Velocity Distribution of Fluid Between Rotating Pipes

Consider a fluid of viscositybetween concentric rotating pipes of radiuswith rotating with angular velocitiesrespectively.

For two dimensional flowin the z – direction the shear stresswhereis the shear strain andis the shear stress.

If the flow is smooth and steadyFor a particle at some andwhereis the angular velocity of the fluid at radius

.Hence

Ifthenand

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 ofsince the flow is steady so there is no angular acceleration and the torque betweenandmust be zero. The torque above must be inependent ofso

Atand at

(1) and

subtraction gives

Then

Then

Usinggives.