The velocity profile in the annular control volume of Example is approximately linear, varying from a velocity of zero at the outer boundary to a value of ωd/2 at the inner boundary. Develop an expression for the fluid velocity, v(r), where r is the distance from the center of the shaft.
Example
A shaft is rotating at constant angular velocity ω in the bearing shown in Figure. The shaft diameter is d and the shear stress acting on the shaft is τ. Find the rate at which energy must be removed from the bearing in order that the lubricating oil between the rotating shaft and the stationary bearing surface remains at constant temperature.
