Stokes’s Law by Dimensional Analysis
Stoke found that the viscous force F depends upon (i) coefficient of viscosity, η of the medium, (ii) terminal velocity, v of the body and (iii) radius, r of the spherical body. This can be written as
F α ηavbrc
or, F = k ηavbrc …....... (i)
Where k is dimensionless constant and a, b and c are the constants to be determined,
In dimensional form, the equation can be written as
[M L T-2] = [M L-1 T-1]a [L T-1]b [L]c
or, [M L T-2] = [Ma L-a+b+c T-a-b]
Applying the principle of homogeneity of dimensions, we get
a = 1 …..... (ii)
-a + b + c = 1 ....… (iii)
and -a-b = -2 ….... (iv)
Solving these three equations, we get
a = 1, b = 1, c =1
substituting the values of a, b and c in equation (i), we get
F = kηrv
For small spheres K is found to be 6π, then
F = 6πηrv
This is known as Stokes’s Law.
Determination of Viscosity of Liquid by Using Stokes’s Law
When a spherical ball falls freely through a viscous medium such as a liquid, its velocity as first goes on increasing. Therefore, the opposing viscous force which acts upwards also goes on increasing. Finally, a stage is reached at which the weight of the ball is just equal to the sum of upthrust due to buoyancy and viscous force. In such a condition, the net force acting on a body is zero and it starts falling with constant velocity (terminal velocity).
Let r be the radius of the spherical ball falling through the viscous fluid of density σ and coefficient of viscosity η. Let ϱ be the density of the material of the ball. The various forces acting on a body are:
- Its weight, W in the downward direction
- Upward thrust, U equal to the weight of the displaced fluid
- The viscous force F, opposite to the direction of the ball
So, the downward force acting on the body = W
Upward force acting on the body = U + F
When the ball is falling with terminal velocity v,
Downward force = Upward force
W = U + F
F = W – U …..... (i)
According to Stokes’s Law, viscous force F = 6πηrv
6πηrv = W – U
Since, weight of the material (W) = mg = ϱ V g (density of the material * volume of the material * g) and Upthrust (U) = weight of the liquid displaced = (volume * density) of fluid * g = σ * V * g
6πηrv = ϱ V g - σ V g
6πηrv = 4/3 π r3ϱ g - 4/3 π r3σ g [.: V = 4/3 π r3]
6πηrv = 4/3 π r3 (ϱ - σ) g
η = 2r2(ϱ - σ) g / 9 v
this is the expression for the coefficient of viscosity of the liquid or fluid
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