13.24: Stokes’ Law

Viscous forces, like friction, are intermolecular forces that resist the relative motion of molecules over each other. When a solid body moves through a liquid, viscous forces drag it in the opposite direction. The force's magnitude depends on the solid's shape and size, as well as its speed and the liquid's coefficient of viscosity, density and temperature.

The expression for the force on a solid spherical object in a fluid is called Stokes' law. Stokes' law is valid only for low Reynolds number fluids, meaning that it is applicable only when the fluid flow is laminar and not turbulent. The force on the solid object is experimentally found to be proportional to the object's radius and speed, along with the liquid's viscosity coefficient. The order of dependence of each variable can be obtained through dimensional analysis. The proportionality constant is obtained experimentally.

The motion of a spherical solid through a liquid can be analyzed using Newton's second law of motion. Initially, there is no viscous drag as the solid is at rest. The net downward force, which is given by the difference in its weight and the liquid's buoyant force, increases its speed downward. However, increasing speed increases the viscous drag. Once there is significant upward viscous drag, there is no net force due to the three forces balancing each other out: the downward weight, the upward buoyancy and the upward viscous drag. The solid then reaches a steady speed, which is called the terminal velocity.

Consider an object moving through a fluid. The fluid layers near the object get dragged along with it and experience relative velocity, resulting in a viscous force on the object. The viscous force depends on the object's shape, size, speed, and fluid properties. 

For a spherical body moving inside a fluid, the viscous force is proportional to the sphere's radius, speed, and fluid viscosity.

Solving the equation dimensionally, gives the dependence of viscous force on these parameters. The expression obtained is Stokes' law, indicating that the viscous force is directly proportional to the object's velocity.

If a sphere is dropped through a fluid, the object's weight pulls it downward, while the viscous and buoyant forces act upward.

As the velocity increases due to acceleration, the viscous force increases. When the viscous force balances the weight and buoyancy, the sphere falls with a constant velocity, called terminal velocity.

For a highly viscous liquid, the terminal velocity can be measured experimentally. Then, using Stokes' law, the viscosity can be obtained.