Question

stokes law derivation
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Answer 1

Answer:

What is Stoke’s Law?

Stoke’s Law is a mathematical equation that expresses the settling velocities of the small spherical particles in a fluid medium. The law is derived considering the forces acting on a particular particle as it sinks through the liquid column under the influence of gravity. The force that retards a sphere moving through a viscous fluid is directly proportional to the velocity and the radius of the sphere, and the viscosity of the fluid. Sir George G. Stokes, an English scientist expressed clearly the viscous drag force F as:

F=6πηrv

Explanation:

Stokes’s law finds application in several areas such as:

Settling of sediment in freshwater

Measurement of the viscosity of fluids

In the next section, let us understand the derivation of Stoke’s Law.

Stoke’s Law Derivation

The viscous force acting on a sphere is directly proportional to the following parameters:

the radius of the sphere

coefficient of viscosity

the velocity of the object

Mathematically, this is represented as

F∝ηarbvc

Now let us evaluate the values of a, b and c.

Substituting the proportionality sign with an equality sign, we get

F=kηarbvc (1)

Here, k is the constant of proportionality which is a numerical value and has no dimensions.

Writing the dimensions of parameters on either side of equation (1), we get

[MLT–2] = [ML–1T–1]a [L]b [LT-1]c

Simplifying the above equation, we get

[MLT–2] = Ma ⋅ L–a+b+c ⋅ T–a–c (2)

According to classical mechanics, mass, length and time are independent entities.

Equating the superscripts of mass, length and time respectively from equation (2), we get

a = 1 (3)

–a + b + c = 1 (4)

–a –c = 2 or a + c = 2 (5)

Substituting (3) in (5), we get

1 + c = 2

c = 1 (6)

Substituting the value of (3) & (6) in (4), we get

–1 + b + 1 = 1

b = 1 (7)

Substituting the value of (3), (6) and (7) in (1), we get

F=kηrv

The value of k for a spherical body was experimentally obtained as 6π

Therefore, the viscous force on a spherical body falling through a liquid is given by the equation

F=6πηrv