Here we will derive the formula of viscosity using Stoke’s law. We have already derived one expression of viscosity using the flow of a viscous fluid between two parallel plates. Now we derive a different equation of viscosity or coefficient of viscosity in terms of terminal velocity.
When a sphere is released and allowed to fall freely in a fluid, it is subjected to three forces: its weight, W, the upthrust, U, and the viscous drag, F.
Initially, a resultant force, R = W – (U + F) will make the sphere accelerate downward. As the velocity of the sphere increases, the viscous drag increases according to Stokes’ law until (U + F) = W. The resultant force then becomes zero, and the sphere continues to fall at a constant velocity known as the terminal velocity.
Derive formula of Viscosity using Stokes’ formula & terminal velocity
By measuring the terminal velocity of a sphere falling through a fluid it is possible to determine the coefficient of viscosity of the fluid. In other words, when terminal velocity is reached we can use the equilibrium equation in this way: upthrust + viscous drag = weight. This equation gives us the formula of viscosity.
For a sphere of volume V, radius r and density ρs falling through a fluid of density ρf and viscosity η with a terminal velocity v, we get the following equilibrium equation:
U + F = W …………….. (1)
U= weight of displaced fluid = mf g = Vρf g = (4/3) πr3 ρf g
F = viscous drag = 6πηrv
W = weight of sphere = ms g = Vρs g = (4/3)πr3 ρs g
[ V = volume of the sphere = volume of displaced fluid ]
Equation (1) now can be written as:
(4/3) πr3ρf g + 6πηrv = (4/3)πr3ρsg …………………….. (2)
which gives the formula of viscosity as:
viscosity η = 2( ρs – ρf) gr2 / (9v) …………… (3)
Terminal velocity equation
We also get one equation of terminal velocity in terms of density of falling object and density of the fluid from equation (3):
Terminal velocity v = 2( ρs – ρf) gr2 / (9 η ) …………….. (4)
This also shows us that the terminal velocity of a falling sphere in a fluid depends on the square of its radius. Hence very small drops of rain and the minute droplets from an aerosol fall slowly through the air.