The effect of the Eckert number on impulsively started pipe flow

The exact analytical solution of the thermal field arising in a pipe of circular section and infinite length is proposed when the incompressible laminar flow, initially at rest, is impulsively started by imposing a constant pressure gradient. The solution has been obtained with the effect of dissipation of kinetic energy also taken into account in the relevant case of Prandtl number equal to one. Four cases have been considered depending on the boundary condition imposed on the wall of the pipe: constant temperature, adiabatic wall, assigned heat loss, assigned constant heat flux.