Last updated on September 26th, 2021 at 03:29 pm
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.
expression of viscosity using the flow of a viscous fluid between two parallel plates