Problem Diagram

Velocity Profile

By assuming that the clearance between the rotating shaft and bearing is small and concentric, this problem can then be adequately modeled using flow between parallel plates where the top plate moves at a velocity U and bottom plate is fixed. The velocity profile of the lubricant is given by

By assuming that there is no pressure gradient in the x-direction, the velocity becomes

     u = Uy/h

where the top plate velocity U = ω (d/2).

The shear stress exerted on the rotating shaft is

     τ = μ(du/dy) = μU/h = μ ω d/2h

The torque is given by multiplying the force by its moment arm. That is,

     T = (τ) (πdL) (d/2) = π μ ω Ld³/4h

It has been shown that the required torque is directly proportional to the viscosity of the lubricant.

For SAE-30 oil, the required torque to rotate the shaft is calculated to be:

Note: Students are encouraged to calculate the Reynolds number and determine if the assumption of laminar flow is valid.