The biomedical applications of the Graetz-Nusselt problem with non-Newtonian properties are closely associated with heat transfer processes in fluid flow systems. Inspired by the ubiquity of complex rheological fluids in the thermal entry flow problem, we extend the classical Graetz-Nusselt problem for Vo ̵́cadlo non-Newtonian fluid. This analysis is also valid for Bingham viscoplastic liquid, power-law and Newtonian fluids. The problem is solved by a semi-analytic technique. For each fluid model, we obtain the closed-form solution of the velocity field for both planner and cylindrical confinements. The separation of variable approach is utilized to solve the energy equation along with the Dirichlet boundary condition. Furthermore, to obtain the refined eigenvalues (from Sturm-Liouville boundary value problem) MATLAB’s built-in bvp4c function is utilized. The viscous dissipation (Brinkman number) and axial conduction (Péclet number) effects on bulk temperature and Nusselt numbers are also incorporated. The plots of Nusselt number and mean temperature are presented for several values of yield stress parameter, power-law index and Brinkman number with their detailed explanation. The local Nusselt number for both geometries achieved fully developed condition for the larger values of axial distance x ̵̅. The current investigation can be applied in the field of nanotechnology, mechanical and biomedical engineering. Designing several thermal types of equipment and microfluidic devices also falls in the scope of this study.