Consider a fluid of viscosity
between concentric rotating pipes of radius
with
rotating with angular velocities
respectively.
For two dimensional flow
in the z – direction the shear stress
where
is the shear strain and
is the shear stress.
If the flow is smooth and steady
For a particle at some
and
where
is the angular velocity of the fluid at radius
.Hence
If
then
and
Similarly
Hence constant shear stress curves may be represented by concentric circles.
The torque acting on an element of fluid is
This torque must be independent of
since the flow is steady so there is no angular acceleration and the torque betweenand
must be zero. The torque above must be inependent ofso
At
and at
(1) and
subtraction gives
Then
Then
Using
gives
.