<p>This work consists of the computational analysis of two-dimensional heat transfer by convection and laminar flow of a fluid. The application was done through a circular pipe with simultaneously developing velocity and temperature profiles, considering two different wall boundary conditions that were alternatively introduced: constant temperature and constant specific heat flux. The physical characteristics of this flow were assumed to be constant and of an incompressible and Newtonian type. The governing equations describing fluid flow, namely, continuity, momentum, energy, and simplifying assumptions, along with associated boundary conditions, have also been presented in detail. These equations, which govern the studied phenomenon, are non-linear partial differential equations (PDE), and we therefore used the explicit finite-difference scheme (EFDS) to integrate these equations by iterations after transforming them into a linear algebraic system. For this purpose, a FORTRAN computer code has been well developed to simulate the thermal problem in a circular pipe and obtain the results presented in both cases. The results were computed in terms of various parameters when the Peclet number was between 0.7 and 100; the number of grid points ranged between 10 and 100, and the distance Z between 0.05 and 0.5. This allowed us to evaluate the rate of heat transfer, to observe the velocity and temperature contours, the distribution of Nusselt number, whether local or average, and to determine the impact of the various factors on the performance and heat transfer characteristics of the fluid flow. Finally, we conducted a critical comparison in order to validate the computer code developed in this study, and our results showed excellent agreement with previous studies. This further strengthens the reliability of the predictive model via digital simulation.</p>