9.5.1. Stokes’ Law

When a body falls through a fluid it drags the layer of the fluid in contact with it. A relative motion between the different layers of the fluid is set and, as a result, the body experiences a retarding force. Falling of a raindrop and swinging of a pendulum bob are some common examples of such motion.

When an object, like a raindrop, falls through air, it doesn't just fall freely. It interacts with the air in a way that the air pushes back against it, creating a force that opposes the fall. This is similar to how you feel resistance when you try to move your hand through water. As the raindrop falls, it causes the air around it to move and creates layers of air with different velocities due to friction and drag. This interaction leads to a retarding force on the falling object.

Imagine trying to push your hand through thick syrup compared to pushing it through air. Your hand meets much more resistance in syrup than in air. Similarly, as a raindrop falls, it meets resistance from the air, which significantly affects its speed and motion.

It is seen that the viscous force is proportional to the velocity of the object and is opposite to the direction of motion. The other quantities on which the force F depends are viscosity η of the fluid and radius a of the sphere.

Stokes’ law states that the force resisting the fall of a sphere in a viscous fluid is directly related to the sphere's velocity and the fluid's viscosity. The higher the viscosity of the fluid, the greater the retarding force on the sphere. This means, for instance, a small ball falling through honey (a viscous fluid) will experience more resistance than one falling through water. Mathematically, this is expressed as: F = 6πηav, where F is the viscous force, η is the fluid's viscosity, a is the radius of the sphere, and v is its velocity.

Think about how easy it is to roll a marble on a flat surface versus rolling it in a pot of thick honey. The marble moves slowly in honey because the viscous force acting against it is significant compared to when it's on a flat surface with little resistance.

Sir George G. Stokes (1819–1903), an English scientist enunciated clearly the viscous drag force F as 6F = avηπ.

Stokes formulated a mathematical equation to describe this resistance force, known as Stokes’ Law. The equation shows that the force due to viscosity increases with the radius of the sphere and the speed at which it is moving through the fluid. This establishes a clear relationship between the size of an object moving through a fluid, its speed, and how sticky or thick the fluid is.

Consider different sizes of balls dropping through a liquid, such as oil. A large ball would experience a greater viscous drag than a smaller one at the same speed due to its larger surface area interacting with the fluid. This principle helps in designing objects like submarines that minimize drag to move efficiently through water.

The sphere (raindrop) then descends with a constant velocity. Thus, in equilibrium, this terminal velocity vt is given by 6πηavt = (4π/3) a3 (ρ-σ)g where ρ and σ are mass densities of sphere and the fluid, respectively.

As the raindrop falls, it accelerates until the upward forces (viscous drag and buoyancy) balance the downward gravitational force. At this point, the raindrop stops accelerating and descends at a constant speed, known as terminal velocity. The equation also shows that larger drops will fall faster due to the factor of a² in the equation, meaning their size greatly influences how fast they can fall through the fluid.

When you drop a small ball and a large ball simultaneously in a viscous fluid, like oil, you'll notice that the larger ball reaches the bottom first. This is because it has a different terminal velocity, dictated by its size, allowing it to overcome the resistance of the fluid more effectively.

Thus, the terminal velocity vt depends on the square of the radius of the sphere and inversely on the viscosity of the medium.

The relationship found in Stokes’ Law indicates that not only does the size of the sphere affect how fast it can fall through a fluid, but also, the thickness of the fluid matters. A more viscous fluid will slow down the descent of the raindrop or object significantly, leading to a lower terminal velocity. Conversely, a larger ball in a less viscous fluid will sink faster.

Think about how easy or difficult it is to run through water versus running through a thick mud. In thick mud (high viscosity), you slow down considerably, while in water you can move more freely. This illustrates how viscosity directly impacts movement speed and terminal velocity in fluids.

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