What is Stokes Law?
Stokes law discusses the active force applied on a body when it is dropped in a liquid. Initially, because of low viscous force, this velocity of the falling body remains low. But, as the spherical body falls with its effective weight, it gains acceleration, and this velocity of the body increases gradually.
This also makes the liquid in contact move with a velocity same as that of this body. The movement of an object in this fluid with increasing velocity causes motion in liquid layers resulting in the development of viscous force. This force increases with the increasing velocity until the point of time when it matches this effective force with which this body moves.
In such a situation, the net force on the body becomes zero, and it attains a constant velocity named terminal velocity (Vt). This idea helped state Stokes law and this derivation of frictional force or Stokes’ drag applies to the interface between the body and fluid.
According to Sir George Stokes, “the force acting between the liquid and falling body interface is proportional to velocity and radius of the spherical object and viscosity of fluid”.
Stoke's law explains the main reason why raindrops falling from the sky do not harm us on the ground. Stokes law is an extremely important concept that is a part of physics
Stokes Law Formula
Stokes came up with this formula in 1851 to calculate this drag force or frictional force of spherical objects immersed in viscous fluids. Here, look at the formula mentioned below.
F = 6 * πηrv
Where,
F is the drag force or frictional force at the interface
η is the viscosity of a liquid
r is the radius of the spherical body
V is the velocity of flow
Stokes Law Derivation
Stokes’ proposition regarding this immersion of the spherical body in a viscous fluid can be mathematically represented as,
F ∝ ηa rb vc
By solving this proportional expression, we can get the Stokes law equation. To change this proportionality sign into equality, we must add a constant to the equation. Let us consider that constant as ‘K’, so the transformed equation becomes
F = K * ηa rb vc ……….. (1)
Now, Let us write down the dimensions of this equation on both sides. Please note, K is a constant which has no dimension.
[MLT-2] = [[ML-1T-1]a . Lb . [LT-1]c ]
We need to simplify this expression by separating each variable as follows
[MLT-2] = [Ma ] [L-a+b+c] [T-a-c]
Comparing the value of length, mass and time, following equations can be found.
a = +1 ……….(2)
-a+b+c = +1 ……....(3)
-a-c= -2 ……………..(4)
On solving equations 2,3, and 4, we get a =1, b=1, c=1. On substituting these values in equation 1, we have
F = K * η1 r1 v1 = K ηrv
Further, the value of K was found to be 6π for spheres through experimental observation. The above calculations helped in Stokes equation derivation along with its fundamental formula.
Terminal Velocity Formula
As explained earlier, terminal velocity is attained at an equilibrium position when the net force acting upon the spherical body and acceleration becomes zero. Here is the formula for terminal velocity derived from Stokes law definition.
Vt = 2a2 (ρ−σ) g / 9η
Where ρ is the mass density of a spherical object and σ is the mass density of a fluid.
Assumptions made in the Stokes Law
Some assumptions were made in the Stokes law. The assumptions made in the Stokes law include the following:
Laminar flow
Particles are spherical
The material is homogeneous or uniform in composition
Surfaces are smooth
Particles do not interfere with each other
Applications of Stoke Law
Stokes law is applicable in many areas. Some applications of stokes law include the following:
It helps in finding the settling of sediment in freshwater
It is also used to measure the viscosity of fluids.
FAQs on Stokes Law Derivation
1. Define Stokes Law with an Example.
According to Stokes law, if a spherical body falls into a viscous liquid then the force acting at the interface is proportional to – Radius of the spherical body, velocity of the sphere, and viscosity of this given fluid.
2.How Can I Derive Stokes Formula?
You can derive this formula by writing the proportionality equation of Stokes law first. Then, you need to write down the proportionality constant and represent this equation with its dimensional formula. Calculate the value of constants and put them in the original equation to get the formula.
3. How does the viscosity of fluid affect the travel of spherical objects?
If the fluid is more viscous, then the speed of the spherical object with which it travels will be low and vice versa. For instance, the air has low viscosity, and hence the spherical objects can move much faster in this case.
4. How is the coefficient of viscosity dependent on temperature?
The coefficient of viscosity of liquids decreases when the temperature is increased and the coefficient of viscosity of gases increases when the temperature is increased. The most important result of the kinetic theory of gases is that the viscosity of the gas does not depend on the density of the gas.
5. What is the importance of Stoke’s law?
Stoke’s law is very important in different fields. Some important fields where it is used include:
It helps in finding the electronic charge by Millikan in his oil drop experiment.
It also helped a man coming down using a parachute
The law also gives an explanation for the formation of clouds.