Introduction
An object moving through a fluid (static or dynamic) will always face a drag force working in the opposite direction. This is caused by the particles of the fluid requiring force to move out of the way so the object can travel past them. The shape of an object traveling through a fluid impacts how much drag force is felt. Certain shapes minimize the effects of the particles moving around them. For example, shapes like the airfoil are used as the cross-sectional shape of airplane wings (picture later), as they are ideal for minimizing drag and maximizing lift.
Fluids are governed by the Navier Stokes, which are coupled partial differential equations that currently only have approximate solutions. Hence, the drag force of a fluid traveling around an object can’t be solved analytically for most shapes (Stoke’s law talks about the only shape that does).
Stoke’s Law
The only shape that has an exact solution for drag force is the sphere. Stoke’s Law describes the following equation for the drag force experienced by a sphere:
Note: The equation above assumes an ideal scenario (laminar flow) where the movement of the object doesn’t drastically change the behavior of the fluid.
The first variable in the equation refers to viscosity, which is a fluid’s resistance to flow. Fluids that appear thicker tend to have higher viscosities. A fluid like honey has a high viscosity, and it doesn’t flow very well when compared to water, which has a relatively low viscosity. If a fluid has high viscosity, it is significantly more difficult to move particles out of the way so the sphere can travel through (more drag force). Although not explicitly written in the equation, viscosity is actually a function of temperature. Most substances will have a lower viscosity at higher temperatures as the intermolecular forces between particles are smaller.
The second variable in the equation refers to the radius of the sphere. If the sphere is big, it has a lot more surface area that comes into contact with the particles of the fluid. This means more particles pushing against the object, creating a higher drag force.
The final variable in the equation refers to the velocity of the object itself inside the fluid. If an object is traveling faster, it comes into contact with more particles, leading to more resistance.
In order to create Strokes law, all of these variables are combined, and a constant (supposedly found through experimental data) is used to tie them all together.
Stoke’s Law to find Velocity
A common way to observe Stoke’s law is to let a ball sink into a cup of liquid. Below is the process used to find the average velocity of the sphere as it travels downward due to gravity.