid, you can apply Stokes' law to calculate the fluid's viscosity coefficient. Since terminal velocity occurs when viscous force balances weight and buoyancy, rearranging Stokes' law allows you to solve for viscosity using the measured velocity and known sphere properties.

Viscous force depends on four primary factors: the sphere's radius, the object's velocity through the fluid, the fluid's viscosity coefficient, and the fluid's density and temperature. Dimensional analysis reveals that force is directly proportional to radius and velocity, while proportionality constants are determined experimentally for specific fluid conditions.

Stokes' law applies exclusively when fluid flow is laminar, not turbulent. At low Reynolds numbers, fluid motion is smooth and orderly, allowing the linear relationship between viscous force and velocity to hold. At high Reynolds numbers, turbulent flow creates complex eddies and nonlinear drag behavior that violates Stokes' law assumptions.

Initially, weight exceeds buoyancy, causing downward acceleration and increasing viscous drag. As velocity rises, upward viscous drag increases until all three forces balance: downward weight equals upward buoyancy plus upward viscous drag. This equilibrium produces terminal velocity, where the sphere moves at constant speed with no net force.