The influence of axial fluid conduction on low Peclet number flows in the thermal entrance region of long circular tubes is investigated in this theoretical study. The convective heat transport of viscous fluids relates to a specific condition under which the first part of the tube (x less than or equal to 0) is insulated and the second part of the tube (x > 0) receives a heat flux of uniform intensity. A conjugate one-dimensional lumped model produces a solution of compressed algebraic form that is able to deliver dependable mean bulk temperatures that are in perfect agreement with those obtained numerically by the formal conjugate two-dimensional distributed model. As a by-product of the succession of algebraic calculations within the platform of the lumped model, the critical Peclet number Pe(cr) has been easily quantified. This number is a figure-of-merit of remarkable importance in the modeling of forced heat convection in tubes because it establishes the threshold between two contrasting situations: one embracing axial fluid conduction (finite Pe) and the other implicating negligible axial fluid conduction (Pe --> infinity). In addition, the local wall temperatures were calculated with an approximate engineering procedure, showing good agreement with those determined numerically by the formal conjugate two-dimensional distributed model.