Stokes Law Derivation - Definition, Formula, FAQs

Introduction
In this article, our focus will be on stokes law its formula and derivation, limitations of stokes law etc.

What is stokes law?

The drag force that stops small spherical particles from falling through a fluid medium is described by Stokes' law, a mathematical equation 6πrηv, where r is the radius of the sphere, v is the velocity of the fall, η is the viscosity of the liquid,

The downward force is equal to , where is the density of the sphere, is the density of the liquid, and g is the acceleration due to gravity. At a constant rate of fall, known as the terminal velocity, the upward and downward forces are in equilibrium. The equation of the two expressions given above and the resolution of v, therefore, gives the required velocity expressed as

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This Story also Contains

1. What is stokes law?
2. Stokes law formula
3. Stokes law derivation
4. Limitation of stokes law
5. Some limitations of Stokes law are:

v =

However, since its validity is limited to conditions where the movement of the particle does not create turbulence in the liquid, various modifications have been made.

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Stokes law formula

r is the radius of the sphere, η is the fluid viscosity, and v is the sphere’s velocity.

Stokes law derivation

The viscous force acting on a sphere is directly proportional to the following parameters, according to Stokes Law viscosity equation:

• the sphere’s radius
• coefficient of viscosity
• the object’s velocity

It is represented as:

………………………………. (i)

K be the constant of proportionality is a numerical value and has no dimensions.

equating the dimensions of parameters on either side of equation (i), we get

Simplifying the above equation, we get,

Equating the superscripts of M, L and T respectively from above equation, we get

a=b=c=1

putting these values in (i)

F = knrv

k= 6for a spherical body

So,

Limitation of stokes law

Stokes' law for the resistance of a falling sphere is not only historically significant due to its use to determine the basic electronic charge, but it is still used today to calculate the viscosity of a liquid. Greenwood et al.³ recently presented a novel method for this. Use Atwood's machine to control the driving force of the balls that fall through the fluid being sought for viscosity. Despite their simplicity, they obtain experimental viscosity values that are systematically higher than the manual N values. The excess is not small and reaches up to 62%. They found with a K generally increasing to increase the diameter of the ball d. The only explanation they offer is that the 0.05mm thick wire being inserted into the liquid may have a greater effect than expected, although they find that explanation unlikely later in the article. Since glycerine is a Newtonian liquid in this range, the viscosity is a constant of the material and it is necessary to account for the apparent viscosity change above. This makes this experiment even more suitable for students because, despite the frankness of the method, the error is surprisingly large, and the experiment ends with a question rather than an answer.

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Some limitations of Stokes law are:

1. Negative density difference in Stokes equation

The Stokes equation is not valid if the density difference in the equation is negative, that is, if the particles are lighter than the scattering medium. This results in floatation or creaming, which is more commonly seen in emulsion systems.

2. High content of dispersed solids

When the solids content of a suspension is high, the Stokes equation may not reflect the actual sedimentation rate. A high solids content gives the system additional viscosity that must be taken into account when determining the correct settling rate. The equation only includes the viscosity of the medium.

3. Dielectric constant

The ignored dielectric constant in the Stokes equation is an important parameter in many situations. The electric potential between two charges has an inverse relationship to the dielectric constant of the medium. Therefore, the zeta potential depends on the dielectric constant of the medium.

It follows that in a vehicle with a low dielectric constant, the double layer is many times thicker than in an aqueous medium, which also leads to a different zeta potential and therefore the sedimentation is different.

4. Brownian motion

Brownian motion, a random (zigzag) motion of suspended particles in a liquid caused by their collision with rapidly moving atoms or molecules in the gas or liquid, is another factor that can affect accuracy of the results in Stokes Equation. Brownian motion counteracts sedimentation to a measurable degree. This can lead to a large deviation of the actual sedimentation rate from that calculated according to the Stokes equation.

Stokes law of sedimentation

Small particles of a certain size range in liquid suspension tend to settle due to various forces acting on them. This behaviour of the particles is known as sedimentation.

Particles with a size less than 75 µm cannot be sieved because the smaller particles carry charges on their surface and tend to adhere to each other and to other particles, even to the sieves or the experimenter's hands. Dealing with such soils would be very difficult.

To analyse soils with these particles, we use the sedimentation analysis method.

A small particle in a liquid suspension, according to this law, tends to settle under its own weight due to gravity's effect. And due to the acceleration due to gravity, its downward speed keeps increasing.

But two forces,

One is the buoyant force that works upwards and due to the pressure difference acts on the body within the liquid.

seconds is the resistance force, which is a resistance force and acts against the direction of movement of the body.

These two forces begin to act on the particle in the opposite direction to the particle's motion and begin to decelerate the particle until it reaches equilibrium. And this makes the speed of the particles constant. That is, the particle now falls at a constant velocity called the terminal velocity.

Writing an equilibrium equation for this particle falling in a liquid yields the term for the terminal velocity.

Particle weight W down, buoyancy B and drag force D up.

As a result, the equilibrium equation will be W = B + D.

B + D = W

The weight of a particle can be calculated by multiplying its volume by its unit weight, gamma s.

The buoyant force is equal to the weight of the liquid, which in our instance is water. This particle has displaced, which is equal to the volume of the particle multiplied by the unit weight of water it displaces.

is the weight of particle of soil

is the weight of water

is the dynamic viscosity

Solving this equation,

Terminal velocity,

By using a simple equation of motion, we can see that the velocity of any particle falling through a height of He centimetre in time t minutes is He upon t.

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NCERT Physics Notes:

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