Important Topic of Physics: Stokes Law

When a small object moves through a fluid, like a tiny droplet falling through the air or a grain of sand sinking in water, it faces resistance. This resistance is called drag, and understanding how it works is important in many fields, from physics to engineering and even biology. Stokes Law, named after Sir George Gabriel Stokes, helps us calculate this drag force for small spherical objects moving at low speeds through a fluid. In this article, we will explore Stokes Law, its formula, applications, and its limitations in simple and clear language.

What is Stokes Law?

Stokes Law explains the force of resistance (drag) that a spherical object experiences when it moves through a fluid (like air or water) at a slow speed. This drag force is caused by the friction between the object’s surface and the fluid. According to Stokes Law, the drag force depends on the size of the object, the speed at which it moves, and the viscosity of the fluid.

In mathematical terms, Stokes Law is expressed as:

F = 6πηrv

Where:

• F = Drag force (in newtons)
• η = Viscosity of the fluid (a measure of how thick or resistant the fluid is to flow, in pascal-seconds)
• r = Radius of the spherical object (in meters)
• v = Velocity of the object relative to the fluid (in meters per second)

This equation tells us how much resistance an object will face while moving through a fluid.

Breaking Down the Formula

Let’s look at the components of Stokes Law to better understand what they mean:

1. Viscosity (η): Viscosity is a property of fluids that describes their thickness. For example, honey is more viscous than water. A higher viscosity means the fluid offers more resistance to an object moving through it.
2. Radius (r): The size of the spherical object matters. A larger radius increases the surface area of the object, leading to more friction and, therefore, more drag force.
3. Velocity (v): The speed at which the object moves through the fluid also plays a role. Faster movement increases the drag force.
4. The Constant (6π): This constant comes from the mathematical derivation of Stokes Law. It accounts for the shape of the object (a sphere) and how it interacts with the fluid.

Conditions for Stokes Law

Stokes Law applies under specific conditions:

• The object must be spherical.
• The motion must be at a low speed (low Reynolds number), meaning the flow of the fluid around the object is smooth (laminar flow) and not turbulent.
• The fluid should be uniform and continuous (no sudden changes in properties like density or viscosity).
• The object must be small compared to the volume of the fluid.

If these conditions are not met, Stokes Law may not accurately describe the drag force.

How Was Stokes Law Derived?

Sir George Gabriel Stokes, a British physicist and mathematician, derived this law in 1851 using fluid dynamics. He analyzed how a sphere interacts with a fluid by solving the Navier-Stokes equations, which describe the motion of fluids. Stokes Law is a simplified result of these complex equations for specific situations.

Applications of Stokes Law

Stokes Law is widely used in science and engineering. Here are some examples:

1. Sedimentation: Stokes Law helps predict how fast small particles (like sand or silt) settle in a liquid. This is important in geology, wastewater treatment, and manufacturing processes.
2. Viscometry: Scientists use Stokes Law to measure the viscosity of fluids. By observing how a tiny sphere falls through the fluid, they can calculate its viscosity using the formula.
3. Aerosol Dynamics: It is used to understand how small particles, such as dust or pollen, move in the air. This is useful in environmental studies and air pollution control.
4. Medical Applications: In medicine, Stokes Law helps in understanding the motion of tiny particles, like blood cells, through liquids. It is also used in drug delivery systems to study how particles move in the bloodstream.
5. Oil Exploration: In the oil industry, Stokes Law is used to study sedimentation and separation processes, helping engineers understand how to remove impurities from crude oil.
6. Physics and Fluid Mechanics Education: Stokes Law serves as a fundamental example of drag forces and fluid motion, helping students understand the basics of fluid mechanics.

Example Problem

Let’s apply Stokes Law to a simple problem:

Problem: A small droplet of oil with a radius of 0.001 m is falling through water at a velocity of 0.02 m/s. The viscosity of water is 0.001 Pa·s. What is the drag force acting on the droplet?

Solution: Using the formula F = 6πηrv, we plug in the values:

• η = 0.001 Pa·s
• r = 0.001 m
• v = 0.02 m/s

F = 6 × 3.14 × 0.001 × 0.001 × 0.02

F = 3.77 × 10⁻⁵ N

So, the drag force is 3.77 × 10⁻⁵ newtons.

Limitations of Stokes Law

While Stokes Law is useful, it has some limitations:

• It only applies to spherical objects. For irregular shapes, the drag force calculation becomes more complex.
• It assumes low-speed motion. At higher speeds, turbulence occurs, and Stokes Law no longer applies.
• It doesn’t account for interactions between multiple objects in the fluid.
• It assumes the fluid is uniform, which may not be true in all real-world scenarios.

Extending Beyond Stokes Law

For objects moving at higher speeds or in turbulent conditions, other formulas, like the drag equation, are used. These equations consider factors like fluid density, object shape, and turbulence to provide a more accurate estimate of drag forces.

Why is Stokes Law Important?

Stokes Law is a cornerstone of fluid mechanics and has a wide range of practical uses. It helps engineers, scientists, and researchers design systems and solve problems involving fluid motion. Whether it’s understanding how pollutants settle in water or designing better medical devices, Stokes Law plays a critical role.