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A lubrication oil with thermal conductivity k fills the annulus (gap) between a long, cylindrical rotating shaft and an outer sleeve, as shown below.
The friction that is due to the rotation generates a uniform energy source within the oil,
where W is the shaft rpm and m is the viscosity of the oil. The sleeve is cooled and the outer temperature is maintained at T_{2}. You may assume the shaft is sufficiently long such that all the heat is conducted in the radial direction.
(a) Derive the model equation for the steady-state temperature distribution in the annulus. For this part you may assume that the shaft is a perfect insulator. Solve and sketch the temperature distribution in terms of the variables of the problem.
(b) Using the temperature profile from (a), solve for the energy flux at outer surface and show that this is consistent with an overall energy balance.
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