Viscous drag, Stokes' law and terminal velocity

Physics · Fluid dynamics. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

Viscosity is the friction between layers of a fluid as they slide over each other. Honey has a high viscosity and water a low one. The viscosity of a liquid falls when it is heated. A body moving through a fluid feels a retarding force called viscous drag, which grows with speed.

When a small sphere is dropped in a fluid it speeds up at first. The drag increases until it equals the weight. The net force is then zero, and the sphere falls at a constant speed called the terminal velocity.

In streamline (laminar) flow every particle passing a point follows the same path with the same velocity. Above a certain speed the flow becomes turbulent, with irregular whirls and eddies.

Rules to remember

Common mistake

Drag depends on r but terminal velocity depends on r2. Doubling the radius doubles the drag at the same speed, but makes the terminal velocity four times as large.

Worked example

A sphere of radius 5.0 mm moves at 0.20 m s−1 through oil of viscosity 0.10 Pa s. Find the viscous drag on it.

  1. Use F = 6πηrv with r = 5.0 × 10−3 m.
  2. ηrv = 0.10 × 5.0 × 10−3 × 0.20 = 1.0 × 10−4 N.
  3. F = 6π × 1.0 × 10−4 = 18.85 × 10−4 N.

Answer: About 1.9 × 10−3 N

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