The problem is 10-5 from the textbook - Deen Analysis of Transport Phenomena 2nd edition. Please only answer if you know the subject well. I will upvote a good answer!

10-5. Heat Transfer in a Hydrodynamic Entrance Region for Pr1 Consider heat transfer in a tube in which the temperature and velocity profiles develop simultaneously and Pr1. The tube of radius R has a constant wall temperature Tw for z>0. Assume that the temperature and velocity are both uniform at z=0, such that T=T0 and vz=U. As suggested in the discussion of case 3 of Fig. 10-7, a good approximation is to set vz=U in the energy equation. (a) Derive an expression for Nu that is valid when the temperature changes are confined to a thin layer of fluid near the tube wall. Approximately how small must z be for this result to be accurate? (b) Repeat part (a) for z large enough that the temperature changes occur over the entire cross section, but still small enough that vzU. Estimate the range of z for which this result will be valid. Figure 10-7. Development of the velocity and temperature fields in a tube (see text)