Name: Viscous Force and Coefficient of Viscosity
Uploaded: 2019-07-19
Duration: 6 min 33 s
Description: Quantitatively, comparing and calculating is an important part of the analysis in fields of science. Let's discuss the mathematics involved in viscosity for...

Question

State stoke's law and deduce it by dimensional methods.

Accepted Answer

Stoke's law: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.

$F = 6 π η r v$

Derivation of Stokes's law:

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

T1)The radius of the sphere

2)Coefficient of viscosity

3)The velocity of the object

Mathematically, this is represented as

$F \propto η a r b v c$

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

Substituting the proportionality sign with an equality sign, we get

$F = k η a r b v c$ .......(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] = M^a⋅ 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

$F = k η r v$

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

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

$F = 6 π η r v$

State stoke's law and deduce it by dimensional methods.

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