Temperature calculation in an incompressible permanent laminar fluid flow through a circular pipe with axial conduction and viscosity

A solution for the energy equation, with a parabolic velocity profile, for an incompressible permanent laminar fluid flow through a circular pipe, including viscous dissipation and axial conduction is shown. The solution is presented as a series of hypergeometric functions. Furthermore, it is shown how to apply this solution to problems involving first, second and third kind boundary conditions. A first kind boundary condition example is examined, for Peclet numbers from 5 to 100.