Question

Derivation of Stoke’s law.

Answer 1

Hint: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.