When a body falls through a viscous fluid, it drags along with it the layers of the fluid in contact with it. But the layers of liquid at a large distance from it remain undisturbed. Thus, the falling body produces a relative motion between the layers of the fluid. Due to this relative motion, a backward dragging force comes into play which opposes the motion of the falling body. This opposing force or dragging force increases with the increase in velocity of the body. It is found that the body after attaining certain velocity starts moving with constant velocity in the fluid. This uniform velocity of the body while moving in a fluid is called its terminal velocity.
Stokes proved that the viscous drag (F) on a spherical body of radius r moving with terminal velocity v in a fluid of viscosity η is given by
F = 6πηrv
This is called Stoke's law
This law can be proved by using method of a dimensional analysis. This viscous drag acting on a sphere depends upon the radius (r) of the sphere, terminal velocity (v) and coefficient of viscosity η.
i.e., F = kravbηc ....(i)
where k is dimensionless constant.
Using the dimensional formulae of various quantities involved in eqn. (i), we have
[MLT-2] = [L]a x [LT-1]b x [ML-1L-1]c
= [McLa+b-cT-b-c]
Comparing the powers of M, L and T (principle of homogeneity), we have
c = 1 ....(ii)
a + b - c = 1 ....(iii)
-b - c = -2 ...(iv)
Using eqn. (ii) in eqn (iv), we have
b = 1
From eqn. (iii)
a = 1 - b + c = 1 - 1 + 1
or, a = 1
Substituting the values of a, b and c in eqn. (i), we get
F = krvη
The value of k found numerically is equal to 6π.
Thus, F = 6πηrv
which is Stroke's formula
