(a) Show that for fully developed laminar Newtonian flow in a circular tube of radius R, the energy equation becomes (11B.2-1) if the viscous dissipation terms are not neglected. Here v, is the maximum velocity in the tube. What restrictions have to be placed on any solutions of Eq. 11B.2-1? (b) For the isothermal wall problem (T= T, at r= R for z>0 and at z-0 for all r), find the as- ymptotic expression for Tr) at large z. Do this by recognizing that &T/az will be zero at large z. Solve Eq. 11B.2-1 and obtain T-T, - (11B.2-2) (c) For the adiabatic wall problem (q, = 0 at r= R for all z > 0) an asymptotic expression for large z may be found as follows: Multiply Eq. 11B.2-1 by rdr and then integrate from r = 0 to r= R. Then integrate the resulting equation over z to get %3! (11B.2-3) in which T, is the inlet temperature at 2 - 0. Postulate now that an asymptotic temperature profile at large z is of the form T-T (4uvm/,R)z + fr) (11B.2-4) Substitute this into Eq. 11B.2-1 and integrate the resulting equation for f(r) to obtain 4ptmen T-T,- (11B.2-5) after determining the integration constant by an energy balance over the tube from 0 to z. Keep in mind that Eqs. 11B.2-2 and 5 are valid solutions only for large z. The complete solu- tions for small z are discussed in Problem 11D.2.