Question

Derivation of Stoke’s law.

Answer 1

Stoke’s Law is defined as the settling velocities of the small spherical particles in a fluid medium. The viscous force $F$is acting on a small sphere of radius $r$moving with velocity $\nu$through the liquid is given by$F = 6\pi \eta r\nu$. Calculate the dimensions of $n$the coefficient of viscosity.

Complete step-by-step solution:
Stoke’s Law Derivation:
The following parameters are directly proportional to the viscous force acting on a sphere.
the radius of the sphere
coefficient of viscosity
the velocity of the object
Mathematically, this is represented as,
$F \propto {\eta ^a}{r^b}{\nu ^c}$
Let us evaluate the values of$a,b,c$.
Substitute the proportionality sign with an equality sign, we get
$F = k{\eta ^a}{r^b}{\nu ^c}.............(1)$
Here, $k$ is the constant value which is a numerical value and has no dimensions.
Now writing the dimensions of parameters on either side of the equation$(1)$, we get
$\left[ {ML{T^{ - 2}}} \right] = {\left[ {M{L^{ - 1}}{T^{ - 1}}} \right]^a}{\left[ L \right]^b}{\left[ {L{T^{ - 1}}} \right]^c}$
Simplifying the above equation, we get
$\left[ {ML{T^{ - 2}}} \right] = {M^a} \cdot {L^{ - a + b + c}} \cdot {T^{ - a - c}}.........\left( 2 \right)$
The independent entities are classical mechanics, mass, length, and time.
Equating the superscripts of mass, length, and time respectively from the equation$\left( 2 \right)$, we get
$a = 1..........\left( 3 \right)$
$- a + b + c = 1...........\left( 4 \right)$
$- a - c = 2$
$a + c = 2\left( 5 \right)$
Using the eq. $\left( 3 \right)$in the eq.$\left( 5 \right)$, we get
$1 + c = 2$
$\Rightarrow c = 1..........\left( 6 \right)$
By putting the values of the eq.$\left( 3 \right)$&$\left( 6 \right)$ in the eq.$\left( 4 \right)$, we get
$\Rightarrow - 1 + b + 1 = 1$
$\Rightarrow b = 1.........\left( 7 \right)$
Substituting the values of the eq.$\left( 3 \right)$, $\left( 6 \right)$ and $\left( 7 \right)$in the eq.$\left( 1 \right)$, we get
$F = k\eta r\nu$
The value of $k$ for a spherical body was experimentally obtained as$6\pi$.
Hence, the viscous force on a spherical body falling through a liquid is given by the equation
$F = 6\pi \eta r\nu$

Note: Stoke’s law is derived from the forces acting on a small particle as it sinks through the liquid column under the influence of gravity.
A viscous fluid is directly proportional to the velocity and the radius of the sphere, and the viscosity of the fluid when the force that retards a sphere is moving.