Table of Contents
Stokes law derivation for laminar flow
Stokes’ law is a mathematical equation used to describe the motion of a small sphere through a viscous fluid. It is named after George Gabriel Stokes, who derived the equation in the 19th century.
The equation is given by:
F = 6πηrv
where:
F is the force required to move the sphere through the fluid at a constant velocity (also known as the drag force) η is the viscosity of the fluid r is the radius of the sphere v is the velocity of the sphere
The equation can be derived by considering the balance of forces on the sphere as it moves through the fluid. The sphere experiences a viscous force from the fluid, which is proportional to the velocity of the sphere and the viscosity of the fluid. The sphere also experiences an opposing force due to its inertia, which is proportional to its mass and acceleration.
Assuming the sphere is moving at a constant velocity, the viscous force and the inertial force must balance each other. This leads to the following equation:
F = 6πηrv
This equation is known as Stokes’ law and can be used to predict the drag force experienced by a small sphere moving through a viscous fluid at a constant velocity.
What is Stokes Law?
Stokes Law is a law that describes the motion of objects through a fluid. It is named after Sir George Stokes, who first derived it in 1851. The law is only valid for objects that are small compared to the size of the fluid, and for fluids that are viscous.
Stokes Law Formula
The Stokes law formula is used to calculate the force required to move a object through a fluid. The formula is:
F = 6 x pi x r x v
where:
F = force required (N)
r = radius of the object (m)
v = velocity of the object (m/s)
pi = 3.14
What is Stokes Law Derivation?
Here is a step-by-step derivation of Stokes’ law:
- Consider a small sphere of radius r moving through a viscous fluid at a constant velocity v.
- The sphere experiences a viscous force Fv from the fluid, which is proportional to the velocity of the sphere and the viscosity of the fluid. The proportionality constant is given by 6πη, where η is the viscosity of the fluid. Thus, we can write:
Fv = 6πηrv
- The sphere also experiences an opposing force Fi due to its inertia, which is proportional to its mass m and acceleration a. We can write:
Fi = ma
- The viscous force and the inertial force must balance each other in order for the sphere to move at a constant velocity. We can write this as:
Fv = Fi 6πηrv = ma
- Solving for the velocity, we get:
v = (ma)/(6πηr)
- Substituting Fv for the viscous force gives us the final form of Stokes’ law:
F = 6πηrv
This equation can be used to predict the drag force experienced by a small sphere moving through a viscous fluid at a constant velocity.
Terminal Velocity Formula in Physics
The terminal velocity of an object is the maximum velocity it can reach as it falls through a fluid, such as air or water. It is the velocity at which the force of gravity pulling the object downward is balanced by the drag force of the fluid opposing the motion.
The terminal velocity of an object can be calculated using the following formula:
v = √((2mg)/(ρACd))
where:
v is the terminal velocity m is the mass of the object g is the acceleration due to gravity ρ is the density of the fluid A is the cross-sectional area of the object facing the fluid Cd is the coefficient of drag of the object
The coefficient of drag is a measure of how resistant the object is to moving through the fluid. It is a function of the shape and surface properties of the object and the viscosity of the fluid.
The terminal velocity of an object depends on its mass, size, and shape, as well as the density and viscosity of the fluid it is moving through. For example, a heavy object with a large cross-sectional area and a low coefficient of drag will have a lower terminal velocity than a lighter object with a small cross-sectional area and a high coefficient of drag.
Assumptions made in the Stokes Law:
There are several assumptions that are made when using Stokes’ law to calculate the drag force experienced by a small sphere moving through a viscous fluid at a constant velocity:
- The sphere is small compared to the dimensions of the fluid. This allows us to ignore any viscous effects at the surface of the sphere.
- The velocity of the sphere is constant. This means that the sphere is moving at a steady state, without accelerating or decelerating.
- The fluid is Newtonian, meaning that the shear stress is directly proportional to the shear rate. This allows us to use the viscosity of the fluid to describe its resistance to flow.
- The flow is laminar, meaning that the fluid flows in smooth, parallel layers. This allows us to use the simple form of the drag force equation given above. If the flow is turbulent, a more complex equation must be used to take into account the additional forces that are present.
- The temperature and pressure of the fluid are constant. This means that the viscosity of the fluid is constant, which simplifies the calculation of the drag force.
- The sphere is a perfect sphere with a smooth surface. If the surface of the sphere is rough or irregular, the drag force may be different from what is predicted by Stokes’ law.
It’s important to note that these assumptions are idealizations that are used to simplify the calculation of the drag force. In real-world situations, it may be necessary to take into account additional factors that can affect the drag force.
Applications of Stoke Law in Daily Life
- Understanding the drag force experienced by a small object moving through a fluid is important in a variety of engineering applications, such as the design of aircraft, boats, and automobiles.
- Stokes’ law is often used to calculate the settling velocity of small particles in a fluid, which is important in fields such as chemical engineering and environmental science.
- The motion of small particles through fluids is also important in the pharmaceutical and food industries, where it is used to design equipment for the separation and purification of materials.
- Stokes’ law can be used to calculate the drag force experienced by a small object moving through air, which is important in the design of sports equipment such as golf balls, baseballs, and arrows.
- The drag force experienced by a small object moving through water is important in the design of underwater vehicles and in the study of ocean currents.
- Stokes’ law is also used to calculate the drag force experienced by a small object moving through a fluid in the human body, such as blood or mucus. This is important in the study of blood flow and the design of medical devices.
- The motion of small particles through fluids is important in the study of atmospheric science, as it can help to understand the transport of pollutants and other particles through the atmosphere.
- Stokes’ law is used in the design of inkjet printers, where it is used to calculate the drag force experienced by small droplets of ink moving through air.
- The drag force experienced by small objects moving through fluids is also important in the study of meteorology, as it can help to understand the motion of raindrops and snowflakes through the atmosphere.
- Stokes’ law is used in the design of consumer products such as vacuum cleaners and air filters, where it is used to calculate the drag force experienced by small particles moving through a fluid.
